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NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering Yingming Mao1,5 , Ximeng Liu2 , Jingyi Cheng2 , Xiyuan Liu3 , Jiashuai Liu1 , Yike Liu1 , Zhen Yao4 , Yuzhou Zhou1 , Siyuan Feng5 , Qiaozhu Zhai1 , Shizhen Zhao2 1 Xi’an Jiaotong University

2 Shanghai Jiao Tong University 4 Huawei

5 Shanghai Innovation Institute

Existing proactive TE methods model uncertainty in two main ways. The first group focuses on worst-case guarantees (e.g., FFC [32]), ensuring bandwidth-guaranteed operation under any predefined failure, typically at the expense of capacity over-reservation. The second group employs probabilistic models: TeaVaR [12] optimizes tail risk (CVaR [41]), while FloMore [26] supports per-flow quantile constraints– albeit at the cost of significantly higher computational complexity through iterative decomposition. These seemingly disparate methods share a common structure that each induces a decision-dependent ordering over failure scenarios and then selects a subset to optimize, which we term the Sort-and-Select paradigm. FFC selects only the top-ranked scenario; CVaR selects the tail; FloMore selects a single rank per flow. This abstraction reveals the root cause of the expressiveness–tractability tradeoff: embedding sorting inside optimization introduces combinatorial structure. Using specific mask types (e.g., CVaR) restores tractability by convexifying the objective, but sacrifices flexibility; allowing per-flow masks captures richer preferences but explodes complexity. As the number of scenarios grows, traditional solvers struggle to run in real-time. For accelerating combinatorial optimization [29], deep learning has emerged as a powerful tool [6, 8, 9, 11, 13, 14, 16, 31, 43, 44]. Instead of using exact algorithms to search for optimal solutions, this paradigm treats optimization as a mapping from problem context to decision variables, and trains a neural network to approximate this mapping. At inference time, the complex search process is reduced to a simple forward pass, drastically cutting down the computation time. This approach has been widely adopted in Traffic Engineering–prior work has learned demand-to-routing mappings (Figret [34]), generalized across topologies (Geminet [35]), and incorporated per-link failure probabilities (FauTE [33]). However, applying these techniques to riskaware TE faces two fundamental barriers. The first is the Feasibility Barrier. Existing Deep TE methods have been largely developed to minimize maximum link utilization (MLU), whose objective is fundamentally misaligned with failure-aware TE. MLU measures network congestion rather than the allocation of limited capacity under

arXiv:2605.12862v1 [cs.NI] 13 May 2026

ABSTRACT In production Wide-Area Networks (WANs), correlated failures dominate availability losses, forcing operators to reserve large safety margins that leave substantial capacity underutilized. Achieving high utilization under strict availability targets therefore requires risk-aware Traffic Engineering (TE) over dozens to hundreds of probabilistic failure scenarios—yet solving this problem at operational timescales remains elusive. We demonstrate that existing risk-aware formulations can be unified under an embedded Sort-andSelect structure, exposing a fundamental trade-off between expressiveness and tractability: classical optimizers either restrict scenario selection for efficiency or incur prohibitive decomposition costs. While deep learning appears promising, prior Deep TE methods mainly target maximum link utilization and rely on scaling-based feasibility, which fundamentally breaks under explicit capacity constraints and scenario-dependent risk. We present NeuroRisk, a physicsinformed deep unrolled optimizer that exploits the structure of Sort-and-Select. NeuroRisk enforces feasibility via gated edge-local reservations and represents scenario sets through permutation-invariant, gradient-aligned cues. Evaluations on production-style WANs show that NeuroRisk achieves small optimality gaps relative to the solver with orders of magnitude speedup (102 − 105 ×) on risk objectives, while outperforming neural baselines on nominal throughput.

1

3 Nanyang Technological University

INTRODUCTION

In Wide-Area Networks (WANs), failures are unavoidable. While operators could overprovision capacity to tolerate such events, the high cost of bandwidth makes this approach economically prohibitive [17, 22]. Instead, the industry relies on Traffic Engineering (TE) [2, 3, 10, 15, 21, 23, 27, 28] to maximize utilization in the presence of failures. However, traditional reactive TE–which re-optimizes flows after a failure is detected–suffers from inevitable control-loop latencies, leading to transient overloads [1, 36]. This has driven a shift toward proactive TE, which prepares the network for uncertainty by optimizing traffic distributions for potential failure scenarios in advance. 1

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resource scarcity. Failure-aware formulations shift from implicit congestion objectives to explicit capacity constraints. Existing approaches claim that post-hoc scaling can generalize to this setting [39]; we find this assumption invalid, leading to collateral damage and systematic short-path bias. The second is the Representation Barrier. Deep TE methods have progressively increased input complexity: from simple demand matrices [39] to concatenated vectors of demands and link failure rates [33], and recently adopting Graph Neural Networks to capture topology features [4]. Yet, none of the methods encode the full probabilistic scenario set Q required by risk-aware TE: correlated multi-link failures 𝑞 ∈ Q with probabilities 𝑝𝑞 , where both the size and composition of Q change at runtime. FauTE’s soft penalty optimizes a surrogate rather than true risk objectives, and naive padding/aggregation erases scenario structure. To address these barriers, we propose NeuroRisk, a physicsinformed deep unrolled optimizer for scenario-dependent Sort-and-Select objectives. Crucially, the structure of these objectives allows the gradient computation to commute with scenario aggregation, naturally motivating a gradient-driven unrolled optimizer rather than a black-box predictor [9, 39, 43]. For feasibility and optimization, we introduce Gated Reservation, which transforms the decision space from globallycoupled flow splits to edge-local reservation ratios, guaranteeing capacity feasibility by construction and enabling stable, high-quality capacity allocation. For representation, a physics engine projects Q into permutation-invariant gradient cues, preserving scenario-dependent structure. Our formulation subsumes deterministic TE (|Q| = 1), making NeuroRisk a general-purpose Deep TE optimizer. Our key contributions are:

2

BACKGROUND AND MOTIVATION

This section introduces the risk-aware TE setting and illustrates why existing formulations lead to a computational bottleneck. We route flows 𝑓 ∈ F with demand 𝐷 𝑓 over candidate tunnels T𝑓 using split ratios x = {𝑥 𝑓 ,𝑡 }, where 𝑥 𝑓 ,𝑡 ≥ 0; Í consistent with prior work [32], we allow 𝑡 𝑥 𝑓 ,𝑡 > 1 (overprovisioning for failover) or < 1 (under-provisioning under capacity scarcity). Uncertainty is captured by scenario set Q, which includes the no-failure (nominal) scenario: each Í scenario 𝑞 occurs with probability 𝑝𝑞 ( 𝑞 ∈ Q 𝑝𝑞 = 1) and disables tunnels via the binary indicator 𝛼𝑡,𝑞 ∈ {0, 1} (1 if tunnel 𝑡 survives in 𝑞, 0 otherwise), inducing per-flow loss ℓ𝑓 ,𝑞 —the fraction of demand undelivered by surviving tunnels. Full notation is in Appendix A.

2.1 An Illustration of Sorted-Loss Paradigm Risk-aware Traffic Engineering (TE) fundamentally deals with uncertainty. Operators must make routing decisions x before knowing which failure scenario 𝑞 ∈ Q will materialize. To illustrate the challenges in risk-aware TE, consider the setup in the left column of Figure 1. The network serves two demands (𝐷𝑆1 𝐷 = 15G, 𝐷𝑆2 𝐷 = 12.5G) destined for 𝐷. Each source has two tunneling options: a high-capacity but risky direct tunnel (e.g., 𝑆1 → 𝐷, prone to failure with probability 0.1) and a safe indirect tunnel via node 𝑀 (𝑆1 → 𝑀 → 𝐷). We present two bandwidth allocation decisions to demonstrate how different risk preferences dictate the optimal choice (see middle column of Figure 1). Decision 1 fully utilizes the direct tunnels, allocating the remaining demand to the safe tunnel via 𝑀. Decision 2 shifts more traffic to the safe tunnel, providing redundancy at the cost of slightly under-serving 𝑆1 in the nominal scenario. Unless otherwise specified, we evaluate decisions based on the flow incompletion rate, where lower values are preferred. We define the loss ratio of flow 𝑓 in scenario 𝑞 as defined in Equation (1):

• Sort-and-Select Risk Modeling: We formalize a unified Sort-and-Select view of risk-aware TE, exposing the root cause of the expressiveness–tractability trade-off in existing optimizers and providing operators a clean knob to express scenario-dependent risk preferences. • Feasibility-Preserving Reparameterization: Scaling is fundamentally flawed for risk-aware objectives. Instead, Gated Reservation reparameterizes routing into edge-local reservations with tunnel-level gating, guaranteeing capacity feasibility by construction. • Structure-Aware Scenario Representation: We encode probabilistic failure scenarios through physics-derived gradient cues aligned with Sort-and-Select objectives, enabling generalization across variable Q, topologies, and tunnel sets while matching solver-quality solutions with orders-of-magnitude speedups.

    ∑︁     ℓ𝑓 ,𝑞 = max 0, 1 − 𝑥 𝑓 ,𝑡 · 𝛼𝑡,𝑞     𝑡 ∈ T𝑓  

(1)

Here ℓ𝑓 ,𝑞 ∈ [0, 1] measures the undelivered fraction due to failures (𝛼𝑡,𝑞 = 0). Notation is summarized in Appendix A. For exposition, we define Í scenario-level loss as the average over flows: ℓ𝑞 = | F1 | 𝑓 ℓ𝑓 ,𝑞 . Following prior work [38], we adopt this loss definition, with the framework agnostic to it. Unifying Diverse Strategies. The right column of Figure 1 shows that these strategies share one structure: first sort scenarios by loss (rank 𝑟 ), then select a subset via a mask. The Robust view uses only the worst rank (dark mask); the Conservative view uses the tail–the few highest-loss ranks 2

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering

1. Context: Topology & Demands 2. Strategies: Allocations x

10G (Dir)

𝐷𝑆 1 𝐷 = 15G

Decision 1: Ignore Failure.

