Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection

Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection Karim Bounja1,*

Lahcen Laayouni2

Boujemaa Achchab1

Abdeljalil Sakat1

arXiv:2608.16473v1 [cs.LG] 17 Aug 2026

Abstract

Neural PDE training yields a finite checkpoint archive, yet its logged energy errors are inaccessible without the exact solution, while loss-based selection does not necessarily recover the logged energy oracle. For admissible neural approximations of symmetric coercive variational problems, we introduce a reference-free selection rule based on minimizing a computable conforming Riesz monitor. The exact residual–energy identity and conforming projection make the monitor an unconditional lower bound converging monotonically to each logged energy error under nested conforming refinement; under saturation, hierarchical enrichment yields a computable upper estimate and hence a lower–upper bracket. A key finding is that archive selection is order-sensitive: unresolved checkpoint-dependent components can reverse the oracle–non-oracle ranking at finite resolution, so checkpointwise recovery alone is insufficient. For finite archives, we prove uniform recovery, yielding convergence to the logged-oracle error and, without saturation, logged-oracle selection at sufficiently fine auxiliary resolution. Under saturation, the bracket gives a computable near-oracle bound and certifies unique logged-oracle selection upon interval separation. We also bound logging-resolution loss and certify oracle inclusion over prescribed comparison trajectories. The resulting criterion replaces inaccessible exact-error minimization by computable, trainingindependent post-training selection on the intrinsic energy-error scale, requiring only the computed candidates and the variational problem. Experiments on diffusion and elasticity, including a non-manufactured perforated plate, demonstrate energy-scale calibration, oracle-level selection, and modest post-processing cost. Keywords: Reference-free checkpoint selection; Neural PDE solvers; Symmetric coercive variational problems; Conforming Riesz reconstruction; Energy-norm error estimation; Oracle selection consistency.

1

Introduction

Neural solvers are increasingly used to approximate solutions of partial differential equations, yet their practical reliability depends on assessing the computed output in a norm intrinsic to the underlying problem [1, 2]. When training yields a finite archive {uθk }k∈Klog , a recorded output is typically retained as the numerical solution. Let (X , ∥ · ∥X ) denote the problem-dependent solution space and the norm in which approximation errors are to be assessed. The reference selection criterion is then to retain the available approximation closest to the exact solution, that is, to identify arg min ∥uθk − u∗ ∥X . k∈Klog 1 Laboratory for Analysis and Modeling of Systems and Decision Support (LAMSAD), Hassan 1st University of Settat, Settat 26000, Morocco. 2 Department of Computer Science, School of Science and Engineering, Al Akhawayn University in Ifrane, Ifrane 53000, Morocco. * Corresponding author: k.bounja.doc@uhp.ac.ma.

1

This ideal error criterion cannot be evaluated directly because the exact solution u∗ is unknown. Accordingly, checkpoint selection on the problem-intrinsic error scale calls for computable diagnostics aligned with the same intrinsic geometry. For the symmetric coercive problems considered here, let A := ub + V0 denote the affine admissible class, where V0 is the space of homogeneous admissible directions. Since the exact solution u∗ and every admissible recorded approximation uθk belong to A, uθk − u∗ ∈ V0 . The symmetric coercive bilinear form a induces the energy norm ∥v∥a := a(v, v)1/2 on V0 . For the finite checkpoint archive, define Ek := ∥uθk − u∗ ∥a ,

∗ KE := arg min Ek . k∈Klog

Although the errors Ek are inaccessible, the quadratic Ritz functional J(u) := 12 a(u, u) − ℓ(u) and the variational residual ⟨R(u), v⟩ := ℓ(v) − a(u, v), v ∈ V0 , satisfy, for every logged checkpoint, J(uθk ) − J(u∗ ) =

1 2 1 Ek = ∥R(uθk )∥2V0,a ′ . 2 2

Thus, the energy-dual residual is exactly calibrated with the target energy error. For a finite-dimensional conforming auxiliary space Vm ⊂ V0 , let zm,k ∈ Vm satisfy a(zm,k , v) = ⟨R(uθk ), v⟩,

v ∈ Vm ,

and define the conforming Riesz monitor ηm,k := ∥zm,k ∥a . Conforming projection gives ηm,k ≤ Ek ,

ηm,k ↑ Ek

along a nested conforming auxiliary hierarchy with dense union in V0 . We therefore select checkpoints by ∗ Km := arg min ηm,k . k∈Klog

The central difficulty for archive-level selection is that checkpointwise recovery does not by itself preserve the target-error ordering at finite resolution. We identify this ordering obstruction and show that checkpointdependent unresolved components may reverse the oracle–non-oracle ranking. For a finite archive, we prove that checkpointwise recovery upgrades to the uniform estimate εm := max |Ek − ηm,k | −→ 0. k∈Klog

∗ ∗ This uniform recovery eventually restores the oracle–non-oracle separation. Consequently, every km ∈ Km satisfies ∗ − E∗ ≤ εm , 0 ≤ Ekm E∗ := min Ek , k∈Klog

and, for all sufficiently large m, ∗ ∗ Km ⊆ KE .

Thus, monitor minimization is consistent in value and eventually recovers the logged energy oracle without any saturation assumption. At finite auxiliary resolution, nested conforming reconstructions provide, under saturation, computable near-oracle bounds and exact logged-oracle certification by interval separation. We further separate auxiliary resolution from logging resolution: relative to a prescribed finite comparison trajectory, competitive-coverage bounds quantify the loss induced by archive subsampling and yield certificates of trajectory-oracle inclusion and unique recovery. The resulting framework replaces inaccessible exact-error minimization by a computable, training-independent post-training selection procedure requiring neither the exact solution nor a reference solve.

2

The present work lies at the intersection of two lines of analysis: error estimates relating residual or variational training objectives to solution errors under suitable stability assumptions [2, 3], and a posteriori constructions providing computable error indicators for neural PDE approximations [4, 5]. For training objectives intended to provide quantitative control of the error with respect to the exact solution, recent work has identified calibration with the problem-intrinsic error as a structural design requirement. Consistent collocation losses providing error control in the target energy norm have been constructed [6], while variationally correct residual losses whose values are uniformly equivalent to the squared solution error in the norm determined by a stable variational formulation have been developed in [7]. Related strategies have also incorporated computable residual-based error information directly into the training objective, including discrete dual residual norms and a posteriori estimators [8, 9]. For archive selection, however, even uniform checkpointwise loss–error equivalence does not by itself guarantee preservation of a minimum-error checkpoint under loss minimization over a finite archive. Indeed, suppose that, for some p > 0 and constants 0 < c < C, cEkp ≤ Lk ≤ CEkp ,

k ∈ Klog .

If two checkpoints satisfy Ei < Ej while CEip > cEjp , these bounds are compatible with the admissible values Li = CEip and Lj = cEjp , for which Li > Lj . Thus, uniform loss–error equivalence controls each candidate checkpointwise but does not, by itself, imply preservation of the target-error ordering or guarantee that every loss minimizer over the archive is a target-error minimizer. Independent post-training error assessment offers a complementary route: rather than inferring target accuracy from the training loss, a posteriori approaches provide computable information on the error of a computed approximation. Representative constructions include a dual-weighted-residual estimator for Deep Ritz approximations of the Laplace and Stokes problems [4], a reliable and efficient energy-norm estimator for VPINNs combining residual, loss, and data-oscillation contributions [5], and rigorous lower and upper error bounds obtained from Riesz representations of residual extensions and restrictions to geometrically simpler domains [10]. A related line constructs neural approximation spaces sequentially, with the resulting updates providing error indicators and stopping criteria in Galerkin and collocation settings [11, 12]; an extended Galerkin framework addresses general boundary-value problems with singular solution structures [13]. For general well-posed variational formulations, checkpointwise error control has been obtained through a computable projected-residual contribution together with a complementary residual contribution admitting reliable a posteriori control [9]. These approaches provide checkpointwise error assessment. The finiteresolution ordering obstruction for the conforming Riesz monitor, analyzed in Proposition 8, however, shows that checkpointwise control alone does not ensure reliable archive-level selection. More generally, target-error order preservation is not guaranteed beyond this setting; see Section 7. The present work therefore makes finite-archive checkpoint selection itself the object of analysis in the symmetric coercive setting, identifying the finite-resolution ordering obstruction and establishing its removal through uniform recovery over the archive. The resulting reference-free rule is ∗ Km := arg min ηm,k . k∈Klog

For finite archives, conforming auxiliary refinement yields eventual logged-oracle selection; under saturation, nested reconstructions provide finite-resolution near-oracle bounds and interval-separation certificates, while logging-resolution control quantifies the loss relative to a prescribed finite comparison trajectory and yields trajectory-oracle certificates. Thus, once optimization has generated sufficiently accurate admissible candidates—the exact Ritz objective being energy-calibrated by Lemma 1—monitor refinement recovers the logged oracle, while logging-resolution control determines whether it is also an oracle of the optimization trajectory. To the best of our knowledge, this passage from checkpointwise error assessment to reference-free archivelevel oracle selection has not previously been analyzed for neural PDE approximations. The framework enables reference-free recovery and certification of energy-optimal checkpoints within the retained archive and relative to a prescribed finite comparison trajectory. The numerical study tests calibration, auxiliary and logging refinement, and oracle selection on manufactured diffusion and elasticity benchmarks and on a non-manufactured perforated plate assessed against

3

an independent refined FEM reference. The results show energy-error calibration at sufficient auxiliary resolution, oracle-level selection, controlled logging loss, and low monitoring overhead. The remainder of the paper is organized as follows. Sections 2–5 develop the variational setting, conforming Riesz reconstruction, logged-oracle guarantees, and logging-resolution control. Section 6 presents the numerical validation, and Section 7 discusses broader implications for archive-level selection.

2

Problem setting

We consider symmetric coercive variational problems posed on an affine admissible class and approximated by parametrized functions belonging to that same class.

2.1

Abstract symmetric coercive setting

Let X be a real Hilbert space and V0 ⊂ X a closed linear subspace representing the homogeneous admissible directions associated with the essential constraints. Given a lifting ub ∈ X of the prescribed essential data, define A := ub + V0 . Thus, admissible functions belong to A, while differences of admissible functions belong to V0 . For homogeneous essential data, ub = 0 and A = V0 . Let a : X × X → R be a continuous symmetric bilinear form whose restriction to V0 × V0 is coercive. Thus, there exist constants 0 < αa ≤ βa < ∞ such that |a(w, v)| ≤ βa ∥w∥X ∥v∥X , a(v, v) ≥ αa ∥v∥2X ,

∀w, v ∈ X, ∀v ∈ V0 .

Let ℓ ∈ X ′ . The exact solution is the unique u∗ ∈ A satisfying a(u∗ , v) = ℓ(v),

∀v ∈ V0 .

(1)

Indeed, subtracting the lifting ub reduces the affine problem to a coercive problem on V0 , to which the Lax–Milgram theorem applies. The bilinear form a induces the energy inner product and norm on V0 , (w, v)a := a(w, v),

∥v∥2a := a(v, v),

w, v ∈ V0 .

For u ∈ X, define the Ritz functional J(u) := 12 a(u, u) − ℓ(u). Its minimization over A has (1) as its first-order optimality condition. Let V0′ denote the topological dual of V0 . For every u ∈ A, the map v 7→ ℓ(v) − a(u, v) is continuous on V0 . We therefore define the variational residual R(u) ∈ V0′ by ⟨R(u), v⟩ := ℓ(v) − a(u, v),

∀v ∈ V0 ,

and measure it in the dual norm induced by the energy norm, ′ ∥F ∥V0,a :=

|⟨F, v⟩| , v∈V0 \{0} ∥v∥a sup

F ∈ V0′ .

The residual–energy identities of Section 3 are formulated on V0 , while the Riesz reconstructions and hierarchical estimators of Subsection 3.3 are defined on conforming subspaces of V0 . The key admissibility property is u − u∗ ∈ V0 , u, u∗ ∈ A, which allows the energy-dual norm of the variational residual to be identified with the energy error. 4

2.2

Admissible neural Ritz approximations

A parametrized approximation uθ is called admissible if uθ ∈ A, and therefore uθ − u∗ ∈ V0 , as required by the residual–energy identity. For concrete boundary-value problems, let ΓD denote the portion of the boundary carrying Dirichlet-type essential conditions. Admissibility may be enforced through a hard boundary ansatz chosen so that its homogeneous correction belongs to V0 . For homogeneous essential data, one may use uθ (x) = g(x)Nθ (x),

g = 0 on ΓD ,

(2)

where Nθ is the free neural network and the boundary factor g is chosen so that gNθ ∈ V0 . For nonhomogeneous essential data, one may use uθ (x) = ub (x) + g(x)Nθ (x),

g = 0 on ΓD .

(3)

The first construction yields uθ = 0 on ΓD , while the second yields uθ = ub there; with the stated choice of boundary factor, both place uθ in A. If the neural trial class is not constrained to lie in A and essential boundary conditions are imposed only through a finite-weight penalty in the training objective, admissibility is not guaranteed. Although the penalty may reduce essential-boundary violations, it does not in general enforce the essential boundary condition exactly. Consequently, uθ − u∗ is not guaranteed to belong to V0 , and the residual–energy identity (5) does not apply directly to the original constrained problem without an admissible correction or a modified formulation. This contrasts with (2)–(3), which enforce admissibility by construction. Natural boundary conditions, such as Neumann data and symmetric Robin terms, instead enter the variational formulation through ℓ(·), or through a(·, ·) and ℓ(·), rather than defining additional affine constraints on uθ .

3

Residual–energy structure and conforming Riesz error assessment

Within the symmetric coercive setting of Section 2, this section develops the residual–energy structure and its computable conforming realization for admissible neural approximations. The analysis combines the quadratic Ritz identity, Fréchet–Riesz representation, conforming Galerkin projection, and nested-space estimation under saturation [14–18]. The resulting lower monitor ηm and, under saturation, the hierarchical bracket (22) provide the analytical basis for the checkpoint-selection and oracle guarantees developed in Section 4.

3.1

Exact residual–energy identities

We first record the exact identities linking the Ritz functional, the energy-dual residual, and the energy error. ′ For admissible u ∈ A, these identities identify ∥R(u)∥V0,a as the residual quantity exactly calibrated with ∥u − u∗ ∥a . Lemma 1 (Energy-gap identity). For every u ∈ A, J(u) − J(u∗ ) = 21 ∥u − u∗ ∥2a . Proof. Set e := u − u∗ ∈ V0 . Using u = u∗ + e, symmetry of a, and ℓ(e) = a(u∗ , e), we obtain J(u) − J(u∗ ) = a(u∗ , e) + 12 a(e, e) − ℓ(e) = 21 a(e, e) = 12 ∥e∥2a .

Proposition 1 (Exact residual–energy identity). For every u ∈ A, ′ ∥R(u)∥V0,a = ∥u − u∗ ∥a .

(4)

J(u) − J(u∗ ) = 12 ∥R(u)∥2V0,a = 12 ∥u − u∗ ∥2a . ′

(5)

Consequently,

5

Proof. Set e := u − u∗ ∈ V0 . For every v ∈ V0 , ⟨R(u), v⟩ = ℓ(v) − a(u, v) = −a(e, v). Hence, by Cauchy–Schwarz, ′ ∥R(u)∥V0,a ≤ ∥e∥a .

If e = ̸ 0, choosing v = −e in the definition of the dual norm gives the reverse inequality. The case e = 0 is immediate, and (5) follows from Lemma 1. Thus, for admissible approximations, the energy-dual residual norm coincides exactly with the energy error. ′ Minimizing ∥R(u)∥V0,a over admissible candidates is therefore equivalent to minimizing the unknown energy error, independently of how the candidates were generated. This identifies the error-calibrated quantity targeted by the conforming Riesz reconstruction. Remark 1 (Ritz ordering and error quantification). Identity (5) also shows that exact Ritz values preserve the ordering of the energy errors. They do not, however, quantify the energy error without the unknown reference level J(u∗ ). The energy-dual residual norm removes this reference dependence: it equals the energy error and admits the computable conforming realization introduced in Subsection 3.3. Remark 2 (Strong residuals as indirect proxies). Suppose that the residual of an admissible approximation u admits an L2 -representation Z ⟨R(u), v⟩ = −

rs (u) · v dx,

∀v ∈ V0 ,

Ω

and that, for some Cemb > 0, ∥v∥L2 (Ω) ≤ Cemb ∥v∥a ,

∀v ∈ V0 .

Then ′ ∥R(u)∥V0,a ≤ Cemb ∥rs (u)∥L2 (Ω) .

In standard elliptic settings, such a bound follows from the relevant Poincaré–Friedrichs, Korn, or normequivalence estimates; see, e.g., [14, 15]. Thus, strong-form L2 residuals may control the energy-dual residual norm through a problem-dependent constant, but they remain indirect proxies rather than the exactly calibrated residual quantity.

3.2

Variational consequences for admissible trial sets

We first record a direct consequence of the exact Ritz–energy identity for admissible trial sets. Proposition 2 (Best energy approximation over arbitrary admissible trial sets). Let Σ ⊂ A be any nonempty admissible trial set, not necessarily linear or convex, and assume that uΣ ∈ arg min J(w). w∈Σ

Then ∥uΣ − u∗ ∥a = inf ∥w − u∗ ∥a . w∈Σ

Proof. Since Σ ⊂ A, every w ∈ Σ satisfies w − u∗ ∈ V0 . By Lemma 1, J(w) − J(u∗ ) = 12 ∥w − u∗ ∥2a ,

w ∈ Σ.

Since J(u∗ ) is independent of w, minimizing J over Σ is equivalent to minimizing ∥w − u∗ ∥a over Σ. Proposition 2 identifies the exact continuous optimization geometry underlying admissible neural Ritz approximation: any exact minimizer of the Ritz functional over an admissible trial set is an energy-best element of that set, namely, a closest element to the exact solution in the energy norm. This exact statement concerns the continuous functional J. If J is replaced by an empirical or quadraturee the exact ordering is not automatically preserved. Indeed, for admissible candidates based approximation J, ui , uj , J(ui ) < J(uj ) ⇐⇒ ∥ui − u∗ ∥a < ∥uj − u∗ ∥a . 6

If e k ) − J(uk ) ≤ εk , J(u

k ∈ {i, j},

then this ordering is guaranteed to be preserved whenever J(uj ) − J(ui ) > εi + εj . When this separation condition is not available, the approximation bounds alone do not guarantee preservation of the energy-error ordering. Consequently, a certified connection between the practical training problem and exact Ritz minimization requires quantitative control of both the functional-approximation error and the optimization error. This motivates a post-training assessment targeted directly at the energy error. By Proposition 1, the energy-dual residual norm provides an exactly calibrated target without access to u∗ ; the next subsection constructs its computable conforming approximation. Remark 3 (Relation to Ritz–Galerkin best approximation). For Σ = ub + Wh , with Wh ⊂ V0 a conforming linear subspace, Proposition 2 reduces to the classical Ritz–Galerkin best-approximation property [14, 15]. Here the same conclusion holds for any nonempty admissible Σ ⊂ A, without linearity or convexity assumptions. Proposition 3 (Near-minimizer control). Let u ∈ A and ε ≥ 0. If J(u) − J(u∗ ) ≤ ε, then ∥u − u∗ ∥a ≤

√

′ ∥R(u)∥V0,a ≤

2ε,

√

2ε.

Proof. The result follows directly from Lemma 1 and Proposition 1. Corollary 1 (Residual convergence and energy-error control). Let (un )n ⊂ A. Then: ′ (i) if ∥R(un )∥V0,a → 0, then un → u∗ in the energy norm; ′ (ii) if u ∈ A and ∥R(u)∥V0,a ≤ ε, then ∥u − u∗ ∥a ≤ ε.

Proof. Both statements follow directly from Proposition 1.

3.3

Conforming Riesz reconstruction and hierarchical error bounds

We now turn the exact residual–energy relation into a computable post-training assessment. By Corollary 1, ′ reducing ∥R(uθ )∥V0,a drives the energy error to zero, while any bound on this residual norm transfers directly to ∥uθ − u∗ ∥a . We therefore approximate the Riesz representative of R(uθ ) in conforming auxiliary subspaces of V0 . A single auxiliary space yields a lower monitor of the resolved energy-error component, while a nested pair quantifies the component revealed by enrichment and, under saturation, provides a conditional upper estimate. 3.3.1

Continuous Riesz representative

For an admissible neural Ritz approximation uθ ∈ A, the residual R(uθ ) belongs to V0′ . By the Riesz representation theorem in the Hilbert space (V0 , ∥ · ∥a ), there exists a unique z ∈ V0 such that a(z, v) = ⟨R(uθ ), v⟩,

∀v ∈ V0 .

(6)

Proposition 4 (Continuous Riesz representative as the signed error). Let uθ ∈ A, and let z ∈ V0 solve (6). Then z = u ∗ − uθ , and consequently ′ . ∥z∥a = ∥uθ − u∗ ∥a = ∥R(uθ )∥V0,a

7

(7)

Proof. Since uθ , u∗ ∈ A, u∗ − uθ ∈ V0 . For every v ∈ V0 , a(z, v) = ⟨R(uθ ), v⟩ = ℓ(v) − a(uθ , v) = a(u∗ − uθ , v). Uniqueness of the Riesz representative gives z = u∗ − uθ . The norm identities follow from Proposition 1. Thus, the exact signed error is the continuous Riesz representative of the residual. The computational problem is therefore to approximate this representative in a conforming auxiliary space. 3.3.2

Single-space conforming realization

Let Vm ⊂ V0 be a finite-dimensional conforming auxiliary subspace. The discrete Riesz representative zm ∈ Vm is defined by a(zm , vm ) = ⟨R(uθ ), vm ⟩, ∀vm ∈ Vm . (8) We define ηm (uθ ) := ∥zm ∥a .

(9)

For clarity, the auxiliary problem approximates the Riesz representative of the residual generated by uθ ; it is not a Galerkin approximation of the primal solution u∗ . Since Vm ⊂ V0 , problem (8) is the conforming Galerkin approximation of the continuous Riesz problem (6). Galerkin orthogonality identifies zm as the a-orthogonal projection of z onto Vm [14, 15]. In the present residual–Riesz setting, this projection yields the computable lower-monitor properties stated below. Proposition 5 (Conforming projection and lower monitor). Let z ∈ V0 and zm ∈ Vm solve (6) and (8), respectively. Then a(z − zm , vm ) = 0, ∀vm ∈ Vm , (10) so that zm is the best approximation of z in Vm : ∥z − zm ∥a =

inf

vm ∈Vm

∥z − vm ∥a .

Moreover, ∥uθ − u∗ ∥2a = ηm (uθ )2 + ∥z − zm ∥2a ,

(11)

′ . ηm (uθ ) ≤ ∥uθ − u∗ ∥a = ∥R(uθ )∥V0,a

(12)

and therefore Proof. Subtracting (8) from (6) gives (10). Hence zm is the a-orthogonal projection of z onto Vm , so ∥z∥2a = ∥zm ∥2a + ∥z − zm ∥2a . Using Proposition 4 and ηm (uθ ) = ∥zm ∥a yields (11) and (12). Thus, solving the finite-dimensional auxiliary problem (8) yields ηm (uθ ) as a computable conforming lower monitor for the energy error, while ∥z − zm ∥a is the component unresolved by Vm . This decomposition motivates nested auxiliary spaces, which quantify the additional component revealed by enrichment. Corollary 2 (Monotone recovery under conforming refinement). Let (Vm )m≥1 be a nested sequence of finite-dimensional conforming subspaces such that [

Vm ⊂ Vm+1 ⊂ V0 ,

∥·∥a

Vm

= V0 .

m≥1

For a fixed uθ ∈ A, let zm ∈ Vm solve (8). Then ηm (uθ ) = ∥zm ∥a is nondecreasing and ′ . ηm (uθ ) ↑ ∥uθ − u∗ ∥a = ∥R(uθ )∥V0,a

8

(13)

Proof. By Proposition 5, ηm (uθ )2 = ∥z∥2a − ∥z − zm ∥2a . Nestedness makes ∥z − zm ∥a nonincreasing, while density and the best-approximation property imply ∥z − zm ∥a → 0. Hence ηm (uθ ) increases to ∥z∥a , yielding (13). Thus, ηm (uθ ) recovers the exact energy error monotonically from below under conforming refinement. Over a finite checkpoint archive, this property provides the basis for the reference-free checkpoint-selection rule and oracle guarantees developed in Section 4. A finite enrichment Vm ⊂ VM is considered next to quantify the additional error component resolved beyond Vm . 3.3.3

Nested-space enrichment and hierarchical gap

Let Vm ⊂ VM ⊂ V0 be two finite-dimensional conforming auxiliary subspaces. In addition to the reconstruction zm ∈ Vm , define the enriched Riesz reconstruction zM ∈ VM by a(zM , vM ) = ⟨R(uθ ), vM ⟩,

∀vM ∈ VM .