Conservative

Robust

Radical

Analysis of Decision 1 Loss (%)

S1

10G

M 𝐷𝑆 2 𝐷 = 12.5G

3. Analysis: Sorted Loss & Selection

Meeting Demands: 𝐷𝑆 1 𝐷 = 15G, 𝐷𝑆 2 𝐷 = 12.5G (Visualized under Normal Scenario)

Base Topology & Total Demands

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7.5G

5G

M 10G (Dir)

S2

2.5G

D

2.5G

(F1,2)

N. No Failure. 𝑝 𝑁 = 0.79

F1. S1-D Direct Fails. 𝑝 𝐹 1 = 0.1

Excess allocated bandwidth sits idle. 10G

S1 F2. S2-D Direct Fails. 𝑝 𝐹 2 = 0.1

4.1G

S2

3.4G

D

3.4G

(F2)

𝑟 =3 (F1)

𝑟 =4 (N)

Analysis of Decision 2 80

72.7%

39.4%

36.3%

20

3.0%

𝑟 =1 (F1,2)

10G

𝑟 =2

Sorted Loss: v = [73.3%, 40.0%, 33.3%, 0%] Robust: 73.3%, Conservative: 73.3%×0.01+40.0%×0.1=4.73% Radical: 0%×0.79+33.3%×0.1=3.33% Rank 𝑟

Loss (%)

50

4.1G

M F1,2. Both Direct Fail. 𝑝 𝐹 1,2 = 0.01

0%

𝑟 =1

Decision 2: Prevent Failure

33.3%

20

10G

All OK

40.0%

50

5G

S2

Failure Scenarios Spectrum

73.3%

D S1

10G

80

10G

𝑟 =2 (F2)

𝑟 =3 (F1)

Sorted Loss: v = [72.7%, 39.4%, 36.3%, 3.0%] Robust: 72.7%, Conservative: 72.7%×0.01+39.4%×0.1=4.67% Radical: 3.0%×0.79+36.3%×0.1=6.00% Rank 𝑟

𝑟 =4 (N)

Figure 1: A Unified Analysis Perspective for Risk-Aware TE Strategies. (Left) A topology with two demands (𝑆1, 𝑆2 → 𝐷). Each demand has a choice between a risky direct tunnel (susceptible to failure) and a safe indirect tunnel (via 𝑀). The failure scenarios and their probabilities are listed below. (Middle) Two distinct allocation strategies: Decision 1 (Radical) maximizes throughput by utilizing risky tunnels, assuming the normal scenario (prob. 0.79); Decision 2 (Robust) proactively throttles traffic and uses safe tunnels to survive failures. (Right) The Sorted Loss Analysis reveals that these strategies–and existing paradigms like FFC (Robust) or TeaVaR (Conservative)–are essentially different “selection masks” prioritizing different ranks in the loss distribution. Sorted loss is the average flow incompletion: e.g., for F1,2 under Decision 1, ℓ𝑆1 = 1 − 5/15 = 66.7%, ℓ𝑆2 = 1 − 2.5/12.5 = 80.0%, average = 73.3%; under Decision 2, ℓ𝑆1 = 1 − 4.1/15 = 72.7%, ℓ𝑆2 = 1 − 3.4/12.5 = 72.8%, average = 72.7%. (blue mask); the Radical view uses the nominal and lowerloss ranks (amber mask). Thus “risk preference” is exactly a choice of selection mask over the sorted loss distribution. Applying these masks to the example shows how risk preference dictates allocation. Under the Robust view (worst-case only), Decision 2 is slightly better due to its more balanced indirect-tunnel allocation. The Conservative strategy, by aggregating over the tail, also favors Decision 2; the Radical strategy, by emphasizing nominal and lower-loss scenarios, clearly favors Decision 1. Figure 1 summarizes this Sort-andSelect view. We next formalize it as a unified optimization problem and identify computational bottlenecks.

2.2

risk preference (which ranks you care about) from physics (capacity and routing), turning previously rigid, objectivespecific formulations into a single, expressive template. Crucially, it also makes the real bottleneck unambiguous: the only nontrivial difficulty is enforcing sorting inside the optimization, which is exactly where prior approaches compromise (by specific masks or aggregating scenarios) and where our solver later removes the complexity. Unlike the scenario-level illustration in §2.1, the formal model sorts losses per flow to enable fine-grained risk control. For each flow 𝑓 , we sort its scenario losses {ℓ𝑓 ,𝑞 }𝑞 ∈ Q in descending order: 𝑣 𝑓 ,𝑟 denotes the 𝑟 -th largest loss (𝑣 𝑓 ,1 ≥ 𝑣 𝑓 ,2 ≥ · · · ≥ 𝑣 𝑓 ,𝑁 ), 𝐼 𝑓 ,𝑟 is a binary mask selecting which ranks contribute, and 𝜋 𝑓 ,𝑟 is the probability mass at rank 𝑟 . Crucially, we distinguish scenario index 𝑞 (identity) from rank index 𝑟 (position after sorting): if scenario 𝑞 lands at rank 𝑟 , then 𝑣 𝑓 ,𝑟 = ℓ𝑓 ,𝑞 and 𝜋 𝑓 ,𝑟 = 𝑝𝑞 . The unified formulation over

Unified Mathematical Formulation

We now view risk-aware TE through a different lens: rather than treating CVaR, chance constraints, and robust TE as separate models, we explicitly model them as the same optimization problem with a tunable selection mask over the sorted-loss ranks (Figure 1). This paradigm shift decouples 3

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graph G(V, E) with 𝑁 =|Q| scenarios is: 𝑁 ∑︁ ∑︁

minimize

𝜋 𝑓 ,𝑟 · 𝐼 𝑓 ,𝑟 ·𝑣 𝑓 ,𝑟 𝑓 ∈ F 𝑟 =1 | {z } Risk Mask

subject to

∑︁ ∑︁

x,𝜹,v,𝝅

𝑓

Case B: Tail-Risk (TeaVaR [12]). Scenario-level; blue mask. Selects the worst 1−𝛽 tail (CVaR): ( 1, if Γ𝑓 ,𝑟 ≤ 1 − 𝛽 𝐼 𝑓 ,𝑟 = (5) 0, otherwise

(2a)

𝐷 𝑓 · 𝑥 𝑓 ,𝑡 ≤ 𝐶𝑒 ,

∀𝑒 ∈ E

(2b)

Case C: Quantile (FloMore [26]). Flow-level. Each flow selects a single rank at the 𝛽-quantile (VaR):

𝑡 :𝑒 ∈𝑡

𝑥 𝑓 ,𝑡 ≥ 0,

(2c)

∀𝑓 , 𝑡 (

𝐼 𝑓 ,𝑟 = 1 ⇐⇒ Γ𝑓 ,𝑟 −1 < 1 − 𝛽 ≤ Γ𝑓 ,𝑟 .

)

ℓ𝑓 ,𝑞 = max 0, 1 −

∑︁

𝑥 𝑓 ,𝑡 · 𝛼𝑡,𝑞 ,

∀𝑓 , 𝑞 (2d)

The amber mask in Figure 1 illustrates a more general “radical” selection of the lower-loss tail; FloMore’s quantile is a special case, selecting only the boundary rank.

𝑡

Sorting & Permutation Constraints (Eq. 3) Here 𝐷 𝑓 is the demand size, 𝐶𝑒 is the capacity of link 𝑒, and 𝑥 𝑓 ,𝑡 defines the split ratios for flow 𝑓 across its tunnels 𝑡 ∈ T𝑓 . Constraint (2b) imposes physical link capacity limits, while (2d) maps a routing decision x to per-scenario loss ℓ𝑓 ,𝑞 via tunnel survival indicators 𝛼𝑡,𝑞 . The objective (2a) aggregates the sorted loss vector via a rank-dependent mask. To realize this, we introduce binary permutation variables 𝛿 𝑓 ,𝑞,𝑟 ∈ {0, 1} (equals 1 iff scenario 𝑞 is ranked 𝑟 -th for flow 𝑓 ) and encode the sorting logic with standard Big-𝑀 constraints: ∑︁

𝛿 𝑓 ,𝑞,𝑟 = 1,

𝑟

∑︁

𝛿 𝑓 ,𝑞,𝑟 = 1,

∀𝑓 , 𝑞, 𝑟

2.3 Motivation: From Combinatorial Search to Neural Approximation The unified formulation in §2.2 reveals that solving riskaware TE in this way (instance-by-instance optimization with explicit sorting constraints) is computationally prohibitive. To overcome this, we propose a fundamental paradigm shift: Optimization as a Mapping. Fundamentally, any solver aims to realize an optimal mapping Ψ∗ that translates the network context (Topology G, Demands D, and Scenarios Q) into an optimal allocation x∗ :

(3a)

𝑞

𝑣 𝑓 ,𝑟 ≤ ℓ𝑓 ,𝑞 + 𝑀 (1 − 𝛿 𝑓 ,𝑞,𝑟 ),

∀𝑓 , 𝑞, 𝑟

(3b)

𝑣 𝑓 ,𝑟 ≥ ℓ𝑓 ,𝑞 − 𝑀 (1 − 𝛿 𝑓 ,𝑞,𝑟 ),

∀𝑓 , 𝑞, 𝑟

(3c)

𝑣 𝑓 ,𝑟 ≥ 𝑣 𝑓 ,𝑟 +1, ∀𝑓 , 𝑟 = 1, . . . , 𝑁 − 1 ∑︁ 𝜋 𝑓 ,𝑟 = 𝑝𝑞 · 𝛿 𝑓 ,𝑞,𝑟 , ∀𝑓 , 𝑟

(3d)

Γ𝑓 ,𝑟 =

𝑞 𝑟 ∑︁

𝜋 𝑓 ,𝑗 ,

∀𝑓 , 𝑟

x∗ = Ψ∗ (G, D, Q) = arg min J (x) x

(3e) (3f)

x = 𝑔𝜃 (G, D, Q) ≈ Ψ∗ (G, D, Q)

Constraint (3a) makes 𝜹 𝑓 a permutation matrix that assigns each scenario 𝑞 to exactly one rank 𝑟 (and vice versa). Constraints (3b)–(3c) force 𝑣 𝑓 ,𝑟 to equal the loss ℓ𝑓 ,𝑞 of whichever scenario is assigned to rank 𝑟 ; (3d) enforces that ranks are ordered by loss (largest to smallest). Here 𝑀 is a sufficiently large constant. Finally, (3e) and (3f) propagate scenario probabilities 𝑝𝑞 into ranked probability masses 𝜋 𝑓 ,𝑟 and their cumulative sum Γ𝑓 ,𝑟 . Prior methods differ in both their rank selection policies and ranking granularity. Scenario-level methods (FFC, TeaVaR) aggregate per-flow losses into a scenario loss ℓ𝑞 , then rank scenarios; all flows share a single mask (𝐼 𝑓 ,𝑟 = 𝐼𝑟 ). Flow-level methods (FloMore) rank each flow independently; each has its own 𝐼 𝑓 ,𝑟 . Case A: Robust (FFC [32]). Scenario-level; dark mask. Only the worst scenario contributes: 𝐼 𝑓 ,𝑟 = 0 ∀𝑓 , 𝑟 > 1.