(14)

1/2

(15)

We set ηM (uθ ) := ∥zM ∥a and define the hierarchical gap δm,M (uθ ) := ηM (uθ )2 − ηm (uθ )2

.

Proposition 6 (Nested-space orthogonal decomposition). Let uθ ∈ A, let z ∈ V0 solve (6), and let zm ∈ Vm and zM ∈ VM solve (8) and (14), respectively. If Vm ⊂ VM ⊂ V0 , then ηm (uθ )2 ≤ ηM (uθ )2 ≤ ∥uθ − u∗ ∥2a . Moreover, δm,M (uθ )2 = ηM (uθ )2 − ηm (uθ )2 = ∥zM − zm ∥2a ,

(16)

and the exact energy error admits the orthogonal decomposition ∥uθ − u∗ ∥2a = ηm (uθ )2 + δm,M (uθ )2 + ∥z − zM ∥2a .

(17)

Proof. Since Vm ⊂ VM , subtracting the two discrete Riesz problems gives a(zM − zm , vm ) = 0,

∀vm ∈ Vm .

Hence ∥zM ∥2a = ∥zm ∥2a + ∥zM − zm ∥2a , which yields (16) and ηm (uθ ) ≤ ηM (uθ ). Moreover, Galerkin orthogonality at level M gives ∥z∥2a = ∥zM ∥2a + ∥z − zM ∥2a . Combining the two identities and using z = u∗ − uθ yields (17) and the remaining inequality. Remark 4 (Hierarchical interpretation). The decomposition in Proposition 6 is the nested-projection geometry underlying hierarchical a posteriori estimation [16, 17, 19]. Here the projected quantity is the signed energy error: ηm (uθ ) measures the component resolved in Vm , δm,M (uθ ) the additional component revealed by enrichment to VM , and ∥z − zM ∥a the remaining unresolved component. Because the auxiliary hierarchy is independent of the network parametrization and training procedure, the same construction can be applied post-training to a finite checkpoint archive. Under saturation, ηm (uθ ) and δm,M (uθ ) then provide the lower–upper information used for the finite-level selection guarantees of Section 4.

9

To control the unresolved component ∥z − zM ∥a through the computable gap δm,M (uθ ), nestedness alone is insufficient. We therefore introduce the standard saturation condition from hierarchical a posteriori estimation [16, 17, 19]. It enters only the conditional upper estimate; the lower monitor and its monotone recovery remain unconditional. Assumption 1 (Saturation). Let uθ ∈ A be fixed, with continuous Riesz representative z = u∗ − uθ . A nested pair Vm ⊂ VM ⊂ V0 satisfies saturation for uθ , with factor q ∈ (0, 1), if ∥z − zM ∥a ≤ q ∥z − zm ∥a .

(18)

Proposition 7 (Conditional upper estimate under saturation). Let uθ ∈ A, and suppose that the nested pair Vm ⊂ VM ⊂ V0 satisfies Assumption 1 for uθ , with factor q ∈ (0, 1). Then ∥uθ − u∗ ∥2a ≤ ηm (uθ )2 +

δm,M (uθ )2 . 1 − q2

(19)

Accordingly, defining  1/2 δm,M (uθ )2 2 Um,M,q (uθ ) := ηm (uθ ) + , 1 − q2

(20)

∥uθ − u∗ ∥a ≤ Um,M,q (uθ ).

(21)

one has Proof. By Proposition 6, δm,M (uθ )2 = ∥z − zm ∥2a − ∥z − zM ∥2a . Assumption 1 therefore gives δm,M (uθ )2 ≥ (1 − q 2 )∥z − zm ∥2a , and hence ∥z − zm ∥2a ≤

δm,M (uθ )2 . 1 − q2

Combining this with ∥uθ − u∗ ∥2a = ηm (uθ )2 + ∥z − zm ∥2a yields (19) and (21). Combining Proposition 5 with Proposition 7 yields the lower–upper bracket ηm (uθ ) ≤ ∥uθ − u∗ ∥a ≤ Um,M,q (uθ ).

(22)

The lower bound is unconditional, whereas the upper bound at a given checkpoint requires Assumption 1 for that checkpoint and the prescribed factor q. In the finite-archive guarantees developed in Section 4, saturation is required only where the corresponding upper estimate is invoked. Remark 5 (Dependence on enrichment and eventual saturation). Nestedness alone does not guarantee saturation for an arbitrary finite pair. For fixed Vm , however, let (VM )M ≥m be a nested enrichment satisfying [

∥·∥a

VM

= V0 .

M ≥m

For each fixed checkpoint k, ∥zk − zM,k ∥a −→ 0, while ∥zk − zm,k ∥a is fixed. Hence, for every prescribed q ∈ (0, 1), saturation holds for all sufficiently large M ; if the coarse projection error vanishes, it holds trivially for every M ≥ m. For a finite checkpoint archive, the corresponding thresholds can be maximized to obtain a common enrichment level. This argument does not provide a computable threshold, so the upper estimate for a prescribed finite pair remains conditional.

10

Algorithm 1 Evaluation of the conforming Riesz monitor and hierarchical estimator Require: Admissible approximation uθ ∈ A; preassembled auxiliary matrices Gm and GM ; data for assembling the residual vectors on Vm and VM ; optional prescribed factor q ∈ (0, 1). Ensure: Lower monitor ηm , enriched monitor ηM , hierarchical gap δm,M , and, when q is prescribed, conditional upper quantity Um,M,q . 1: for h ∈ {m, M } do 2: Assemble the checkpoint-dependent residual vector rh . 3: Solve Gh c(h) = rh . 4: Compute
1/2 ηh (uθ ) = rh⊤ c(h) . 5: end for 6: Compute

δm,M (uθ ) = ηM (uθ )2 − ηm (uθ )2

1/2

.

7: if q ∈ (0, 1) is prescribed then 8: Compute

 Um,M,q (uθ ) =

δm,M (uθ )2 ηm (uθ ) + 1 − q2 2

1/2 .

9: end if 10: return ηm (uθ ), ηM (uθ ), δm,M (uθ ), and, when computed, Um,M,q (uθ ).

Operational interpretation of the hierarchical bracket Under the saturation condition, the bracket implies 0 ≤ ∥uθ − u∗ ∥a − ηm (uθ ) ≤ Um,M,q (uθ ) − ηm (uθ ). Thus, its width bounds the possible underestimation of the energy error by the lower monitor. A wide bracket indicates that further auxiliary refinement may be required. Applied over a finite checkpoint archive, these intervals provide the finite-level information used in Section 4. 3.3.4

Matrix realization (h)

Nh For h ∈ {m, M }, let {ϕi }i=1 be a basis of Vh , and write

zh =

Nh X (h) (h) cj ϕ j . j=1

Define the auxiliary stiffness matrix Gh ∈ RNh ×Nh and the checkpoint-dependent residual vector rh ∈ RNh by (h) (h)
(h)
(h)
(Gh )ij = a ϕj , ϕi , (rh )i = ℓ ϕi − a uθ , ϕi . The discrete Riesz problem is then equivalent to Gh c(h) = rh .

(23)

By symmetry and coercivity of a, Gh is symmetric positive definite. Once (23) has been solved, the monitor is obtained from     ηh (uθ ) = ∥zh ∥a = (c(h) )⊤ Gh c(h)

1/2

= rh⊤ c(h)

1/2

.

(24)

The matrices Gm and GM depend only on the bilinear form and the auxiliary spaces and may therefore be preassembled and factorized once. For each checkpoint, only the checkpoint-dependent residual vectors are assembled and the corresponding linear systems solved. Applying the construction at h = m and h = M yields ηm (uθ ) and ηM (uθ ), from which δm,M (uθ ) and, for a prescribed q ∈ (0, 1), Um,M,q (uθ ) follow as summarized in Algorithm 1. The identities above refer to the exact auxiliary variational forms; their numerical realization under quadrature is assessed separately in Section 6 and C. 11

4

Reference-free checkpoint selection and logged-oracle guarantees

This section develops the reference-free selection rule by minimizing the conforming Riesz monitor over a finite archive. We identify the finite-resolution ordering obstruction, show that conforming refinement removes it uniformly and yields eventual logged-oracle selection, and derive, under saturation, a computable near-oracle bound and a unique-oracle certificate.

4.1

Monitor minimization and finite-resolution ordering obstruction

Let Klog denote the finite set of recorded checkpoints and, for brevity, write uk := uθk . Define Ek := ∥u∗ − uk ∥a ,

k ∈ Klog .

By Proposition 5 and Corollary 2, for every fixed k ∈ Klog , ηm,k ≤ Ek ,

ηm,k −→ Ek

as m → ∞.

(25)

These checkpointwise properties motivate the computable selection rule ∗ km ∈ arg min ηm,k . k∈Klog

The corresponding consistency question is whether, for all sufficiently fine auxiliary resolutions, arg min ηm,k ⊆ arg min Ek . k∈Klog

k∈Klog

At a fixed auxiliary level, checkpointwise recovery alone does not guarantee this ordering. Let i be an energy-oracle checkpoint and j a non-oracle checkpoint, so that Ei < Ej . Although both monitors are valid lower bounds, one may still have ηm,j < ηm,i , because the unresolved error component may differ across checkpoints. To characterize this obstruction, define zk := u∗ − uk ,

ρm,k := ∥(I − Pm )zk ∥a ,

where Pm denotes the a-orthogonal projection onto Vm . Then 2 Ek2 = ηm,k + ρ2m,k .

Proposition 8 (Finite-resolution ordering obstruction). For any two checkpoints i, j ∈ Klog ,

2 2 ηm,j − ηm,i = Ej2 − Ei2 − ρ2m,j − ρ2m,i . Consequently, if Ei < Ej , then the monitor ordering is reversed, ηm,i > ηm,j , if and only if ρ2m,j − ρ2m,i > Ej2 − Ei2 . Moreover, if {0} ⊊ Vm ⊊ V0 , then such a reversal is compatible with the present assumptions: there exist admissible approximations ui , uj ∈ A such that Ei < Ej and 0 < ηm,j < ηm,i .

12

Proof. For each checkpoint k, the a-orthogonal decomposition zk = Pm zk + (I − Pm )zk gives 2 ηm,k = Ek2 − ρ2m,k .

Subtracting the identities for i and j yields the first claim and, when Ei < Ej , the stated characterization of rank reversal. To show that such a reversal can occur, choose a-unit vectors x ∈ Vm ,

y ∈ Vm⊥a ∩ V0 ,

zj := γx + βy,

ui := u∗ − zi ,

and let 0 < γ < α < β. Define zi := αx,

uj := u∗ − zj .

Then ui , uj ∈ A, and Ei = α <

p

γ 2 + β 2 = Ej .

Since Pm zi = αx,

Pm zj = γx,

one has ηm,i = α > γ = ηm,j > 0.

Thus, the finite-resolution obstruction is caused by checkpoint-dependent projection defects, not by failure of the lower-bound property itself. At fixed Vm , the unresolved component ρm,k depends on the direction of the error zk relative to the auxiliary space, not only on its norm. Different checkpoints may therefore have different resolved and unresolved error components, allowing the unresolved-component difference to overcome the exact energy gap and reverse the ordering.

4.2

Logged-oracle consistency under conforming refinement

We now show that the finite-resolution obstruction identified in Proposition 8 cannot persist under a common nested conforming refinement of a finite checkpoint archive. The key point is that checkpointwise recovery becomes uniform over finitely many checkpoints. Define the logged energy-oracle level and oracle set by ∗ KE := arg min Ek .

E∗ := min Ek , k∈Klog

k∈Klog

∗ Since u∗ is unknown, E∗ and KE are unavailable in practical computations. At auxiliary level m, define ∗ Km := arg min ηm,k . k∈Klog

∗ ∗ A reference-free checkpoint may therefore be selected as any km ∈ Km . This rule uses only the unconditional lower monitor and is independent of saturation, which enters only the finite-resolution guarantees developed later.