(7)

where J (x) denotes the chosen risk objective (e.g., a tailfocused metric over scenario losses). Instead of computing this mapping implicitly by solving a combinatorial problem for every new input, we propose to approximate it using a parameterized neural network 𝑔𝜃 :

𝑗=1

𝐼 𝑓 ,1 = 1,

(6)

(8)

where 𝜃 are learnable parameters. Intuitively, 𝑔𝜃 takes the network, demands, and scenario distribution as input and outputs a feasible routing decision without explicitly solving the problem online. This shift from solving for variables to learning a mapping bypasses two barriers in the MILP formulation: • Sorting. The MILP optimizes over the permutation (which scenario gets which rank) via 𝑂 (|F | · 𝑁 2 ) binary variables 𝜹—the main bottleneck. NeuroRisk does not: 𝑔𝜃 outputs x directly; ranking is only used when computing the loss (e.g., sort scenario losses then aggregate by the risk objective). • Linearization. Tail-Risk and Quantile objectives (Mask × Loss) require Big-M linearization in MILP, adding many auxiliary variables. NeuroRisk optimizes the non-linear risk surface directly.

(4) 4

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering

Í Í Í penalty 𝑒 𝑝𝑒 · 𝑓 𝑡 :𝑒 ∈𝑡 𝐷 𝑓 · 𝑥 𝑓 ,𝑡 . Collapsing scenarios into scalar probabilities loses correlation structure (e.g., links sharing conduits often fail together) and restricts optimization to expectations. Existing risk objectives (e.g., CVaR) all require reasoning over the loss distribution across discrete scenarios. These two barriers are fundamental mismatches between existing Deep TE architectures and risk-aware optimization. Overcoming them necessitates a holistic redesign of both the output space (for feasibility) and the input representation (for scenario encoding).

Collapsed

Balanced 𝑒 1 (𝐶𝑒1 =1)

𝑒1

𝑒 2 (𝐶𝑒2 =1)

𝑒 2 +0.5

𝑒2

𝑒 3 (𝐶𝑒3 =1)

+0.5

𝑒3 100

𝑒 4 (𝐶𝑒4 =100) 𝛾 =1

𝑒4

−33

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67

𝛾 = 1.5

Figure 2: GS optimization fragility. Consider bottleneck edges 𝑒 1 –𝑒 3 (capacity 𝐶𝑒 =1) and a large edge 𝑒 4 (𝐶𝑒4 =100). The optimizer must keep all 𝑒𝑖 loads at their capacity limits. When 𝑒 2 ’s load exceeds by just 0.5, 𝛾 rises to 1.5 and all edges are scaled–𝑒 4 loses 33 units to accommodate 𝑒 2 ’s 0.5 excess. This disproportionate collateral damage makes convergence difficult.

4 SYSTEM DESIGN 4.1 Overview To dismantle these barriers, NeuroRisk fundamentally rethinks the TE decision space and the optimization paradigm. To address the Feasibility Barrier, we shift the decision variables from globally-coupled tunnel-level allocations x to edge-local reservation ratios y, which guaranties capacity feasibility by design without post-hoc scaling. To overcome the Representation Barrier, we exploit a key property of Sort-and-Select objectives: the total gradient Í decomposes additively across scenarios, i.e., ∇y J = 𝑞 𝜌𝑞 ∇y J𝑞 , where 𝜌𝑞 = 𝑝𝑞 ·𝑚𝑞 combines the scenario probability 𝑝𝑞 with the risk-specific selection mask 𝑚𝑞 (e.g., 𝑚𝑞 = 1 for tail scenarios in CVaR, 0 otherwise), and J𝑞 is the risk contribution from scenario 𝑞. Crucially, this property holds for the gradient with respect to the decision variables y, not the neural parameters 𝜃 . This distinction dictates a fundamental shift from one-shot prediction to iterative optimization:

3 CHALLENGES OF DEEP LEARNING FOR RISK-AWARE TE Deep learning has achieved remarkable success in Traffic Engineering [4, 34, 35, 39]: near-optimal routing with ordersof-magnitude speedup, generalizing across demands and topologies. These methods, however, are primarily designed for and benchmarked on Maximum Link Utilization (MLU), which is ill-suited for risk-aware settings: under failures, capacity becomes scarce and MLU provides no guidance on how to prioritize flows or distribute limited bandwidth. Riskaware TE also often requires over-provisioning (split ratios >1) for failover, which the MLU formulation (split ratios sum to 1) does not model. Moreover, risk-aware TE reasons over a scenario set Q (which failures, with what probabilities), not just nominal demands and topology. Risk-aware objectives therefore require two capabilities that existing architectures lack: explicit capacity feasibility and scenario-level failure information. We elaborate on each next. The Feasibility Barrier (output). Existing methods use global scaling (GS) to satisfy constraints (2b): divide all alloÍ Í cations by 𝛾 max = max𝑒 ( 𝑓 𝑡 :𝑒 ∈𝑡 𝐷 𝑓 · 𝑥 𝑓 ,𝑡 /𝐶𝑒 ). GS creates a “whack-a-mole” pathology (Figure 2): the model must keep every bottleneck edge at or below capacity; overshooting any one (e.g., 𝑒 2 by 0.5) collapses throughput network-wide (𝑒 4 loses 33 units). Furthermore, GS obscures the true ‘offending’ edges by normalizing them to feasibility, which inadvertently encourages the model to increase their allocations. This triggers a cycle of shifting bottlenecks and throughput oscillations, fundamentally stalling convergence. The Representation Barrier (input). Risk-aware TE requires scenario-level failure information: which failures, with what probabilities. Existing Deep TE methods accept demands and topology but lack a mechanism to ingest a scenario set Q. To our knowledge, FauTE [33] is the only attempt, encoding per-link failure probabilities 𝑝𝑒 as a soft

• The “Blank Slate” Dilemma (One-shot): A direct predictor lacks an intermediate decision state. Without a concrete y, the gradient ∇y J𝑞 is undefined. Consequently, the network must implicitly encode the combinatorial interactions of all scenarios into its weights–a massive representation bottleneck. • The “State Sensitivity” Advantage (Iterative): NeuroRisk maintains an explicit decision anchor y𝑘 at each step. This allows us to compute the state sensitivity— how the risk in each scenario 𝑞 responds to changes in the current plan. These per-scenario sensitivities are independent and additive: scenario 𝑞 1 pushes y𝑘 one way; scenario 𝑞 2 pushes it another. NeuroRisk simply aggregates these local signals, decoupling the architecture from the complexity of Q. Leveraging there insights, NeuroRisk is formulated as a learnable dynamical system [18, 35, 40, 47] (Figure 3) that evolves a latent strategy (Z, W) over 𝐾 iterations. Each step 𝑘 executes a synchronized Generate-Perceive-Update cycle: (1) Generate: A gated projection Sgate maps the latent state (Z𝑘 , W𝑘 ) to a physically feasible reservation y𝑘 ; 5

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CONSTRAINED PHYSICAL DOMAIN Environment Topology Demands D Scenarios

UNCONSTRAINED LATENT DOMAIN

Final Loss Calculation k=K Flow Calculation

k<K

Module I: Edge-Centric Gated Reservation ( )

Module II: Dual-Variable Decomposition Latent Strategy State Upate

Module III: Physics-Aware Neural Optimizer Physical Perception (

)

Policy & Aggregation(

)

∑ Scenario Agg.

Risk precursors Physics Precursors

Initialization ( )

Feature Vectors Shared-Weight MLP

Saturation Precursors

∑ Scenario &Path Gradient Flow

Figure 3: NeuroRisk framework architecture. The system separates into two domains: the Unconstrained Latent Domain (right, orange) where optimization occurs freely, and the Constrained Physical Domain (left, blue) where feasibility is enforced. Module I maps latent variables (Z, W) to strictly feasible reservations y via per-edge Softmax. Module II maintains two coupled latent variables: local contention Z and tunnel-level gating W. Module III extracts gradient-proxy features from (y, Q) and computes updates via a shared-weight MLP, aggregated across scenarios. (2) Perceive: A physics engine Φphy computes current allocations and extracts gradient-aligned features s𝑘,𝑞 for each scenario 𝑞 ∈ Q; (3) Update: A shared neural policy Ψ𝜃 processes these features to propose local updates, which are aggregated using risk weights 𝜌𝑞 to refine the global strategy.

introduces Gated Reservation (Sgate ), addressing the Feasibility Barrier; while §4.3 describes the Physics Engine (Φphy ), addressing the Representation Barrier.

The resulting computation graph is fully differentiable, enabling end-to-end training where the policy 𝜃 learns to minimize the terminal risk J by backpropagating:

As analyzed in §3, Global Scaling (GS) suffers from optimization fragility: a single bottleneck’s micro-fluctuation triggers network-wide collapse. A natural fix is Local Scaling (LS): instead of a shared denominator 𝛾 max , each tunnel 𝑡 is scaled Í Í by its own bottleneck factor 𝛾𝑡 = max𝑒 ∈𝑡 ( 𝑓 𝜏:𝑒 ∈𝜏 𝐷 𝑓 · 𝑥 𝑓 ,𝜏 /𝐶𝑒 ). This localizes the impact–adjusting one tunnel no longer directly affects others. Unfortunately, LS introduces a subtler but equally fatal problem: gradient distortion. LS enforces feasibility via a projection operator 𝜋 (·) that maps an infeasible solution back onto the constraint boundary. The key issue is that each tunnel variable 𝑥 𝑓 ,𝑡 appears in multiple capacity constraints (one per edge along its path). As the optimizer explores, different constraints become active depending on the current solution, and the effective projection rule switches accordingly. Figure 4 illustrates this Projection Discontinuity Trap: a small step that appears beneficial in the unconstrained domain (𝐴→𝐵) can cross the Ridge and trigger a jump in the projection (𝐴′ = 𝜋 (𝐴) to

min

J (y𝐾 ; G, D, Q)

4.2

(9a)

𝜃

s.t.

y𝑘 = Sgate (Z𝑘 , W𝑘 ) (9b) ∑︁
(ΔZ𝑘 , ΔW𝑘 ) = 𝜌𝑞 · Ψ𝜃 Φphy (y𝑘 , G, D, 𝑞) (9c) 𝑞∈ Q

Z𝑘+1 = Z𝑘 + ΔZ𝑘 , W𝑘+1 = W𝑘 + ΔW𝑘

(9d)

Notably, Ψ𝜃 operates exclusively on physics-aware cues s𝑘,𝑞 and never observes raw topology or scenario data. By learning the underlying update dynamics rather than memorizing instance-specific mappings, NeuroRisk achieves zero-shot generalization to unseen network scales and failure distributions. The following sections detail each component: §4.2 6

Gated Reservation: Decision Space Transformation

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering

where 𝑧𝑡,𝑒 is a learnable Í logit for tunnel 𝑡 on edge 𝑒. By construction, 𝑦˜𝑡,𝑒 ≥ 0 and 𝑡 𝑦˜𝑡,𝑒 = 1—capacity is never violated, no scaling needed. This eliminates both GS’s collateral damage (no shared 𝛾 max ) and LS’s projection trap (no pertunnel scaling factor). The bandwidth allocation ratio can be recovered as:

Ridge 𝑒1

𝐴

Infeasible region

𝐵 𝐴′ 𝑑𝐴

𝑥∗

𝑑 𝐵 > 𝑑𝐴

𝐵′

SIGCOMM’26, Colorado, USA

𝑒2

𝑥 𝑓 ,𝑡 = (min 𝐶𝑒 · 𝑦˜𝑡,𝑒 )/𝐷 𝑓 𝑒 ∈𝑡

However, this naive approach has a critical flaw: the ShortBoard Trap. A tunnel’s delivered bandwidth is min𝑒 ∈𝑡 𝐶𝑒 ·𝑦˜𝑡,𝑒 — the minimum across all edges. If the optimizer increases 𝑦˜𝑡,𝑒1 on one edge while others remain low, the extra allocation is “wasted” (throughput unchanged since the bottleneck is elsewhere). Competing tunnels on 𝑒 1 then push to reclaim the unused capacity. Path-level growth—raising all edges of a tunnel together—is prohibitively difficult with purely edgelocal 𝑧𝑡,𝑒 updates. Each edge’s Softmax is an independent competition; there is no mechanism for “raise my share on all my edges simultaneously.” To escape this trap, we need to restore tunnel-level coordination—but without reintroducing the projection discontinuities of LS.

Feasible region

Figure 4: Mechanism of Projection Discontinuity. 𝐵 ′ = 𝜋 (𝐵)), leading to worse outcomes and a bounce-back that prevents progress toward 𝑥 ∗ . In our experiments, LS-based training exhibits a strong short-path preference: when both short and long paths are feasible, LS tends to concentrate traffic on shorter (often direct) routes, since longer paths traverse more edges and are more likely to activate tight constraints and incur projection penalties. For risk-aware TE, this bias is catastrophic: backup paths are essential for absorbing failures, yet LS systematically under-utilizes them. Both GS and LS fail because they attempt to enforce feasibility via post-hoc scaling. The root cause is that flow-level variables are globally coupled: each 𝑥𝑡 appears in multiple constraints, so any scaling operation must navigate this entanglement. The solution is not a better scaling rule–it is to transform the decision space so that feasibility is guaranteed by construction, without any scaling.

4.2.2 Module II: Tunnel-Level Gating. To escape the ShortBoard Trap, we introduce a tunnel-level gate 𝑤𝑡 that coordinates all edges of a tunnel. Combined with the edge-level Softmax from Module I, this defines the gated projection Sgate formalized in the overview. The final reservation ratio 𝑦𝑡,𝑒 is computed as: exp(𝑧𝑡,𝑒 + 𝑤𝑡 ) 𝑦𝑡,𝑒 = Í , ∀𝑡 ∈ T𝑒 . (12) 𝜏 ∈ T𝑒 exp(𝑧𝜏,𝑒 + 𝑤 𝜏 ) The two latent variables serve complementary roles: Local contention (𝑧𝑡,𝑒 ) controls how aggressively tunnel 𝑡 competes on edge 𝑒, relative to other tunnels sharing that edge; Global gating (𝑤𝑡 ) is a tunnel-wide priority score shared across all edges of 𝑡. Because 𝑤𝑡 appears in the Softmax on every edge along tunnel 𝑡, increasing 𝑤𝑡 simultaneously boosts 𝑡’s share on all its edges—exactly the “raise all boards together” signal that pure 𝑧𝑡,𝑒 cannot express. Think of a tunnel as a barrel: its throughput is limited by the shortest board (bottleneck edge). With edge-local 𝑧𝑡,𝑒 alone, raising one board while others remain short wastes capacity—the water still spills. The tunnel-level gate 𝑤𝑡 acts as a lifting mechanism that raises all boards simultaneously. Mathematically, the gradient 𝜕J /𝜕𝑤𝑡 aggregates feedback from all edges along tunnel 𝑡: 𝜕J ∑︁ 𝜕J 𝜕𝑦𝑡,𝑒 = · (13) 𝜕𝑤𝑡 𝑒 ∈𝑡 𝜕𝑦𝑡,𝑒 𝜕𝑤𝑡 Í Crucially, the summation 𝑒 ∈𝑡 in Eq. (13) acts as a voting mechanism: globally efficient tunnels accumulate positive votes across all their edges and grow uniformly; inefficient detours accumulate negative votes and shrink. In LS, the

4.2.1 Module I: Edge-Centric Reservation. We implement this decision-space transformation via Bandwidth Reservation (BR): by directly optimizing per-edge reservation ratios, feasibility is guaranteed by construction, eliminating the need for post-hoc scaling. Traditional TE asks: “What fraction of flow 𝑓 ’s demand goes on tunnel 𝑡?” This produces variables 𝑥 𝑓 ,𝑡 that must satisfy capacity constraints Í 𝑓 ,𝑡 :𝑒 ∈𝑡 𝐷 𝑓 · 𝑥 𝑓 ,𝑡 ≤ 𝐶𝑒 for every edge. The problem: each 𝑥 𝑓 ,𝑡 appears in multiple constraints (one per edge along the tunnel), so adjusting one variable ripples through many constraints. We invert the perspective: “What fraction of edge 𝑒’s capacity is reserved for tunnel Í𝑡?” This produces variables 𝑦𝑡,𝑒 ∈ [0, 1] with constraint 𝑡 ∈ T𝑒 𝑦𝑡,𝑒 = 1 for each edge, where T𝑒 = {𝑡 : 𝑒 ∈ 𝑡 } denotes the set of tunnels traversing edge 𝑒. Now each 𝑦𝑡,𝑒 appears in exactly one constraint–its own edge’s partition. Edges allocate capacity independently; there is no global coupling. A natural implementation is per-edge Softmax over learnable logits: exp(𝑧𝑡,𝑒 ) , 𝜏 ∈ T𝑒 exp(𝑧𝜏,𝑒 )

𝑦˜𝑡,𝑒 = Í

∀𝑡 ∈ T𝑒 ,

(11)

(10) 7

SIGCOMM’26, Colorado, USA Category

Feature

captures exactly the information needed to improve the objective. By the chain rule, the gradient with respect to the edge-level latent variable 𝑧𝑡,𝑒 factorizes into three physically interpretable components (Risk × Physics × Saturation):

Physical Interpretation

ℓ𝑓 ,𝑞 ℓ𝑞

Tunnel survival status (1 if path connects, 0 if broken) Flow 𝑓 ’s unserved demand ratio under 𝑞 Scenario severity (global loss)

Physics

𝑥 𝑓 ,𝑡 𝑚𝑡,𝑒 𝐶𝑒 𝐷 𝑓 (𝑡 )

Split ratios of tunnel 𝑡 for flow 𝑓 Bottleneck Margin: 𝐶𝑒 𝑦𝑡,𝑒 − 𝑏𝑡 Edge capacity (normalized) Demand magnitude

Saturation

𝑦𝑡,𝑒

Current Softmax share (gradient saturation proxy)

𝛼𝑡,𝑞 Risk

Mao et al.