Lemma 2 (Uniform recovery over the logged set). Under the assumptions of Corollary 2, define
εm := max Ek − ηm,k . k∈Klog

Then εm = max |Ek − ηm,k | −→ 0 k∈Klog

13

as m → ∞.

Proof. For every fixed k ∈ Klog , Corollary 2 gives 0 ≤ Ek − ηm,k −→ 0. Since Klog is finite, the maximum of these finitely many convergent defects also tends to zero. Thus, 0 ≤ Ek − ηm,k ≤ εm

∀k ∈ Klog ,

with εm → 0. Combined with monitor minimality, this uniform control yields value consistency. ∗ ∗ Proposition 9 (Value consistency of monitor-based selection). Let km ∈ Km be any monitor-minimizing checkpoint. Then ∗ − E∗ ≤ εm . 0 ≤ Ekm

Consequently, ∗ −→ E∗ Ekm

as m → ∞.

∗ ∗ Proof. Let kE ∈ KE . By monitor minimality, the lower-bound property, and the definition of εm , ∗ ≤ ηm,k ∗ + εm ≤ ηm,k ∗ + εm ≤ E∗ + εm . Ekm m E ∗ , the stated bound follows. Since E∗ ≤ Ekm

Thus, monitor minimization is asymptotically consistent in value. Finiteness of the archive strengthens this to eventual logged-oracle selection. Corollary 3 (Eventual logged-oracle selection). Under the assumptions of Corollary 2, there exists m0 such that ∗ ∗ Km ⊆ KE ∀m ≥ m0 . If the logged energy oracle is unique, ∗ ∗ KE = {kE },

then ∗ ∗ Km = {kE }

∀m ≥ m0 .

∗ Proof. If KE = Klog , the conclusion is immediate. Otherwise, finiteness of the archive gives the positive gap
δE := min ∗ Ek − E∗ > 0. k∈Klog \KE

∗ ∗ By Lemma 2, there exists m0 such that εm < δE for every m ≥ m0 . For any km ∈ Km , Proposition 9 gives ∗ − E∗ ≤ εm < δE . Ekm

∗ ∗ ∗ No checkpoint outside KE can satisfy this inequality. Hence km ∈ KE . The unique-oracle case follows immediately.

Remark 6 (Scope of logged-oracle recovery). Proposition 9 and Corollary 3 concern the oracle over the prescribed archive Klog . If this archive contains a minimizer of the energy error over the computed training trajectory, then its logged oracle is also trajectory-wide. Whether such a minimizer is retained is the distinct archive-coverage question addressed in Section 5. ∗ Remark 7 (Geometric removal of the ordering obstruction). If KE = Klog , there is no oracle–non-oracle ordering to resolve. Otherwise, define
γE := min ∗ Ek2 − E∗2 > 0. k∈Klog \KE

∗ ∗ For any s ∈ KE and k ∈ Klog \ KE , the orthogonal decomposition gives 2 2 ηm,k − ηm,s ≥ γE − max ρ2m,j . j∈Klog

14

Since the archive is finite and ρm,j → 0 for every fixed j, max ρ2m,j −→ 0.

j∈Klog

Hence, for all sufficiently large m, ηm,s < ηm,k

∗ ∀s ∈ KE ,

∗ ∀k ∈ Klog \ KE .

Thus, conforming refinement makes the unresolved projection defects uniformly too small to offset the oracle–non-oracle energy gap, eliminating the finite-resolution rank-reversal mechanism. The preceding results provide an asymptotic consistency statement: under common conforming refinement of the finite archive, monitor minimization eventually selects only logged energy-oracle checkpoints. The proof of Corollary 3 makes the mechanism explicit. Whenever εm < δE ,

δE :=

min

∗ k∈Klog \KE

(Ek − E∗ ),

one has ∗ ∗ Km ⊆ KE .

This asymptotic guarantee does not provide a directly verifiable finite-resolution criterion. Both εm and δE depend on the inaccessible exact energy errors Ek , and no computable convergence rate for εm is assumed. Hence, observing the same monitor minimizer over finitely many auxiliary levels does not imply that εm < δE ; the ordering obstruction may disappear only under further refinement. Such stabilization nevertheless provides an empirical indication of selection robustness under the tested auxiliary refinements. In experiments with reference information, the ratios ηm,k /Ek quantify how closely the monitors recover the corresponding energy errors. Without reference information, however, stabilization alone is not a certificate, motivating the computable finite-resolution guarantees derived next from the hierarchical lower–upper intervals.

4.3

Finite-resolution near-oracle control and logged-oracle certification

Throughout this subsection, the finite archive Klog is prescribed before selection and remains fixed under auxiliary refinement. Changing the archive changes the target oracle and therefore defines a distinct selection problem. Write ∗ Elog := min Ek . k∈Klog

Fix a nested auxiliary pair Vm ⊂ VM and a prescribed factor q ∈ (0, 1). The lower-monitor selection rule is independent of q, which enters only through the conditional upper estimates. For each k ∈ Klog , define Um,M,q,k := Um,M,q (uk ). Whenever saturation holds at checkpoint k, ηm,k ≤ Ek ≤ Um,M,q,k . Proposition 10 (Finite-resolution near-oracle control). Let ∗ ∗ km ∈ Km ∗ be selected by the lower monitor. Assume that saturation holds at km and that ∗ > 0. ηm,km

Then

∗ ∗ Um,M,q,km Ekm ≤ . ∗ ∗ Elog ηm,km

15

∗ ∗ Proof. Let kE ∈ KE . By monitor minimality and the lower-bound property, ∗ ∗ ≤ ηm,k ∗ ≤ E ηm,km log . E ∗ ∗ ∗ > 0, one has E Since ηm,km log > 0. Saturation at km gives ∗ ≤ Um,M,q,k ∗ . Ekm m

Dividing the two bounds yields the result. Corollary 4 (Exact logged-oracle certification by interval separation). Let s ∈ Klog , and assume that saturation holds at s. If Um,M,q,s < min ηm,k , k∈Klog k̸=s

then s is the unique logged energy-oracle checkpoint: Es < Ek

∀k ∈ Klog ,

k ̸= s.

Proof. For every k ∈ Klog \ {s}, Es ≤ Um,M,q,s < ηm,k ≤ Ek .

Operational selection procedure At a given auxiliary resolution, evaluate ηm,k over the prescribed archive and select ∗ km ∈ arg min ηm,k . k∈Klog

The hierarchical upper estimate U ∗ km , the separation condition

∗ m,M,q,km

is then computed at the selected checkpoint. Under saturation at

∗ < min ηm,k Um,M,q,km

k∈Klog ∗ k̸=km

∗ as the unique logged energy oracle. certifies km ∗ > 0, the same saturation hypothesis yields the computable factor If separation is not attained and ηm,km ∗ := Γlog,km

with

∗ Um,M,q,km , ∗ ηm,km

∗ Ekm ∗ . ≤ Γlog,km ∗ Elog

∗ If this bound is sufficiently sharp for the intended use, km may be retained; otherwise, the lower auxiliary space is refined, the monitors are reevaluated over the same archive, and the procedure is repeated with a further nested enrichment.

Scope and contribution In the present setting, selection is restricted to candidates generated by the optimization run and retained in the prescribed archive; it cannot compensate for a trajectory that fails to contain a sufficiently accurate approximation. The symmetric coercive Ritz structure nevertheless yields an important alignment: J(uk ) − J(u∗ ) = 12 Ek2 . Hence the exact Ritz functional and the energy error induce the same ordering over admissible candidates. Practical empirical or quadrature-based evaluations may, however, perturb this ordering unless the associated approximation errors are sufficiently controlled; see the discussion following Proposition 2. The contribution is therefore to recover, without a reference solution, the minimum-energy-error candidate contained in the prescribed archive. Conforming refinement yields eventual logged-oracle recovery, while the hierarchical bounds provide finite-resolution near-oracle control and, under saturation, exact certification by interval separation. Whether the archive omits a better iterate from a denser computed trajectory is the distinct coverage question addressed next. 16

5

Logging resolution and trajectory-wide guarantees

A logging schedule determines the finite archive over which checkpoint selection is performed. Consequently, the corresponding logged oracle depends on the logging resolution: a coarser archive may exclude lower-energy iterates and thereby increase the smallest attainable energy error among the retained checkpoints. For a uniform logging stride h dividing T , let Eh∗ := min Ek .

Kh := {0, h, 2h, . . . , T }, If h2 is a multiple of h1 , then

k∈Kh

Eh∗1 ≤ Eh∗2 .

Kh2 ⊆ Kh1 ,

Thus, coarser logging cannot improve the best energy-error level available within the archive. To quantify this coverage effect, let Ktraj be a prescribed finite comparison trajectory, which may in particular consist of a denser set of iterates from the same optimization run, and fix an archive Klog ⊆ Ktraj . Define ∗ Elog := min Ek ,

∗ Etraj := min Ej . j∈Ktraj

k∈Klog

Then ∗ ∗ 0 ≤ Elog − Etraj ,

and this difference measures the loss induced by archive subsampling relative to the comparison trajectory.

5.1

Competitive archive coverage and trajectory-wide control

Assume that ηm,j is available for every j ∈ Ktraj , and select s ∈ arg min ηm,k . k∈Klog

Fix a nested pair Vm ⊂ VM and a prescribed q ∈ (0, 1), and assume that saturation holds at s. Define Us := Um,M,q,s ,

Es ≤ Us .

The available lower–upper information allows trajectory iterates that cannot improve upon s to be excluded. Define Kcomp,s := {j ∈ Ktraj : ηm,j ≤ Us } . (26) Indeed, if j ∈ / Kcomp,s , then Ej ≥ ηm,j > Us ≥ Es , so j cannot improve upon the selected checkpoint. Conversely, if j ∗ ∈ arg minj∈Ktraj Ej is a trajectory oracle, then ∗ ηm,j ∗ ≤ Etraj ≤ Es ≤ Us . Hence every trajectory oracle belongs to Kcomp,s , and trajectory-wide coverage may be restricted to this competitive set. Define the competitive coverage radius by rcomp,s :=

min ∥uj − uk ∥a .

max

j∈Kcomp,s k∈Klog

(27)

It measures the largest energy distance from a competitive trajectory iterate to its nearest retained checkpoint. These pairwise distances do not require access to u∗ .

17

Proposition 11 (Trajectory-wide control through competitive coverage). Under the preceding assumptions, ∗ ∗ 0 ≤ Elog − Etraj ≤ rcomp,s .

(28)

Moreover, ∗ 0 ≤ Es − Etraj ≤

Us − ηm,s | {z }

+

auxiliary-resolution gap

rcomp,s | {z }

.

(29)

archive-coverage loss

The trajectory-oracle level satisfies ∗ Etraj ≥ ηm,s − rcomp,s .

(30)

Hence, if rcomp,s < ηm,s , then Γtraj,s :=

Us ηm,s − rcomp,s

(31)

is well defined and satisfies Es ≤ Γtraj,s . ∗ Etraj

(32)

If, in addition, Us <

min

ηm,k ,

(33)

∗ 0 ≤ Es − Etraj ≤ rcomp,s .

(34)

k∈Klog \{s}

then s is the unique logged energy oracle and ∗ Es = Elog ,

∗ Proof. Let j ∗ ∈ Ktraj attain Etraj . Since j ∗ ∈ Kcomp,s , choose k ∗ ∈ Klog such that

∥uj ∗ − uk∗ ∥a ≤ rcomp,s . The triangle inequality gives ∗ Ek∗ ≤ Etraj + rcomp,s . ∗ ∗ Since Etraj ≤ Elog ≤ Ek∗ , (28) follows. ∗ ∗ . Monitor minimality and the lower-bound property give Let klog attain Elog ∗ ∗ ηm,s ≤ ηm,klog ≤ Elog .

Together with (28), this yields ∗ Etraj ≥ ηm,s − rcomp,s ,

proving (30). Since Es ≤ Us ,

∗ Es − Etraj ≤ Us − ηm,s + rcomp,s ,

which gives (29). If rcomp,s < ηm,s , division by the positive lower bound in (30) yields (32). ∗ Finally, (33) and Corollary 4 certify s as the unique logged energy oracle. Hence Es = Elog , and (34) follows from (28). The trajectory-wide suboptimality admits the exact decomposition

∗ ∗ ∗ ∗ Es − Etraj = Es − Elog + Elog − Etraj . The first term is the loss incurred by monitor-based selection within the prescribed archive, whereas the second is the loss caused by archive subsampling relative to the comparison trajectory. Proposition 11 bounds these two contributions by ∗ 0 ≤ Es − Elog ≤ Us − ηm,s ,

∗ ∗ 0 ≤ Elog − Etraj ≤ rcomp,s .

Hence (29) separates the trajectory-wide suboptimality into an auxiliary-resolution contribution and an archive-coverage contribution. 18

5.2

Competitive coverage evaluation and path-length bounds

When the required pairwise energy distances are available, the competitive coverage radius can be evaluated directly. Define
da j, Klog := min ∥uj − uk ∥a . k∈Klog

Then rcomp,s =

max

j∈Kcomp,s

da j, Klog .

Direct evaluation may, however, require many pairwise energy-distance computations when the comparison trajectory or retained archive is large. Successive trajectory increments provide an alternative based only on local inter-iterate distances or their upper bounds, for example when these are available from densely recorded consecutive checkpoints or from bounds on successive iterate displacements. Suppose that Ktraj = {0, . . . , T },

Klog = {t0 , . . . , tN },

0 = t0 < · · · < tN = T,

and that j = 0, . . . , T − 1.

∥uj+1 − uj ∥a ≤ dj ,

Corollary 5 (Competitive path-length bound). Under the preceding assumptions, rcomp,s ≤ rcomp,s , where rcomp,s :=

max

0≤i<N j∈Kcomp,s ∩{ti ,...,ti+1 }

min

(35)

 j−1 X

ti+1 −1

dℓ ,



ℓ=ti

X

dℓ

ℓ=j

 

,

(36)



with empty sums interpreted as zero. For a uniform stride h and dj ≤ dmax ,   h rcomp,s ≤ dmax . 2

(37)

Proof. For j ∈ {ti , . . . , ti+1 }, the triangle inequality gives ∥uj − uti ∥a ≤

j−1 X

ti+1 −1

dℓ ,

∥uj − uti+1 ∥a ≤

X

dℓ .

ℓ=j

ℓ=ti

Since both endpoints are retained, the distance from uj to the archive is bounded by the smaller path length. Maximizing over the competitive set gives (35). For uniform stride h, every index is at most ⌊h/2⌋ increments from a retained endpoint, yielding (37). The path-length construction yields the valid upper bound rcomp,s ≤ rcomp,s , and, for uniform logging, makes the dependence on the stride h explicit through (37). Direct evaluation of rcomp,s , when available, avoids this additional relaxation, whereas the path-length bound may therefore be less sharp. More generally, any valid upper bound rbcomp,s ≥ rcomp,s may replace rcomp,s in the trajectory-wide guarantees, yielding correspondingly conservative bounds. Thus, evaluating or bounding rcomp,s quantifies the price of archive subsampling. This enables the logging stride to be chosen by balancing sparsity against trajectory-wide accuracy.

19

5.3

Logging-resolution design with trajectory-wide control

Coarser logging reduces the number of retained checkpoints and can make the interval-separation condition (33) easier to satisfy by reducing the competitor set; however, it may also omit lower-energy trajectory iterates. The objective is therefore to determine how far the logging stride can be increased while keeping the resulting trajectory-wide loss under quantitative control. Let H be a finite family of candidate uniform strides, chosen so that the corresponding archives (Kh )h∈H are nested, with larger h corresponding to sparser logging. All candidates are assessed relative to the same prescribed comparison trajectory Ktraj . For each h ∈ H, select sh ∈ arg min ηm,k , k∈Kh

and, assuming saturation at sh , define Uh := Um,M,q,sh ,

ah := Uh − ηm,sh ,

rh := rcomp,sh ,

where rh is evaluated relative to Klog = Kh . Proposition 11 gives ∗ 0 ≤ Eh∗ − Etraj ≤ rh ,

∗ 0 ≤ Esh − Etraj ≤ ah + rh .

Thus, rh bounds the loss induced by archive coarsening, whereas ah + rh bounds the trajectory-wide suboptimality of the checkpoint selected from that archive. If only a valid upper bound rbh ≥ rh is available, it may replace rh in these guarantees conservatively. Whenever rh < ηm,sh , the corresponding relative guarantee is Esh Uh ≤ Γtraj,sh := . ∗ Etraj ηm,sh − rh For prescribed tolerances τlog > 0 and τtot > 0, the preceding bounds yield h∗cov := max {h ∈ H : rh ≤ τlog } ,

(38)

h∗sel := max {h ∈ H : ah + rh ≤ τtot } .

(39)

and Whenever the corresponding feasible sets are nonempty, these are the sparsest tested resolutions satisfying, respectively, ∗ ∗ Eh∗∗cov ≤ Etraj + τlog , Esh∗ ≤ Etraj + τtot . sel

A relative alternative is to retain the largest tested h satisfying Γtraj,sh ≤ 1 + ε, which guarantees ∗ Esh ≤ (1 + ε)Etraj .

Candidate strides can be assessed by deterministic subsampling of a single densely recorded comparison run, without retraining or modifying the optimization schedule. The quantities rh , ah + rh , and Γtraj,sh quantify how far logging can be coarsened: without a prescribed accuracy criterion they expose the sparsity–accuracy trade-off, whereas with one they identify the sparsest tested resolution satisfying it.

5.4

Exact oracle inclusion and trajectory-oracle certification

The preceding analysis quantifies the accuracy loss induced by archive coarsening and thereby provides a basis for choosing the logging resolution. A stronger question is whether a coarsened archive has incurred any loss at all, that is, whether it still contains a trajectory oracle. We therefore derive exact certificates for trajectory-oracle inclusion and, more strongly, for the selected checkpoint itself to be a trajectory oracle. When Ktraj contains the full recorded optimization trajectory, these certificates respectively show that sparse 20

logging has preserved, or that the selected checkpoint attains, the best energy-error level reached along that trajectory. To compare the best retained and omitted trajectory iterates, define Lout h :=

min

j∈Ktraj \Kh

ηm,j ,

with the convention Lout h = +∞ if Kh = Ktraj . When valid hierarchical upper estimates Uk := Um,M,q,k are available for all k ∈ Kh , define Bh := min Uk . k∈Kh

Then Lout is a lower bound on the best energy error among omitted trajectory iterates, whereas Bh is an h upper bound on the best energy error attained within the archive. Proposition 12 (Exact trajectory-oracle certificates). For a candidate archive Kh , the following statements hold. 1. If valid upper estimates Uk are available for all k ∈ Kh and Bh < Lout h ,

(40)

then Kh contains a trajectory oracle, and hence ∗ Eh∗ = Etraj .

(41)

2. If Uh is a valid upper estimate at sh and Uh <

min

j∈Ktraj \{sh }

ηm,j ,

(42)

then sh is the unique trajectory oracle, so that ∗ Esh = Eh∗ = Etraj .

(43)

Proof. Choose k̄ ∈ Kh such that Uk̄ = Bh . Under (40), every j ∈ Ktraj \ Kh satisfies Ek̄ ≤ Uk̄ = Bh < Lout h ≤ ηm,j ≤ Ej . Thus no omitted checkpoint can attain the trajectory minimum, proving (41). Under (42), Esh ≤ Uh < ηm,j ≤ Ej ∀j ∈ Ktraj \ {sh }. Hence sh is the unique trajectory oracle, which yields (43). For nested candidate archives assessed with the same valid checkpointwise upper estimates, Bh cannot increase under archive refinement, whereas Lout cannot decrease. Hence (40), once satisfied, remains valid h under further refinement. The sparsest tested archive certified to contain a trajectory oracle is therefore  h∗incl := max h ∈ H : Bh < Lout , (44) h whenever the feasible set is nonempty. If a valid upper estimate is available only at the selected checkpoint, the selected-only condition Uh < Lout h still certifies ∗ Eh∗ = Etraj .

Likewise, the sparsest tested archive for which the selected checkpoint is certified as the unique trajectory oracle is   h∗cert := max h ∈ H : Uh <

min

j∈Ktraj \{sh }

ηm,j

,

(45)

whenever the feasible set is nonempty. Together, these criteria yield a hierarchy from quantitative control to exact certification: rh ≤ τlog controls the loss caused by archive coarsening, ah + rh ≤ τtot controls the trajectory-wide suboptimality of the selected checkpoint, Bh < Lout certifies preservation of a trajectory oracle, and (42) certifies the selected checkpoint h itself as the unique trajectory oracle. 21

5.5

Joint logging and auxiliary refinement

The preceding results provide finite-resolution guarantees and exact certificates. We now consider the complementary asymptotic regime in which logging and auxiliary resolutions are refined simultaneously. Corollary 6 (Trajectory-oracle recovery under joint refinement). Consider a sequence of logging and auxiliary configurations, all assessed relative to the same finite comparison trajectory. Let sn be the selected checkpoint, let Un be a valid upper estimate at sn , and define an := Un − ηmn ,sn ,

rn := rcomp,sn .

If an −→ 0,

rn −→ 0,

then ∗ Esn −→ Etraj . ∗ If Etraj > 0, then Γtraj,sn is well defined for all sufficiently large n and

Γtraj,sn −→ 1. Proof. Proposition 11 gives ∗ 0 ≤ Esn − Etraj ≤ an + rn , ∗ so Esn → Etraj . Moreover,

ηmn ,sn ≤ Esn ≤ Un ,

Un − ηmn ,sn = an ,

hence ∗ ηmn ,sn −→ Etraj , ∗ If Etraj > 0, then

∗ Un −→ Etraj .

∗ ηmn ,sn − rn −→ Etraj > 0,

and therefore Γtraj,sn =

Un −→ 1. ηmn ,sn − rn

Corollary 6 yields trajectory-oracle consistency for the complete selection procedure relative to the prescribed comparison trajectory. The auxiliary-resolution gap an controls selection within the logged archive, while rn controls the loss due to archive coverage; their joint decay is sufficient for the selected checkpoint to recover the trajectory-oracle level asymptotically.

6

Numerical validation

This section evaluates the conforming Riesz monitor ηm as a post-training error-assessment and checkpointselection criterion. The experiments examine energy-scale calibration, reference-free oracle selection, auxiliary refinement, logging resolution and trajectory coverage, and applicability to a non-manufactured benchmark. Reference solutions are used only for validation and enter neither training, Riesz reconstruction, checkpoint selection, nor certification criteria. We denote by ( Equad := ∥uθ − u∗ ∥a,quad , manufactured benchmarks, Eref = EFEM := ∥uθ − uref perforated plate, FEM ∥a , the numerical validation energy error. Thus ηm /Eref measures monitor calibration. For manufactured problems it approximates the fraction of the exact energy error resolved by the conforming reconstruction; for the perforated plate it provides an external calibration ratio against the independent refined FEM reference.