𝜕J𝑞 𝜕J𝑞 𝜕𝑥 𝑓 ,𝑡 𝜕𝑦𝑡,𝑒 · = · 𝜕𝑧𝑡,𝑒 𝜕𝑥 𝑓 ,𝑡 𝜕𝑦𝑡,𝑒 𝜕𝑧𝑡,𝑒 |{z} |{z} |{z} Risk

Physics Saturation

The gradient with respect to the tunnel-level gate 𝑤𝑡 has the same Í structure, summed over all edges along the tunnel ( 𝑒 ∈𝑡 ). (1) Risk Sensitivity captures how loss responds to allocation changes; (2) Physical Coupling (𝜕𝑥/𝜕𝑦) captures the “Short-Board” logic—it is non-zero only if edge 𝑒 is the bottleneck, motivating the Bottleneck Margin feature 𝑚𝑡,𝑒 ; (3) Saturation State captures the Softmax derivative. The Physics Engine Φphy uses these components not to regress the gradient directly, but to construct a physics-aware state representation (summarized in Table 1; derivation in Appendix C) that enables the policy to learn update directions. Update Dynamics. The neural policy Ψ𝜃 is a lightweight MLP applied to each feature tuple s𝑡,𝑒,𝑞 . Per-scenario outputs are aggregated using the risk weights 𝜌𝑞 defined in §4.1:

Table 1: Physics-Aware Feature Vector s𝑡,𝑒,𝑞 . These features directly correspond to the components in Eq. (14).

decision variables are already tunnel-level (flow/tunnel bandwidth allocations). Feasibility is enforced by a per-tunnel scaling/projection factor 𝛾𝑡 , which is computed from edge loads via a non-smooth max over constraints. This creates projection discontinuities. In contrast, 𝑤𝑡 is a learnable tunnel-level coordination variable that lives inside the smooth Softmax, ensuring stable gradient propagation. Summary: The Best of Both Worlds. Gated Reservation synthesizes the strengths of BR and LS. It inherits the strictly feasible, smooth optimization landscape of the edge-centric design (Module I), while 𝑤𝑡 restores the vital tunnel-level coordination of LS. This synergy avoids the optimization traps of post-hoc scaling, effectively enabling the “raise all boards together” capability—a collective growth mechanism that neither approach can achieve in isolation.

4.3

(14)

Δ𝑧𝑡,𝑒 =

∑︁

𝜌𝑞 · Ψ𝜃(𝑧 ) (s𝑡,𝑒,𝑞 )

(15a)

𝑞∈ Q

Δ𝑤𝑡 =

∑︁ ∑︁

𝜌𝑞 · Ψ𝜃(𝑤 ) (s𝑡,𝑒,𝑞 )

(15b)

𝑒 ∈𝑡 𝑞 ∈ Q

Eq. (15a) updates local contention using edge-specific sigÍ nals, while Eq. (15b) sums across the tunnel ( 𝑒 ∈𝑡 ) to adjust the tunnel-wide gate. By operating on these physics-derived features rather than raw topology, the learned policy Ψ𝜃 captures the invariant update dynamics of the optimization process, enabling zero-shot generalization to unseen topologies and scenario distributions.

Physics-Aware Neural Optimizer (Module III)

Module III functions as the neural update policy within the iterative optimization loop. As illustrated in Figure 3, it comprises two coordinated sub-components: (1) a Physics Engine (Φphy ) that extracts gradient-relevant features from the physical state, and (2) a Shared-Weight Policy (Ψ𝜃 ) that maps these features to optimal update steps. At each iteration 𝑘, this module observes the feasible reservation y𝑘 and failure scenarios Q, then outputs incremental updates (ΔZ𝑘 , ΔW𝑘 ) to evolve the latent strategy. Physics Engine: Gradient-Grounded Feature Design. As established in §4.1, maintaining an explicit state y𝑘 allows us to compute the state sensitivity—how each scenario’s risk responds to changes in the latent variables (Z, W). We now detail how to translate this analytical structure into learnable features. Rather than learning from raw data, we ground our per-scenario features s𝑡,𝑒,𝑞 in the analytical gradient, which

5

EVALUATION

We outline methodology (§5.1), compare the quality and runtime of the solution (§5.2 – §5.3), evaluate generalization across scenarios, demands, tunnels and topologies (§5.4), and provide ablation analysis (§5.5).

5.1

Methodology

Topologies. Our evaluation covers five WAN topologies representing diverse scales and connectivity patterns: B4, IBM, GEANT, GERMANY50, and TATANID. For B4 and IBM, we use the well-known optical-layer topologies also used in Semi-Oblivious TE [30] and generate IP-layer topologies using the distributions described in ARROW [48]. For GEANT, 8

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering

0.03

PreTE TeaVaR FFC

Solver Excessive UB-LB Gap !!!

0.02

Speedup

Relative Error

0.04

Solver Excessive UB-LB Gap !!!

0.01

10

6

10

5

10

4

10

3

639934×

B4

IBM

GEANT

10

GERMANY50

PreTE TeaVaR FFC

38038×

312×

0.00

SIGCOMM’26, Colorado, USA

456× 184×

323×

194×

2

B4

IBM

335× 109×

GEANT

154×

GERMANY50

(b) End-to-end speedup (log-scale).

(a) Relative error vs. Gurobi optimum.

Figure 5: Overall performance summary. (a) Relative error of NeuroRisk versus the Gurobi optimum for each objective (PreTE/TeaVaR/FFC). PreTE on GEANT and GERMANY50 is omitted (N/A) because the solver fails to obtain a non-trivial solution within a 1-hour time budget (unmet ratio remains 1.0). (b) End-to-end speedup of NeuroRisk inference over solver-based baselines, averaged per topology and objective (including model construction time). For GERMANY50, GPU latency is measured with scenario chunking to cap peak memory. GERMANY50 and TATANID, we source optical-layer topologies from the TopologyBench Dataset [37] and derive IPlayer topologies via a rule-based optical-to-IP mapping. Unless otherwise stated, candidate tunnels are generated by 𝐾𝑠𝑝 -shortest paths (KSP) with 𝐾𝑠𝑝 =3 per source-destination pair. Table 2 summarizes the information for each topology. Topology B4 IBM GEANT GERMANY50 TATANID

#Nodes

#Edges

#Tunnels

12 17 22 50 142

55 85 36 88 180

396 816 1,386 7,350 60,066

NeuroRisk is implemented in PyTorch 2.4.1 (CUDA 12.1) with Python 3.8.0; key hyperparameters are in Appendix B.

5.2

Overall Performance

We summarize end-to-end solution quality and speed across topologies and risk objectives, and compare against FauTE, a neural baseline that collapses the scenario set. Takeaway. Figure 5(a) shows that NeuroRisk achieves small optimality gaps across objectives and topologies where solver baselines are tractable, while Figure 5(b) shows orders-ofmagnitude end-to-end speedups that make frequent reoptimization practical. For flow-level objectives (PreTE) on larger topologies (GEANT, GERMANY50), the solver times out (1 hour) with a trivial solution (zero throughput); NeuroRisk produces a non-trivial feasible routing in ∼18–68 ms. Why scenario collapse is insufficient (FauTE). We compare against FauTE [33], a neural baseline that collapses Q into link-local failure probabilities and optimizes a surrogate objective; for a fair comparison, we evaluate both methods on expected throughput over Q and use the LP optimum from Gurobi as ground truth. Table 3 summarizes results on B4 and IBM, revealing substantial performance gaps for the surrogate-based approach.

Table 2: Network topologies in our evaluation.

Workloads, scenarios, and setup. For B4 and IBM, we use traffic matrices from PreTE [38]; for GEANT, GERMANY50, and TATANID, we generate traffic matrices using a standard gravity model [5, 42]. We follow PreTE [38] to generate a probabilistic multi-link scenario set Q with pruning cutoff 𝜏, and parameterize each scenario configuration by a pair (𝑐, 𝑠), where the cutoff is set to 𝜏 ≜ 𝑐 × 10−5 and 𝜆 ≜ 𝑠 × 10−3 is the Weibull scale. We train NeuroRisk on (100, 2), (50, 2), and (50, 4), and reserve all other (𝑐, 𝑠) combinations for testing. We compare against FFC [32], PreTE [38], TeaVaR [12], and FauTE [33]; solver-based baselines are solved using Gurobi 9.5.1 [20] and we report TotalTime. Unless otherwise stated, we set 𝛽=0.95 (quantile for PreTE, CVaR level for TeaVaR). Experiments run on two servers: Gurobi optimization on an Intel® Xeon® Gold 6530 CPU (1.5 TB RAM); DL Models inference on an Intel® Xeon® Platinum 8558 CPU (2.0 TB RAM) with an NVIDIA H200 GPU (141 GB VRAM).

5.3

Runtime Analysis

We evaluate the scalability of NeuroRisk against solver-based methods along two critical dimensions: the number of failure scenarios (𝑁 ) and the topology size. Scaling with Scenarios (𝑁 ). Figure 6 illustrates the impact of scenario count 𝑁 =|Q| on solve time using the IBM topology. Solver-based methods scale roughly linearly or superlinearly with 𝑁 , because each additional scenario introduces a replicated block of constraints (and auxiliary variables) to the underlying LP formulation. Notably, PreTE exhibits 9

SIGCOMM’26, Colorado, USA Topology

Mao et al.

Method

Expected Throughput

Rel. errors

B4

Solver (Gurobi) NeuroRisk (Ours) FauTE [33]

73.21k 72.94k 48.73k

0 0.28% 33.37%

IBM

Solver (Gurobi) NeuroRisk (Ours) FauTE [33]

94.54k 93.40k 27.71k

0 1.23% 70.50%

on GEANT and GERMANY50 due to this multiplicative explosion of the state space. Practical Deployment (Inference). In contrast to the solver’s intractability on large graphs, NeuroRisk remains highly efficient for real-time deployment. On a GPU, it solves the largest instances in under 70 ms. Crucially, Table 4 demonstrates that even on a single CPU thread, NeuroRisk produces feasible solutions for GEANT in just 7 seconds. This confirms that our lightweight model is deployable on commodity control plane hardware without requiring specialized accelerators. Training Overhead. While NeuroRisk requires offline training, this is a manageable one-time cost. Even for GERMANY50, training converges in under 4 hours on a single GPU (see Appendix H). Given that training occurs infrequently (e.g., only upon major topology changes), this offline overhead is negligible compared to the continuous, millisecond-level efficiency gains during online operation.

Solve Time (s)

Table 3: Performance comparison of NeuroRisk against FauTE and the Gurobi optimum on B4 and IBM.

10

4

10

2

10

0

10

−2

10

−4

PreTE - Solver TeaVaR - Solver FFC - Solver

PreTE - Our TeaVaR - Our FFC - Our

5.4

10

20

30

40

Number of Scenarios (N)

Generalization

Generalization across scenario distributions. We first evaluate scenario-distribution generalization by testing on unseen scenario configurations (𝑐, 𝑠) (different cutoff thresholds and Weibull scales) while keeping model weights fixed. Figure 7(a) shows a representative example on IBM. Across objectives, NeuroRisk generalizes well to unseen test configurations (e.g., (1, 2) and (5, 2)), indicating robustness to changes in scenario pruning and failure severity. Robustness to demand perturbations. To test robustness to traffic dynamics, we perturb the input demand by injecting zero-mean Gaussian noise whose standard deviation is proportional to the original demand and then measure the resulting change in relative error. Importantly, the training set and model weights remain fixed: we do not retrain or fine-tune on perturbed data. The perturbation is applied only to the test instances at inference time, evaluating out-of-distribution generalization under demand shifts. Concretely,  for each
2 demand 𝐷 𝑓 , we sample a noise term 𝜂 𝑓 ∼ N 0, 𝜎𝐷 𝑓

50

Figure 6: Solve time vs. number of scenarios (𝑁 ) on IBM. Solver-based methods (dashed lines) degrade significantly as 𝑁 grows, with PreTE spiking to 103 s due to decomposition overhead. In contrast, NeuroRisk (solid lines) maintains a flat, millisecond-level inference time.

non-smooth runtime “jumps” (rising to 102 –103 s), suggesting that the solver frequently enters harder combinatorial regimes (e.g., heavy branching or decomposition steps) once the formulation crosses certain complexity thresholds. In contrast, NeuroRisk maintains a flat scaling profile, effectively decoupling inference latency from 𝑁 by parallelizing scenario processing within the neural architecture. Scaling with Topology. Importantly, the computational burden is not governed by 𝑁 alone, but by the scenarioexpanded problem size, which scales with the product of 𝑁 and the routing decision space (number of tunnels/edges). As shown in Table 2, the number of tunnels grows rapidly from IBM to GEANT and GERMANY50. Consequently, on these larger topologies, each additional scenario replicates constraints over thousands of tunnels, quickly pushing the solver beyond practical memory and iteration limits. This explains the sharp transition observed in Table 4: while the solver handles IBM (∼200 s), it consistently times out (>1 h)

Topology B4 IBM GEANT GERMANY50

Solver (PreTE)

NeuroRisk (Ours)

(Time / Status)

GPU CPU (1-thread)

53.0 s 7 ms 196.4 s 11 ms Timeout (>1h) 18 ms Timeout (>1h) 68 ms

151 ms 660 ms 7.0 s 177 s

Table 4: Inference latency comparison. For large topologies where the solver times out due to scenarioexpanded constraint explosion, NeuroRisk remains feasible even on a single CPU thread. (NeuroRisk includes model construction time; 𝐾=7 unroll iterations). 10

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering FFC

TeaVaR

Train Set

Relative Error

Relative Error

PreTE 0.03

0.02

0.01

0.04

SIGCOMM’26, Colorado, USA

Noise Level 5% 15% 30%

0.02

0.00 (1, 2)

(5, 2)

(10, 2)

Scenario

(50, 2)

(1, 2)

(100, 2)

PreTE

FFC

Relative ErrorError Relative

Relative ErrorError Relative

Train Set (a) Scenario distribution shift.

0.03 0.025

TeaVaR

0.020 0.02 0.015 0.01 0.010 0.005

(5, 4)

(1, 2)

(5, 2)

PreTE

(10, 4)

(50, 4)

(100, 4)

(10, 2)

(50, 2)

(100, 2)

Scenario Scenario

(10, 2)

(50, 2)

Scenario

(100, 2)

(b) Demand perturbation.

0.02 0.04

0.00 0.02

(1, 4)

(5, 4)

B4

IBM

(10, 4)

(50, 4)

(100, 4)

GEANT

GERMANY50

TATANID

Scenario

Test topology

(d) Cross-topology (TeaVaR).

(c) Tunnel-set shift. FFC TeaVaR

Relative Error

0.025

(1, 4)

0.04 0.06

Noise Level 5% 15% 30%

(5, 2)

Figure 0.020 7: Generalization summary. NeuroRisk generalizes across multiple distribution shifts without retraining: (a) unseen scenario configurations (𝑐, 𝑠) on IBM, (b) demand-proportional perturbations on B4 (TeaVaR), (c) changes 0.015 to candidate tunnel sets on B4, and (d) zero-shot transfer to unseen topologies for the TeaVaR objective. For 0.010 readability, subfigures (a)–(d) show cropped views; Appendix E provides the full, uncropped plots and additional variants across objectives and topologies. 0.005  (1, 4) (5, 4) (10, 4)′ (50, 4) (100, 4) and form the perturbed demand 𝐷 𝑓 = max 0, 𝐷 𝑓 + 𝜂𝑓 , Scenario where the truncation enforces the physical constraint that demands are nonnegative. We evaluate three noise levels 𝜎 ∈ {5%, 15%, 30%}, corresponding to standard deviations of 0.05𝐷 𝑓 , 0.15𝐷 𝑓 , and 0.30𝐷 𝑓 , respectively. Figure 7(b) shows the effect of demand perturbations on B4 with the TeaVaR objective. Even under 30% demand noise, NeuroRisk maintains relative errors below 5% in most configurations. The degradation is graceful: the relative error increases approximately linearly with the noise level, indicating stable gradient propagation through the Gated Reservation mechanism. This robustness is crucial for practical deployment, where runtime traffic matrices inevitably deviate from the historical distributions used during training. Generalization across tunnel configurations. NeuroRisk is trained with KSP candidate sets of size 𝐾𝑠𝑝 =3 per sourcedestination pair. To assess robustness to changes in the action space, we perform a cross-𝐾𝑠𝑝 evaluation: we keep the trained model fixed and test on instances whose candidate tunnel sets are generated with 𝐾𝑠𝑝 =2. Figure 7(c) shows that NeuroRisk remains accurate under this tunnel set shift. This robustness follows from the edge-centric reservation parameterization: since each edge’s Softmax operates over whichever tunnels

traverse it, adding or removing tunnels simply changes the Softmax dimension without requiring architectural modifications. This flexibility is valuable for operators who may dynamically adjust routing options according to conditions. Cross-topology generalization. We next evaluate whether a single TeaVaR model can generalize across unseen network topologies. We train one model on a mixture of IBM and GEANT instances (with unified padding and masking to support multi-topology batching), select the checkpoint using validation instances from the same two training topologies, and then test zero-shot on held-out instances from IBM/GEANT as well as unseen B4/GERMANY50/TATANID. Throughout, we treat B4, GERMANY50, and TATANID as pure target domains: we perform no retraining or fine-tuning on these topologies. This isolates cross-topology transfer and avoids per-topology training overhead, which can be expensive on large graphs (e.g., TATANID). Figure 7(d) summarizes the relative error compared to the optimum in each test topology. The gap in unseen B4 is minimal, and even on the much larger TATANID, relative error remains low (< 8%), confirming robust transfer across diverse scales. Generalization across iteration counts. Because shared weights are reused at every unroll step, a model trained at 11

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Mao et al.

𝐾=7 can be deployed at any 𝐾1 ≤ 𝐾 without retraining, allowing operators to trade solution quality for latency at runtime. On B4 (TeaVaR), relative error degrades gracefully from 1.35% (𝐾1 =7) to 7.72% (𝐾1 =1), with latency scaling linearly in 𝐾1 (Appendix F).

that GR achieves 97% of optimal throughput on GEANT and 96% on B4, whereas the widely used GS method yields only 13% on GEANT and 17% on B4. These results underscore that GR’s feasibility-by-construction design is effective for general TE, not only for risk-aware objectives.

5.5

6

Ablation and Module Analysis

Feasibility enforcement comparison. We conduct an ablation study to evaluate the effectiveness of our Gated Reservation (GR) mechanism compared to other feasibility enforcement methods. Specifically, we compare: (1) Global Scaling (GS): Post-hoc scaling by a network-wide factor 𝛾 max (§3). (2) Local Scaling (LS): Post-hoc scaling by per-tunnel factors 𝛾𝑡 (§4.2). (3) Bandwidth Reservation (BR): Per-edge Softmax without tunnel-level gating (Eq. (10)). (4) Gated Reservation (GR): Our full mechanism with tunnel-level gating (Eq. (12)). Figure 8 presents the results on B4. GS and LS exhibit signifi-

Relative Error

0.005 0.004 0.003

GR BR LS GS

0.25

0.010

0.20

0.008

0.15

0.006

0.002

0.10

0.004

0.001

0.05

0.002

0.000

0.00

PreTE

TeaVaR

0.000

DISCUSSION

Deployment flexibility. As shown in §5.4, operators can reduce the number of unroll iterations at deployment time to trade solution quality for latency without retraining. Only training requires a GPU; inference runs efficiently on CPU (§5.3), enabling deployment on SDN controllers or edge servers without dedicated accelerators. For unseen topologies, an existing model can serve as a warm-start (§5.4); once a topology-specific model is trained, it can be swapped in. Scalability to large scenario sets. The Sort-and-Select formulation decouples scenarios: loss and gradient computation proceed independently per scenario, then aggregate via weighted summation (§4.3). During training, we leverage gradient checkpointing to avoid storing intermediate computation graphs across scenarios; this makes memory consumption independent of |Q|, so increasing the scenario count only affects training time. When exhaustive enumeration |𝐸 |
ˆ is infeasible (e.g., 𝑘-failure models with 𝑂 ( 𝑘ˆ ) scenarios), practitioners can prune to high-probability or high-impact scenarios without altering the algorithm.