22

Table 1: Experimental protocols. All networks use hyperbolic tangent activations and Adam with learning rate 10−3 ; each run uses a fixed independent validation set of 2 × 104 points for Jval , and each benchmark is repeated over five seeds 0, . . . , 4. The notation 644 denotes four hidden layers of width 64, and Mop denotes the operational auxiliary level. Family

Network

Scalar diffusion Elasticity L-shaped domain Perforated plate

4

Steps Log Training points

2−64 −1 2−804 −2 2−804 −1 2−804 −2

2000 3000 4000 4000

50 100 100 100

4

Validation reference

Auxiliary hierarchy

2

10 12000 15000 15000

Equad : 96 , 3×3 Gauss 6, 12, 24, 48, Mop = 24 Equad : 962 , 3×3 Gauss 6, 12, 24, 48, Mop = 24 Equad : 1282 , 3×3 Gauss 8, 16, 32, 64, 128, Mop = 64 FEM 256 (check: 192) 24, 48, 96, Mop = 48

We use the hierarchical gap δm,M and conditional upper estimate Um,M,q from Subsection 3.3, with q = 0.9 unless stated otherwise. Monitor-based selection is independent of q; sensitivity of the conditional certificates and upper estimates to q is reported in A. For validation and refinement diagnostics we also use !1/2 2 2 Equad − ηM δM,L Eref,kc , γm,M,L := qobs := , Dc := , 2 2 Equad − ηm δm,M Eref,koracle whenever defined. Here qobs ≤ q checks saturation against the manufactured reference at the tested checkpoint, while γm,M,L < 1 indicates decay of successive newly resolved Riesz components but does not itself verify saturation. The factor Dc measures the energy deterioration of criterion c relative to the oracle of the same logged archive; Dc = 1 denotes oracle-level selection with respect to the validation reference. When Equad 2 2 1/2 is available, we also report ρquad /Equad as an external measure of the unresolved Riesz m,rel := (Equad − ηm ) component. The experimental protocols are summarized in Table 1. Unless stated otherwise, reported values of the form x ± y denote the mean and one sample standard deviation over the five seeds. For the manufactured square-domain tests, the auxiliary spaces are the conforming Q1 spaces  VmQ1 = vm ∈ C 0 (Ω) ∩ H01 (Ω) : vm |K ∈ Q1 (K) ∀K ∈ Tm , with [VmQ1 ]2 in elasticity; the L-shaped benchmark uses the same construction on its active quadrilateral cells. To realize Algorithm 1 on the nested auxiliary hierarchies, we use the common finest-level Galerkin ⊤ ⊤ form Gm = Pm,L GL Pm,L and rm = Pm,L rL , so nestedness and monitor monotonicity hold algebraically at the assembled discrete level. The finest levels are L = 48 for the standard scalar and elasticity hierarchies, L = 96 for the additional κ4 certification hierarchy, and L = 128 for the L-shaped hierarchy. P1 For the perforated plate, let Vm,◦ denote the conforming piecewise-linear space satisfying the homogeneous essential condition on the outer boundary. We use the exactly nested vector-valued hierarchy P1 2 P1 2 P1 2 [V24,◦ ] ⊂ [V48,◦ ] ⊂ [V96,◦ ] ,

obtained by straight-sided uniform refinement of a common conforming triangular mesh fitted to the fixed polygonal hole. All levels therefore represent the same polygonal domain. The discrete-hierarchy and quadrature-order audit is reported in C. All approximations are trained with the Deep Ritz energy [20], with essential boundary conditions imposed strongly. Jtrain and Jval denote energy evaluations on the fixed training and independent validation sets, respectively. The strong-form residual is used only as an L2 -type baseline. Neural training and automatic differentiation use single precision, whereas auxiliary assembly, Riesz solves, reference quadratures, and FEM computations use double precision. The operational levels Mop in Table 1 are those used for checkpoint ranking; enriched levels are used only for post-training refinement, calibration, and certification diagnostics.

6.1

Manufactured-reference validation benchmarks

6.1.1

Scalar variable-coefficient diffusion

We first consider −∇ · (κ∇u) = f

in Ω = (0, 1)2 , 23

u = 0 on ∂Ω,

Value

Deterministic energy error Equad

Deterministic energy error Equad Riesz monitor 24 L 2 strong residual

101

100

10 1 0

100

10 1

Logged iterates Fit = 1

10 1

250 500 750 1000 1250 1500 1750 2000 Iterations

Riesz monitor m

100

(b) Calibration against Equad .

(a) Training diagnostics.

Figure 1: Representative κ1 run (seed 0): temporal tracking and calibration of the operational monitor η24 against Equad . Table 2: Scalar diffusion: checkpoint selection over five seeds. κ1

Selection criterion Energy oracle min Equad Riesz monitor min η24 Strong residual L2 Training Ritz loss min Jtrain Validation Ritz loss min Jval

κ2

κ3

κ4

Step

Dc

Step

Dc

Step

Dc

Step

Dc

1390 ± 201 1390 ± 201 1390 ± 201 2000 ± 0 1370 ± 182

1.000 ± 0.000 1.000 ± 0.000 1.000 ± 0.000 1.723 ± 0.368 1.036 ± 0.067

1420 ± 175 1420 ± 175 1250 ± 332 2000 ± 0 1390 ± 156

1.000 ± 0.000 1.000 ± 0.000 1.105 ± 0.223 1.778 ± 0.294 1.020 ± 0.029

1410 ± 222 1410 ± 222 1230 ± 355 2000 ± 0 1420 ± 160

1.000 ± 0.000 1.000 ± 0.000 1.074 ± 0.163 1.603 ± 0.389 1.030 ± 0.040

1480 ± 289 1480 ± 289 1300 ± 237 2000 ± 0 1460 ± 268

1.000 ± 0.000 1.000 ± 0.000 1.049 ± 0.109 1.529 ± 0.358 1.012 ± 0.020

with V0 = H01 (Ω) and ai (u, v) =

R Ω

κi ∇u · ∇v dx. The first three cases use the common manufactured solution

u∗i (x, y) = sin(πx) sin(πy),

i = 1, 2, 3,

with κ1 = 1 + 12 sin(2πx) sin(2πy),

κ2 = 1 + 41 sin(6πx) sin(6πy),

κ3 = 1 + 0.45 sin(10πx) sin(10πy). The fourth combines the more oscillatory coefficient κ4 = 1 + 0.49 sin(14πx) sin(14πy) with

u∗4 (x, y) = sin(πx) sin(πy) + 0.35 x(1 − x)y(1 − y) 
 × exp −80 (x − 0.35)2 + (y − 0.65)2 (1 + 0.5 sin(2πx) sin(3πy)) .

In each case, fi = −∇ · (κi ∇u∗i ). Figure 1 shows that η24 tracks the energy error on its natural scale and attains its minimum in the same late-training regime. The strong L2 residual evolves on a different scale, as expected from Remark 2. Across all twenty scalar runs (Table 2), η24 selects the same logged checkpoint as the Equad -oracle. The validation Ritz loss remains close to the oracle, whereas the training loss selects the final iterate and the strong residual departs from the oracle in the three more demanding cases. This does not contradict the exact Ritz–energy identity, which applies to the continuous functional J; the discrete evaluations Jtrain and Jval may perturb its ordering, as discussed after Proposition 2. None of these baselines carries the energy-dual ∗ ∈ arg mink∈Klog ηm,k . ordering guarantee used by the Riesz criterion km At the selected checkpoints (Table 3), η24 /Equad = 0.975–0.983, increasing to η48 /Equad = 0.994–0.996. The observed saturation factors are near 0.5, and the conditional upper estimates remain within about 5%–7% of the reference error for q = 0.9. 24

Table 3: Scalar diffusion: calibration and hierarchical diagnostics at the η24 -selected checkpoint. Coefficient κ1 κ2 κ3 κ4

Monitor calibration

Refinement / estimator diagnostics

η24 /Equad

η48 /Equad

γ12,24,48

qobs

U24,48,0.9 /Equad

0.981 ± 0.007 0.975 ± 0.011 0.983 ± 0.008 0.979 ± 0.013

0.995 ± 0.002 0.994 ± 0.003 0.996 ± 0.002 0.995 ± 0.004

0.519 ± 0.004 0.535 ± 0.019 0.528 ± 0.013 0.525 ± 0.028

0.503 ± 0.001 0.506 ± 0.004 0.506 ± 0.003 0.509 ± 0.011

1.052 ± 0.018 1.070 ± 0.029 1.048 ± 0.020 1.057 ± 0.033

Table 4: κ4 : logged-oracle selection and interval certification for q = 0.9. Match uses Equad only for external validation; certified counts are conditional on saturation. Auxiliary pair

h

|Kh |

Match

Certified

Γlog,h

V24 ⊂ V48

50 100 200 400

41 21 11 6

5/5 5/5 4/5 5/5

0/5 2/5 4/5 4/5

1.079 ± 0.048 1.080 ± 0.048 1.088 ± 0.065 1.094 ± 0.066

V48 ⊂ V96

50 100 200 400

41 21 11 6

5/5 5/5 4/5 5/5

2/5 5/5 4/5 5/5

1.021 ± 0.014 1.021 ± 0.014 1.025 ± 0.023 1.027 ± 0.023

Thus, in these scalar tests, the level-24 monitor already resolves the archive ordering needed for oracle selection and incurs below-1% observed overhead (Table 10), supporting its use as the operational monitor; enrichment further sharpens energy-scale calibration. Across all four coefficients, auxiliary refinement increases ηm /Equad toward unity (Figure 3a), consistent with Corollary 2. Having established energy-scale calibration, we next test finite-resolution certification of the selected archive minimizer. Finite-level interval certification We use κ4 to test finite-resolution certification across auxiliary resolutions and logging strides. The dense log is K50 = {0, 50, . . . , 2000}, and the nested subarchives are Kh = {0, h, 2h, . . . , 2000} for h ∈ {50, 100, 200, 400}. For the monitor-selected checkpoint sh , let Uh := Um,M,q,sh . We report the conditional near-oracle factor Γlog,h = Uh /ηm,sh ; interval separation certifies sh as the unique oracle of Kh , conditional on saturation. Auxiliary enrichment improves certifiability (Table 4): with V48 ⊂ V96 , all five K100 selections are certified and match the observed Equad -oracle. Higher certification counts are observed on several coarser subarchives, although the dependence on h is not monotone. On K50 , all five selections still match but only two are certified, illustrating that failure of interval separation is inconclusive rather than evidence of incorrect selection. The single mismatch observed on K200 is not certified and has only 0.72% deterioration. No certified mismatch occurs. Trajectory-wide archive control Certification within Kh does not control checkpoints omitted by logging. We therefore take K50 as the prescribed finite comparison trajectory and apply Section 5 to h ∈ {100, 200, 400} using V48 ⊂ V96 and q = 0.9. For external validation we report   Equad,sh traj δsh := 100 −1 . minj∈K50 Equad,j The hatted quantities below are deterministic-quadrature realizations; they are reference-free diagnostics relative to K50 , not certified continuous-energy-norm bounds unless quadrature error is controlled. 25

Table 5: Trajectory-wide κ4 diagnostics relative to K50 . Oracle inclusion is reported as observed/certified; in these runs the selected-trajectory-oracle counts are identical. The last column gives mean/max observed deterioration. h

|Kh | Oracle inclusion

100 200 400

21 11 6

r bh

b ah + r bh

btraj,h Γ

mean/max δstraj

0.012 ± 0.013 0.015 ± 0.015 1.127 ± 0.111 0.032 ± 0.026 0.036 ± 0.029 1.397 ± 0.334 0.061 ± 0.041 0.065 ± 0.045 2.197 ± 1.148

3/5 / 2/5 1/5 / 1/5 0/5 / 0/5

h

0.22%/0.91% 3.38%/7.91% 8.15%/23.89%

Table 6: Elasticity: checkpoint selection over five seeds. C1

Selection criterion Energy oracle min Equad Riesz monitor min η24 Strong residual L2 Training Ritz loss min Jtrain Validation Ritz loss min Jval

C2

Cinc

Step

Dc

Step

Dc

Step

Dc

960 ± 114 960 ± 114 680 ± 84 3000 ± 0 940 ± 89

1.000 ± 0.000 1.000 ± 0.000 1.249 ± 0.141 6.597 ± 2.770 1.011 ± 0.024

960 ± 55 960 ± 55 700 ± 71 3000 ± 0 940 ± 55

1.000 ± 0.000 1.000 ± 0.000 1.206 ± 0.127 6.819 ± 2.047 1.000 ± 0.001

400 ± 122 420 ± 130 320 ± 110 3000 ± 0 500 ± 245

1.000 ± 0.000 1.000 ± 0.000 1.011 ± 0.016 12.287 ± 3.022 1.059 ± 0.089

The trajectory-wide results (Table 5) separate within-archive certifiability from trajectory coverage. Although K400 admits a logged-oracle certificate in all five runs with V48 ⊂ V96 , it contains no K50 -oracle. By contrast, K100 halves the archive from 41 to 21 checkpoints and limits the maximum observed trajectory deterioration to 0.91%. Coarser logging increases the reported trajectory-wide bounds and the observed deterioration. All trajectory-wide certificates obtained here satisfy the external check qobs ≤ 0.509 < 0.9. Thus auxiliary refinement controls selection within a fixed archive, whereas logging refinement controls coverage of the prescribed trajectory. We next test the same construction in the vector-valued energy geometry of linear elasticity and under localized material contrast. 6.1.2

Plane-strain elasticity and smoothed high-contrast inclusion

We consider plane-strain elasticity on Ω = (0, 1)2 , with homogeneous Dirichlet conditions, V0 = H01 (Ω; R2 ), and Z a(u, v) = 2µ(x)ε(u) : ε(v) + λ(x) div u div v dx, Ω

where ε(v) = 12 (∇v + ∇v ⊤ ),

σ(v) = 2µε(v) + λ div(v)I.