FFC

Figure 8: Ablation study on B4. Comparison of feasibility enforcement methods (GS, LS, BR, GR) across different objectives. GR consistently achieves the lowest relative error by combining construction-based feasibility with tunnel-level coordination.

7

RELATED WORK

TE Optimization and Demand Robustness. Centralized TE optimizes utilization via global programs (e.g., SWAN [21] and B4 [22]), while scalability motivates decomposition methods such as NCFlow [1] and SSDO [36]. Recent learningbased systems further accelerate TE optimization and improve generalization (e.g., Teal [46], HARP [4], RedTE [19]). Robustness to demand uncertainty has been studied via worst-case formulations (e.g., Oblivious Routing [7] and COPE [45]) and semi-oblivious routing (e.g., SMORE/SemiOblivious [30]); complementary approaches incorporate prediction errors or burstiness more explicitly (e.g., Figret [34]). Failure Robustness and Risk-Aware TE. Proactive methods plan across failure scenarios to enforce availability or tailrisk objectives (e.g., FFC [32], PCF [25], TeaVaR [12]), while complementary systems emphasize fast reaction and crosslayer restoration (e.g., Flexile [24], Arrow [48], PreTE [38]). FauTE [33] applies deep learning with a probability-weighted penalty; in contrast, NeuroRisk enforces feasibility by construction and optimizes scenario-aware risk objectives, avoiding scaling/projection pitfalls. Moreover, NeuroRisk naturally extends to demand robustness: by treating distinct traffic matrices as scenarios in Q, the same architecture can optimize over demand uncertainty without modification.

cantly higher relative errors due to the optimization fragility and projection discontinuities discussed in §3. BR (Eq. (10)) eliminates scaling pitfalls but suffers from the Short-Board Trap (§4.2.1), as it lacks a mechanism for tunnel-level coordination. The GR mechanism (Eq. (12)) achieves near-optimal performance by restoring tunnel-level coordination within a smooth, feasibility-by-construction framework. The results confirm that both construction-based feasibility and tunnellevel gating are essential. Route diversity analysis. As hypothesized in §4.2, the LS approach exhibits a systemic bias toward short tunnels. We validate this by measuring the direct ratio—the fraction of carried traffic allocated to 1-hop tunnels—on B4. As shown in Table 9 (Appendix G), LS yields a direct ratio of 0.989, compared to 0.883 for GR, confirming that LS collapses route diversity toward direct tunnels. Nominal (no-failure) comparison. To demonstrate that GR’s benefits extend beyond failure scenarios, we evaluate throughput under the nominal (failure-free) setting. The detailed results, summarized in Table 10 (Appendix G), show 12

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering

8

CONCLUSION

SIGCOMM’26, Colorado, USA

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Risk-aware traffic engineering must jointly handle expressive objectives beyond MLU, strict capacity feasibility, and a probabilistic, variable-sized failure scenario set as a runtime input. We propose NeuroRisk, a physics-informed neural optimizer that achieves feasibility by construction via Gated Reservation (edge-local reservations with tunnel-level coordination) and supports general Sort-and-Select risk objectives using a physics-aware update network that is invariant to scenario ordering. Across WAN topologies and risk formulations, NeuroRisk matches solver solution quality with orders-of-magnitude speedups, and it generalizes to demand shifts and candidate tunnel changes without retraining; future work will extend evaluation to additional topologies and traffic sources and further integrate online forecast updates. Our implementation will be made available on GitHub to support further research in AI-driven network optimization. This work does not raise any ethical concerns.

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APPENDIX A FULL NOTATION To facilitate a deeper understanding of the NeuroRisk framework, we provide a comprehensive summary of the notation used throughout this paper in Table 5. Our notation is organized into four logical groups: (i) Network Topology and Traffic: Variables such as G, 𝐶𝑒 , and 𝐷 𝑓 describe the underlying physical graph and the input traffic demands. (ii) Routing and Allocation Decisions: 𝑥 𝑓 ,𝑡 and 𝑦𝑡,𝑒 represent the target traffic fractions and link-level reservation ratios, respectively, which are the primary decision variables optimized by our framework. (iii) Failure Scenarios and Risk Metrics: To characterize network reliability, we use 𝑞 and 𝑝𝑞 to denote probabilistic failure scenarios, and ℓ𝑓 ,𝑞 to capture the resulting loss ratio. The variables 𝑣 𝑓 ,𝑟 and 𝜋 𝑓 ,𝑟 are specifically defined for the ranking-based risk assessment (e.g., CVaR computation). (iv) Neural Optimization Components: Φphy and Ψ𝜃 denote the core functional blocks of our unrolled optimization architecture, responsible for physicsinformed feature extraction and iterative state updates. Symbol

Meaning

G(V, E) 𝑓 ∈F T𝑓 T𝑒 𝑞 ∈ Q, 𝑝𝑞 𝐶𝑒 𝐷𝑓 , D 𝑥 𝑓 ,𝑡 , x

Network graph with nodes V and links E. Traffic demand (flow). Candidate tunnels for flow 𝑓 . The set of tunnels traversing edge 𝑒. Failure scenario and its probability. Capacity of link 𝑒. Demand of flow 𝑓 and the demand vector. Target fraction of 𝐷 𝑓 to be routed on tunnel 𝑡 (allocation decision). Indicator that tunnel 𝑡 survives in scenario 𝑞. (1 if tunnel 𝑡 survives in 𝑞, 0 otherwise) Realized loss ratio of flow 𝑓 under scenario 𝑞. 𝑟 -th largest loss for flow 𝑓 after sorting scenarios. Probability mass of the scenario ranked 𝑟 for flow 𝑓 . Í Cumulative probability up to rank 𝑟 : 𝑟𝑗=1 𝜋 𝑓 ,𝑗 . Binary selection mask encoding the risk preference. Scenario-level selection mask (e.g., 1 for tail scenarios in CVaR, 0 otherwise). Scenario risk weight: 𝜌𝑞 = 𝑝𝑞 · 𝑚𝑞 . Indicator that scenario 𝑞 is ranked 𝑟 for flow 𝑓 . Latent contention logit and resulting reservation ratio on link 𝑒 for tunnel 𝑡. Tunnel-wide priority score shared across all edges of 𝑡. Physics feature extractor and learned update network.

𝛼𝑡,𝑞 ℓ𝑓 ,𝑞 𝑣 𝑓 ,𝑟 𝜋 𝑓 ,𝑟 Γ𝑓 ,𝑟 𝐼 𝑓 ,𝑟 𝑚𝑞 𝜌𝑞 𝛿 𝑓 ,𝑞,𝑟 𝑧𝑡,𝑒 , 𝑦𝑡,𝑒 𝑤𝑡 Φphy , Ψ𝜃

Table 5: Notation used throughout the paper.

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B

Mao et al. Physics term 𝜕𝑥 𝑓 ,𝑡 /𝜕𝑦𝑡,𝑒 . We have 𝑥 𝑓 ,𝑡 = 𝑏𝑡 /𝐷 𝑓 (𝑡 ) with 𝑏𝑡 = min𝑒 ′ ∈𝑡 𝐶𝑒 ′ 𝑦𝑡,𝑒 ′ . So 𝑥 𝑓 ,𝑡 is determined by the bottleneck edge along tunnel 𝑡. The derivative 𝜕𝑥 𝑓 ,𝑡 /𝜕𝑦𝑡,𝑒 is nonzero only when edge 𝑒 is the bottleneck for 𝑡 (i.e., 𝐶𝑒 𝑦𝑡,𝑒 = 𝑏𝑡 ). It depends on: (i) the current satisfaction 𝑥𝑡 = 𝑏𝑡 /𝐷 𝑓 (𝑡 ) ; (ii) the bottleneck margin 𝑚𝑡,𝑒 = 𝐶𝑒 𝑦𝑡,𝑒 − 𝑏𝑡 (zero at the bottleneck, positive elsewhere); (iii) capacity 𝐶𝑒 and demand 𝐷 𝑓 (𝑡 ) (scale the derivative). Hence the Physics features: 𝑥𝑡 , 𝑚𝑡,𝑒 , 𝐶𝑒 , 𝐷 𝑓 (𝑡 ) . Saturation term 𝜕𝑦𝑡,𝑒 /𝜕𝑧𝑡,𝑒 . Under the gated Softmax 𝑦𝑡,𝑒 = Í exp(𝑧𝑡,𝑒 + 𝑤𝑡 )/ 𝑘 ∈ T𝑒 exp(𝑧𝑘,𝑒 + 𝑤𝑘 ), we have 𝜕𝑦𝑡,𝑒 /𝜕𝑧𝑡,𝑒 = 𝑦𝑡,𝑒 (1 − 𝑦𝑡,𝑒 ). So the sensitivity of the reservation to the logit is determined by the current 𝑦𝑡,𝑒 . Hence the Saturation feature: 𝑦𝑡,𝑒 . Summary. The chain rule thus identifies exactly the eight quantities in Table 1: three Risk, four Physics, and one Saturation. Feeding these as s𝑡,𝑒,𝑞 allows Ψ𝜃 to approximate the effect of an update on the risk objective without computing gradients explicitly.