The manufactured displacement is ∗

u (x, y) =

! sin(πx) sin(πy) , 1 2 sin(2πx) sin(πy)

−∇ · σ(u∗ ) = f.

With E0 = 1, ν = 0.30, and the corresponding baseline Lamé parameters (λ0 , µ0 ), we set λ(x) = s(x)λ0 , µ(x) = s(x)µ0 and consider s1 = 1, s2 = 1 + 0.45 sin(6πx) sin(4πy), plus the smoothed high-contrast inclusion sinc (x, y) = 1 + 19χε (x, y),

" !# p (x − 0.55)2 + (y − 0.52)2 − 0.18 1 χε (x, y) = 1 − tanh . 2 0.035

The three cases are denoted by C1 , C2 , and Cinc . Figure 2 shows that the vector-valued monitor remains aligned with the elastic energy error, including in the high-contrast inclusion case. The Riesz criterion is oracle-level in all three material cases (Table 6); in one inclusion seed it selects a neighboring checkpoint with negligible deterioration (Dc = 1.000077). The training Ritz loss always selects 26

103

Deterministic energy error Equad

Deterministic energy error Equad Riesz monitor 24 Strong residual L 2

Value

102 101 100

100

Logged iterates Linear fit α = 1.5

0

500

1000

1500 2000 Iterations

2500

3000

100

Riesz monitor ηm

(b) Calibration against Equad .

(a) Training diagnostics.

Figure 2: Representative high-contrast inclusion run: temporal tracking and calibration of the vector-valued monitor η24 . Table 7: Elasticity: calibration and hierarchical diagnostics at the η24 -selected checkpoint. Material C1 C2 Cinc

Monitor calibration

Hierarchical estimator

η24 /Equad

η48 /Equad

qobs

U24,48,0.9 /Equad

0.971 ± 0.007 0.973 ± 0.008 0.986 ± 0.004

0.993 ± 0.002 0.993 ± 0.002 0.996 ± 0.001

0.507 ± 0.002 0.507 ± 0.002 0.509 ± 0.002

1.079 ± 0.019 1.074 ± 0.020 1.040 ± 0.011

the final logged iterate with mean deterioration factors Dc = 6.60–12.29, while independent validation is substantially closer to the oracle. As in the scalar benchmarks, the comparison with Jval is empirical, since finite-set evaluations may perturb the ordering of the continuous Ritz functional. At the operational level (Table 7), the mean calibration ratio η24 /Equad ranges from 0.971 to 0.986; level 48 raises the mean ratio to 0.993–0.996. The external saturation check again gives qobs ≃ 0.5, while the mean q = 0.9 upper-estimate ratio ranges from 1.040 to 1.079. Across the three elasticity cases, ηm /Equad again increases toward unity under enrichment (Figure 3b), in agreement with the refinement behavior established in Corollary 2. Thus oracle-level selection and tight energy-scale calibration persist in vector-valued elasticity with localized material contrast. A complementary spatial validation in Figure 6 shows that the Riesz-projected density captures the dominant energetic concentration associated with the localized inclusion. We next stress the auxiliary-resolution requirement using a reentrant-corner singularity. 6.1.3

L-shaped-domain stress test

The final manufactured benchmark uses
ΩL = (−1, 1)2 \ [0, 1) × (−1, 0] , with ∗

2

2

u (x, y) = (1 − x )(1 − y )r

2/3

−∆u = f,   2θ sin , 3

u = 0 on ∂ΩL ,

f = −∆u∗ .

Here (r, θ) are polar coordinates centered at the reentrant corner, with θ ∈ [0, 3π/2]. The reentrant-corner singularity tests checkpoint selection when uniform auxiliary spaces require substantially greater resolution. Figure 4 shows that all monitored levels track the evolution of Equad , while auxiliary enrichment improves their energy-scale calibration. The stronger resolution requirement is consistent with the reentrant-corner singularity.

27

ηm/Equad

1.0

1.0

1.0

0.8

0.8

0.8

0.6

0.6

0.6 κ1 κ2 κ3 κ4

0.4 6

12

24

1 2 inc

0.4

48

6

12

Auxiliary level m

(a) Scalar diffusion.

24

0.4

48

8

16

Auxiliary level m

32

64

128

Auxiliary level m

(b) Plane-strain elasticity.

(c) L-shaped domain.

Figure 3: Auxiliary refinement at the operationally selected checkpoints for the manufactured benchmarks. Curves report the mean ηm /Equad over five seeds, with error bars denoting one standard deviation. Within each run, the neural checkpoint is held fixed as m varies; the dashed line marks unit calibration. 100

100

Equad

32, = 1 48, = 1 64, = 1 128, = 1

Value

Equad

32 48 64 128

10 1

10 1 0

10 1

500 1000 1500 2000 2500 3000 3500 4000 Iterations

100 m

(b) Calibration against Equad .

(a) Training diagnostics.

Figure 4: L-shaped-domain monitor comparison for a representative run. The conforming Riesz monitors at increasing auxiliary resolutions track the reference energy error along the logged trajectory and approach the energy-error scale under enrichment. Table 8: L-shaped domain: checkpoint selection over five seeds. The strong residual uses the cutoff r > 0.035 around the reentrant corner. Selection criterion

Selected step

Dc

min η32 min η48 min η64 min η128

1900 ± 1086 1980 ± 1031 1960 ± 1031 1960 ± 1031

1.003 ± 0.005 1.000 ± 0.001 1.000 ± 0.000 1.000 ± 0.000

Strong residual L2 (r > 0.035) Training Ritz loss min Jtrain Validation Ritz loss min Jval

1640 ± 767 4000 ± 0 1800 ± 1251

1.007 ± 0.006 1.181 ± 0.095 1.092 ± 0.169

The light monitor η32 is already near-oracle, while η64 and η128 recover the logged oracle in all five runs (Table 8). Exact logged-oracle recovery across all five runs is therefore reached only at a finer auxiliary resolution in this benchmark than in the smooth square-domain tests, consistent with Corollary 3. At the η64 -selected checkpoint (Table 9), the captured fraction rises from 0.801 ± 0.044 at level 32 to 0.965 ± 0.009 at level 128, while the unresolved fraction decreases accordingly. For V64 ⊂ V128 , the external check gives qobs = 0.637 ± 0.007 < 0.9 and the conditional upper estimate is 1.165 ± 0.035 times Equad . For the reported L-shaped runs, we use η64 as the operational selector and η128 as the enriched post-training check. 28

Table 9: L-shaped domain: selected auxiliary-resolution diagnostics. ηm /Equad

ρquad m,rel

γ or qobs

U/Equad

32 × 32 64 × 64 128 × 128

0.801 ± 0.044 0.911 ± 0.022 0.965 ± 0.009

0.595 ± 0.061 0.409 ± 0.050 0.260 ± 0.034

– – –

– – –

(32, 64, 128) (64, 128)

– –

– –

0.729 ± 0.037 0.637 ± 0.007

– 1.165 ± 0.035

Level/diagnostic

Table 10: Representative cost of the operational and enriched Riesz reconstructions. Overhead is measured against the pure optimizer-update time between logged checkpoints. Family

Mop Operational call (s) Operational overhead Enriched level Enriched overhead

Scalar diffusion Elasticity/inclusion L-shaped domain

24 24 64

0.034–0.045 1.881–2.045 0.131 ± 0.009

0.58–0.76% 8.01–8.72% 0.74 ± 0.03%

48 48 128

12.32–15.30% 8.27–9.01% 4.04 ± 0.74%

Refinement remains monotone and progressively improves energy-scale calibration (Figure 3c), consistently with Corollary 2. Notably, η64 /Equad = 0.911 ± 0.022 already recovers the logged oracle in all five runs. Thus, in this benchmark, oracle recovery precedes near-unit calibration: the archive ordering is already resolved sufficiently for oracle recovery before near-exact recovery of the error scale is reached. A complementary spatial validation in Figure 7 shows that the enriched Riesz reconstruction captures the dominant energy concentration around the reentrant corner. We next quantify the cost of the operational auxiliary levels. 6.1.4

Operational cost of checkpoint selection

The operational monitor incurs less than 1% overhead in the scalar and L-shaped tests and 8.01%–8.72% in elasticity (Table 10). These costs support its practical use for checkpoint ranking along the logged trajectory, while enriched reconstructions are reserved for tighter post-training assessment and certification. We finally test the selector without a closed-form solution, using FEM only for external post-training validation.

6.2

Perforated plate: reference-free selection with external FEM validation

We consider plane-strain elasticity on Ω = (0, 1)2 \ P96 (c, r),

c = (1/2, 1/2),

r = 0.16,

where P96 (c, r) is a fixed polygonal approximation of a circular hole. The material is homogeneous with E = 1, ν = 0.30, zero body force, and outer-boundary displacement
uD (x, y) = ε0 x + γ0 y, −νε0 y , ε0 = 5 × 10−2 , γ0 = 2.5 × 10−2 , while the hole boundary is traction-free. The neural approximation satisfies the prescribed outer-boundary displacement strongly, and we use the nested auxiliary hierarchy V24 ⊂ V48 ⊂ V96 ⊂ V0 described above. No closed-form solution is available. Both external FEM references are conforming vector-valued P1 Galerkin solutions on fitted triangular meshes of the same fixed polygonal domain. The level-256 reference, used only for external post-training validation, has 179 882 displacement degrees of freedom, while the level-192 check has 109 608. The latter yields the same FEM-reference oracle in all five runs and changes checkpoint errors by at most 6.3 × 10−4 relatively. Neither FEM solve enters training, Riesz reconstruction, or checkpoint selection. Figure 5 shows that both η48 and η96 remain closely aligned with EFEM along the logged trajectory.

29

η48 (η/EFEM = 0.994 ± 0.003) unit slope y = x

η96 (enriched monitor)

η48 (operational monitor)

mean minimum (k = 2700)

FEM-reference error EFEM

Energy-scale value

η96 (η/EFEM = 0.998 ± 0.001)

EFEM (external reference)

10−2

0

500

1000

1500

2000

2500

3000

3500

10−2

10−2

4000

Iterations

Riesz monitor

(a) Training diagnostics.

(b) Calibration against the FEM reference.

Figure 5: Perforated plate: external FEM validation of the operational η48 monitor over five runs. Table 11: Perforated plate: checkpoint selection against the external FEM reference. Criterion c

Step

DcFEM

final min EFEM min η48 min η96 min Jtrain min Jval min Rstrong

4000 ± 0 2640 ± 873 2640 ± 873 2640 ± 873 3900 ± 100 2720 ± 1064 360 ± 152

1.204 ± 0.138 1.000 ± 0.000 1.000 ± 0.000 1.000 ± 0.000 1.139 ± 0.161 1.018 ± 0.024 4.146 ± 0.274

Table 12: Perforated plate: hierarchical diagnostics at the η48 -selected checkpoint. Ratios involving EFEM are external validation quantities; U48,96,0.9 /η48 is the reference-free conditional bracket factor. Quantity

Value

η24 /EFEM η48 /EFEM η96 /EFEM

0.977 ± 0.002 0.993 ± 0.001 0.998 ± 0.000

δ48,96 /EFEM γ24,48,96

0.093 ± 0.005 0.517 ± 0.001

U48,96,0.9 /η48 U48,96,0.9 /EFEM

1.023 ± 0.002 1.016 ± 0.002

Both η48 and η96 select the FEM-reference oracle in all five runs (Table 11). The final iterate is 20.4% worse on average, validation Ritz selection remains close to the oracle, and the strong residual selects a much less accurate checkpoint. We next assess the sharpness of the reference-free hierarchy at the selected checkpoint. The auxiliary hierarchy is tightly calibrated on the FEM-reference scale (Table 12): ηm /EFEM increases from 0.977 ± 0.002 at level 24 to 0.998 ± 0.000 at level 96. Independently of the FEM reference, the conditional q = 0.9 bracket has upper-to-lower factor 1.023 ± 0.002; the FEM comparison gives the external check U48,96,0.9 /EFEM = 1.016 ± 0.002. To assess the mechanical impact of checkpoint selection, we compare the η48 -selected output with the final iterate at the solution level. The late-iterate degradation is also visible in the solution-level errors (Table 13): relative to the η48 selected output, the final iterate increases the displacement, strain, von Mises stress, and sampled maximumdisplacement errors by factors 1.405, 1.140, 1.206, and 1.275, respectively.

30

Table 13: Perforated plate: solution-level comparison with the refined FEM reference. Here ref ref ref eu,L2 = ∥uθ − uref FEM ∥L2 /∥uFEM ∥L2 , eε = ∥ε(uθ − uFEM )∥L2 /∥ε(uFEM )∥L2 , and evm = ∥σvm (uθ ) − ref σvm (uref FEM )∥L2 /∥σvm (uFEM )∥L2 , where σvm denotes the plane-strain von Mises equivalent stress. The sampled maximum-displacement error is eu,max := maxx∈S ∥uθ (x) − uref FEM (x)∥2 , where S is the union of the reference-mesh nodes and the degree-two triangle quadrature points; the final row reports final/selected ratios. Selected output

EFEM

min η48 final

−3

final / min η48

eu,L2

eε

evm

sampled eu,max

−4

4.27×10 ± 2.65×10 0.006 ± 0.001 0.085 ± 0.005 0.061 ± 0.002 1.22×10−3 ± 1.50×10−4 −3 −4 5.15×10 ± 7.10×10 0.008 ± 0.002 0.097 ± 0.011 0.073 ± 0.012 1.56×10−3 ± 2.30×10−4 1.204

1.405

1.140

1.206

1.275

Table 14: Perforated plate: external FEM-solve cost and Riesz monitoring cost. FEM times are one-time reference solves used only for validation; Riesz times are mean costs per logged checkpoint, with overhead measured against the pure optimizer-update time between logs. Quantity

DOFs

Time (s)

Overhead

FEM ref. 256 FEM check 192

179882 109608

53.39 34.47

– –

η24 η48 η96

2642 10536 42080

0.212 ± 0.055 0.959 ± 0.250 3.621 ± 0.677

0.93% 4.19% 15.84%

The operational η48 monitor adds 4.19% overhead in this benchmark (Table 14); the more expensive η96 reconstruction is used only for post-training qualification. Thus the reference-free selector operates at moderate cost without requiring the refined FEM solve in the selection pipeline. Taken together, the manufactured benchmarks validate energy-scale calibration, refinement, and oracle-level selection, while the perforated-plate experiment supports the use of the Riesz monitor as a lightweight, reference-free post-training selector when no exact solution is available.

7

Discussion and outlook

This work developed a reference-free archive-level checkpoint-selection framework for admissible neural approximations of symmetric coercive variational problems. Conforming Riesz reconstruction converts the exact residual–energy geometry into a computable, training-independent monitor, making the logged energy oracle recoverable without the exact solution or a reference solve. Monitor refinement yields eventual loggedoracle recovery, while, under saturation, nested reconstructions provide finite-resolution near-oracle bounds and interval-separation certificates. Logging-resolution control further quantifies the loss relative to a prescribed finite comparison trajectory and provides trajectory-oracle certificates. Across the numerical benchmarks, the monitor approaches the reference energy error under refinement and recovers oracle-level checkpoints once sufficiently resolved. On the perforated plate, this behavior is confirmed against an independent FEM reference, with modest observed post-processing cost. The key mechanism is preservation of the oracle–non-oracle ordering. Checkpointwise recovery alone does not ensure archive-level selection because finite-dimensional projection defects can reverse this ordering; finitearchive uniform recovery eventually restores it. This ordering issue becomes more fundamental beyond the symmetric coercive setting. Indeed, residual–error norm equivalence alone does not preserve the target-error ordering. Let X = Y = R2 with the Euclidean norm and B = diag(1, 2),

ei = (0, 1),

ej = (3/2, 0).

Then ∥e∥2 ≤ ∥Be∥2 ≤ 2∥e∥2

∀e ∈ R2 ,

but ∥ei ∥2 < ∥ej ∥2 ,

∥Bei ∥2 > ∥Bej ∥2 . 31

Thus, even exact norm equivalence does not in general preserve the ordering relevant to archive selection. In the present setting, by contrast, ′ ∥R(uθk )∥V0,a = Ek , so the continuous residual and energy-error orderings coincide; the ordering obstruction analyzed here is therefore introduced by the finite-dimensional reconstruction. Natural extensions are saturation-free finite-resolution certification and archive-level selection beyond symmetric coercive formulations. Explicit quadrature-error control could further upgrade the trajectory-wide diagnostics to fully certified continuous-energy-norm bounds. The framework concerns selection among candidates generated by the optimizer. Viewed from this archivelevel perspective, a further natural direction is to carry the order-preservation principle upstream into the optimization procedure. We leave this direction for future work.

A

Sensitivity to the prescribed saturation parameter

2 2 The prescribed saturation parameter q enters only the conditional upper estimate Um,M,q = (ηm + δm,M /(1 − 2 1/2 q )) and therefore does not affect monitor-based checkpoint selection. We assess the sensitivity of the finite-resolution certificates and upper estimates over the range q ∈ {0.70, 0.80, 0.90, 0.96}. Table 15 reports the resulting interval-separation certification counts on κ4 over five independent trajectories.

Table 15: Sensitivity of interval-separation certification on κ4 . Each entry is the number of certified runs out of five. V24 ⊂ V48 q 0.70 0.80 0.90 0.96

V48 ⊂ V96

K50 K100 K200 K400

K50 K100 K200 K400

2/5 1/5 0/5 0/5

3/5 3/5 2/5 0/5

5/5 4/5 2/5 0/5

4/5 4/5 4/5 2/5

5/5 5/5 4/5 4/5

5/5 5/5 5/5 3/5

4/5 4/5 4/5 4/5

5/5 5/5 5/5 5/5

The certification pattern remains stable over a broad range of q, particularly for V48 ⊂ V96 , with the expected gradual loss of certificates as the upper bounds become more conservative. Larger counts on coarser subarchives reflect greater checkpoint separation rather than improved trajectory coverage. For the perforated plate, Table 16 reports the conditional upper estimate relative to the independent FEM-reference error at the η48 -selected checkpoint. Table 16: Sensitivity of the conditional upper estimate on the perforated plate. q

U48,96,q /EFEM

0.70 0.80 0.90 0.96

1.002 ± 0.000 1.005 ± 0.001 1.016 ± 0.002 1.048 ± 0.005

The upper estimate remains close to the FEM-reference error throughout the tested range, reaching only 1.048 ± 0.005 at q = 0.96. Thus the qualitative conclusions remain stable over the tested range of q, with the expected increase in conservatism as q grows.

B

Spatial diagnostics of the conforming Riesz reconstruction

For two manufactured stress tests, we compare the reference energy-error density with the spatial density of the conforming Riesz reconstruction. These comparisons provide a qualitative view of the dominant energetic regions captured by the reconstruction. Figure 6 shows that the Riesz-projected density identifies the same dominant energetic region associated with the localized material transition as the reference density. Figure 7 shows that both densities concentrate primarily near the reentrant corner while retaining nonzero contributions away from it, consistent with the observed need for global auxiliary refinement. 32

17.5

0.6

12.5

15.0

2

0.6

4

0.4

7.5 5.0

0.2

2.5

0.0 0.0

0.2

0.4

x

0.6

0.8

0.4

0.0 0.0

1.0

0.2

0.4

x

0.6

0.8

2 4

10

0.2

12

0.0 0.0

8

0.2

0

0.8 0.6

6

y

y

10.0

0.8

Riesz-projected error density

6

y

0.8

1.0

0

1.0

0.4

8

log10 (ea, 48 + )

Local energy-error density

1.0

log10 (ea + )

Material scale sinc(x, y)

1.0

10 12

0.2

0.4

x

0.6

0.8

1.0

1.0

0.5

0.0

x

0.5

1.0

Local energy-error density 0 1 2 3 4 5 6 7

1.0

0.5

0.0

x

0.5

1.0

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

Riesz-projected density 0 1 2 3 4 5 6 7

log10 (ea, 128 + )

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

y

L-shaped domain

log10 (ea + )

1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00

y

y

Figure 6: Spatial energy diagnostics for the smoothed high-contrast inclusion. Left: material scale sinc (x, y). Middle: reference energy-error density ea (x) = 2µ|ε(uθ − u∗ )|2 + λ| div(uθ − u∗ )|2 . Right: conforming Riesz-projected density ea,48 (x) = 2µ|ε(z48 )|2 + λ| div z48 |2 . Both densities are displayed as log10 ( · + ε).

1.0

0.5

0.0

x

0.5

1.0

Figure 7: Spatial diagnostics for the L-shaped-domain stress test. Left: domain geometry. Middle: reference energy-error density ea (x) = |∇(uθ − u∗ )(x)|2 . Right: conforming Riesz-projected density ea,128 (x) = |∇z128 (x)|2 on the 128 × 128 auxiliary space. Densities are shown as log10 ( · + ε).

C

Discrete-hierarchy and quadrature-refinement audit

All audits below are post-training and use the same trained trajectories and saved checkpoints as the main experiments.

C.1

Monitor-hierarchy audit

For the scalar-diffusion and L-shaped benchmarks, we recompute the reported monitor diagnostics at the saved operationally selected checkpoints on the same auxiliary levels using the common finest-level Galerkin restrictions of Section 6. No monotonicity violation is observed. Increasing the quadrature order of the common Galerkin assembly from 3 to 5 changes the monitor values by at most 1.314 × 10−6 for scalar diffusion and 6.486 × 10−4 for the L-shaped problem.

C.2

Validation-oracle stability

As noted after Proposition 2, a quadrature approximation can perturb the ordering induced by an exact energy criterion. For the manufactured benchmarks, Equad numerically defines the validation oracle, and we assess whether the reported ordering is stable under quadrature refinement. Without retraining, the manufactured-reference energies are reevaluated at tensor-product Gauss orders Q = 3, 5, 7. Table 17 reports the largest checkpointwise relative changes. The reference-energy evaluations are stable under refinement. The largest sensitivity occurs for the singular L-shaped problem and decreases from Q = 3 → 5 to Q = 5 → 7. On the complete κ4 archive, the oracle and runner-up remain unchanged in all five trajectories, while the smallest relative oracle–runner-up gap, 1.986 × 10−3 , remains well above the observed quadrature variations. The trajectory-oracle inclusion counts 33

Table 17: Manufactured-reference quadrature-refinement audit. The first three rows concern saved decisionrelevant outputs; the κ4 row uses the complete 205-checkpoint comparison archive. Benchmark Scalar diffusion L-shaped domain Elasticity κ4 full archive

Audited outputs

Maximum Q = 3 → 5

Maximum Q = 5 → 7

20 5 16 205

1.494 × 10−7

1.494 × 10−7 5.721 × 10−4 1.237 × 10−7 4.759 × 10−8

2.932 × 10−3 1.043 × 10−7 2.263 × 10−7

are also unchanged: K100 : 3/5, K200 : 1/5, and K400 : 0/5. Thus the validation ordering underlying the reported κ4 oracle and logging-resolution conclusions is insensitive to the tested quadrature refinements. For the perforated plate, refining the independent FEM reference from level 192 to 256 likewise preserves the FEM-reference oracle in all five runs, with maximum relative checkpoint-error change 6.26 × 10−4 and rank correlation at least 0.999826. Together, these audits support numerical stability of the reported selection and logging-resolution conclusions under the tested quadrature and FEM refinements. Remark 8 (Continuous-energy oracle certification). The refinement audit above establishes numerical stability ek its numerical with respect to the tested discretizations. Let Ek denote the exact continuous energy error and E e approximation for k in a finite archive K. If maxk∈K |Ek − Ek | ≤ ε and the unique numerical minimizer e k∈K ej − E ee ) > 2ε, then e has oracle–competitor gap minj∈K\{ek} (E k is also the unique continuous-energy oracle, k ej − E ee − 2ε > 0 for every j ∈ K \ {e since Ej − Eek ≥ E k}. Thus certification of an Equad -based validation k oracle against the exact continuous energy would follow from a suitable uniform numerical-integration error bound; such bounds are left to future work.

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