HYPERPARAMETER SETTINGS

Table 6 summarizes the representative hyperparameter configuration for our model. These settings were determined via a systematic grid search covering learning rate 𝜂 ∈ {10−4, 5×10−4, 10−3 }, hidden layer dimensions ∈ {32, 64, 128}, and the number of unrolled iterations 𝐾. The selected 𝐾 = 7 strikes an optimal balance between the quality of the optimization solution and inference latency. The update network Ψ𝜃 is implemented as a 2-layer MLP to ensure minimal per-iteration computational overhead. We employ the Adam optimizer with an early stopping patience of 10 epochs, which provides sufficient training stability across various network scales. To manage the memory footprint during the processing of a large number of failure scenarios |Q|, our framework includes an optional scenario chunking mechanism. This feature is disabled by default to maximize throughput on standard benchmarks. For large-scale topologies, we enable chunking with a size ranging from 1 to 20, depending on the available GPU VRAM and the specific complexity of the topology. This mechanism ensures that the training and inference remain feasible on a single commercial GPU without altering the gradient integrity. Hyperparameter

Value

Architecture Update MLP Hidden dimension Input dimension Unroll iterations 𝐾 Training epochs Early stopping patience Batch size Learning rate Scenario chunking

Unrolled MLP-based Optimization 2-layer MLP (ReLU) 64 8 7 30 10 16 10−3 Disabled by default (1–20 for large)

D

SCENARIO SET STATISTICS Topology

𝑠

𝑐=1 𝑐=5 𝑐=10 𝑐=50 𝑐=100

B4

2 4

33 34

22 34

19 22

10 16

10 11

GEANT

2 4

80 40 239 108

33 46

23 27

18 23

GERMANY50

2 4

242 107 671 294

76 134

58 68

39 53

IBM

2 4

37 52

26 45

23 28

14 17

11 13

TATANID

2 871 227 4 2787 551

159 236

107 127

72 100

Table 6: Key hyperparameters. Table 7: Scenario counts after pruning (sorted by 𝑐). Number of retained scenarios |Q| for each topology.

C

DERIVATION OF PHYSICS ENGINE FEATURES

Table 7 indicates that |Q| is driven by (i) topology scale, which increases the number of correlated multi-link failure combinations, and (ii) scenario pruning/severity (𝑐, 𝑠), which controls how much tail mass survives. Across all topologies, smaller 𝑐 (weaker cutoff) monotonically increases |Q| by admitting more low-probability combinations; larger 𝑠 (heavier tail) further increases |Q| by pushing more multilink events above the cutoff. This scaling causally explains solver runtime jumps: scenario-enumerating baselines expand in constraints/subproblems with |Q|, so larger graphs or heavier-tail (𝑐, 𝑠) push the solver into harder decomposition/branching regimes.

This appendix derives why the eight features in Table 1 suffice for the update network Ψ𝜃 to produce effective increments (ΔZ, ΔW). The argument follows the chain rule for 𝜕J𝑞 /𝜕𝑧𝑡,𝑒 and 𝜕J𝑞 /𝜕𝑤 𝑡 (Eq. (14) in §4.3). Risk term 𝜕J𝑞 /𝜕𝑥 𝑓 ,𝑡 . The scenario objective J𝑞 is a function of per-flow losses ℓ𝑓 ,𝑞 . For
flow 𝑓 and tunnel 𝑡 ∈ T𝑓 , Í ℓ𝑓 ,𝑞 = max 0, 1 − 𝑡 ′ 𝑥 𝑓 ,𝑡 ′ 𝛼𝑡 ′ ,𝑞 , so 𝜕J𝑞 /𝜕𝑥 𝑓 ,𝑡 depends on: (i) whether tunnel 𝑡 survives in 𝑞, i.e. 𝛼𝑡,𝑞 ; (ii) the flow’s loss under 𝑞, i.e. ℓ𝑓 ,𝑞 ; and (iii) the scenario-level loss ℓ𝑞 (e.g., for CVaR-style masks). Hence the Risk features: 𝛼𝑡,𝑞 , ℓ𝑓 ,𝑞 , ℓ𝑞 . 16

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering

GENERALIZATION RESULTS ON ADDITIONAL TOPOLOGIES AND OBJECTIVES

PreTE

Relative Error

E

This appendix complements §5.4 by providing the full, uncropped versions of the plots that are summarized in Figure 7. In the main paper, we crop subfigures to improve readability under tight space constraints and to keep a consistent visual scale across panels; here we report the complete figures and additional variants across objectives and topologies for completeness and reproducibility. Figures 9–11 report scenarioPreTE

FFC

TeaVaR

FFC

TeaVaR

Train Set

0.03

0.02

0.01

(1, 2)

(5, 2)

(10, 2)

Scenario

(50, 2)

(100, 2)

Train Set

Relative Error

Relative Error

0.025

SIGCOMM’26, Colorado, USA

Train Set

0.020 0.015

0.03

0.02

0.01

0.010 (1, 4)

(5, 4)

0.005

(10, 4)

Scenario

(50, 4)

(100, 4)

0.000 (1, 2)

(5, 2)

(10, 2)

Scenario

(100, 2)

Figure 10: Scenario-distribution generalization on IBM. Same as above.

Train Set

0.020

0.04

0.015

Relative Error

Relative Error

0.025

(50, 2)

0.010 0.005 0.000 (1, 4)

(5, 4)

(10, 4)

Scenario

(50, 4)

(100, 4)

FFC

TeaVaR

Train Set

0.03

0.02

0.01 (1, 2)

Figure 9: Scenario-distribution generalization on B4. Relative error against solver baselines (FFC/PreTE/TeaVaR) across scenario configurations. Bars marked with yellow background indicate training configurations.

(5, 2)

(10, 2)

Scenario

Relative Error

0.04

configuration generalization on B4, IBM, and GEANT, respectively. Each plot evaluates fixed model weights on unseen scenario configurations (𝑐, 𝑠) (cutoff thresholds and Weibull scales), and compares the relative error against solver baselines (FFC/PreTE/TeaVaR). Bars marked with yellow background indicate the configurations used during training, while the remaining bars correspond to held-out test configurations. These full plots are the uncropped counterparts of the view shown in Figure 7(a). Figures 12–14 provide the complete demand-perturbation results on B4 for TeaVaR, FFC, and PreTE objectives, respectively. Across objectives, we apply the same perturbation protocol described in §5.4: we inject zero-mean Gaussian noise with standard deviation proportional to each demand entry and clip negative demands to zero, while keeping the training set and model

(50, 2)

(100, 2)

Train Set

0.03

0.02

0.01 (1, 4)

(5, 4)

(10, 4)

Scenario

(50, 4)

(100, 4)

Figure 11: Scenario-distribution generalization on GEANT. Same as above. checkpoint fixed. These figures extend the representative TeaVaR-only view shown in Figure 7(b) and illustrate that the robustness trend is consistent across different risk preferences. Figure 15 reports the full version of the tunnel-set shift experiment corresponding to Figure 7(c). It evaluates the same trained model under altered candidate tunnel sets (generated with a different KSP parameter), demonstrating 17

0.04

Mao et al.

Noise Level 5% 15% 30%

0.010

Relative Error

Relative Error

SIGCOMM’26, Colorado, USA

0.02

0.00 (10, 2)

Scenario

(50, 2)

Noise Level 5% 15% 30%

0.010 0.008 0.006

(5, 2)

(5, 4)

(10, 4)

Scenario

(50, 4)

(50, 2)

(100, 2)

(10, 4)

(50, 4)

(100, 4)

Noise Level 5% 15% 30%

0.004 0.002

(100, 4)

(1, 4)

Figure 12: Demand-perturbation robustness on B4 (TeaVaR). Relative error under demand-proportional Gaussian noise with 𝜎 ∈ {5%, 15%, 30%}; demands are clipped to be nonnegative.

(5, 4)

Scenario

Figure 14: Demand-perturbation robustness on B4 (PreTE). Same as above.

Relative Error

0.025

Noise Level 5% 15% 30%

PreTE

FFC

TeaVaR

0.020 0.015 0.010 0.005

0.006 (1, 2)

(5, 2)

(10, 2)

Scenario

0.004

0.010 0.008

(5, 2)

(10, 2)

Scenario

(50, 2)

0.025

(100, 2)

Relative Error

(1, 2)

Relative Error

(10, 2)

Scenario

0.000 (1, 4)

Relative Error

0.002

(1, 2)

0.00

0.008

0.004

(100, 2)

Relative Error

Relative Error

(5, 2)

0.02

0.010

0.006

0.000 (1, 2)

0.04

0.008

Noise Level 5% 15% 30%

Noise Level 5% 15% 30%

PreTE

FFC

(50, 2)

(100, 2)

(50, 4)

(100, 4)

TeaVaR

0.020 0.015 0.010 0.005

0.006 (1, 4) 0.004 (1, 4)

(5, 4)

(10, 4)

Scenario

(50, 4)

(5, 4)

(10, 4)

Scenario

(100, 4)

Figure 15: Tunnel-set generalization on B4. Relative error under candidate tunnel-set shifts (cross-𝐾𝑠𝑝 KSP), evaluated with fixed model weights.

Figure 13: Demand-perturbation robustness on B4 (FFC). Same as above.

F that the BR reservation parameterization remains compatible with changes in the action space. Since each edge performs a Softmax over the set of tunnels that traverse it, adding or removing tunnels only changes the local Softmax support without requiring architectural modifications.

ITERATION COUNT GENERALIZATION

Table 8 provides the full iteration-count generalization results referenced in §5.4. We deploy a model trained at 𝐾=7 unroll iterations with varying 𝐾1 ∈ {1, 2, 3, 5, 7} on B4 (TeaVaR objective) without retraining. 18

NeuroRisk: Physics-Informed Neural Optimization for Risk-Aware Traffic Engineering 𝐾1

1

2

3

5

7

Rel. Error (%)

7.72

4.53

3.18

1.74

1.35

GEANT

Table 8: Iteration count generalization on B4 (TeaVaR). A model trained at 𝐾=7 is deployed at 𝐾1 ≤ 𝐾 without retraining. Relative error degrades gracefully as 𝐾1 decreases, enabling operators to trade quality for speed at runtime.

G

This appendix provides supplementary diagnostics for the ablation study in §5.5, focusing on two aspects: (1) route structure under risk-aware objectives, and (2) performance in the nominal (no-failure) setting. Route structure under TeaVaR objective. Table 9 reports the fraction of carried traffic allocated to direct (1-hop) tunnels versus multi-hop tunnels on B4. LS exhibits the highest direct ratio (0.989), confirming that LS induces a strong short-tunnel bias: nearly all traffic is routed via direct tunnels, leaving multi-hop capacity underutilized. In contrast, GR achieves a lower direct ratio (0.883), indicating better exploitation of tunnel diversity.

GR BR LS GS

Input dim

Direct

Multi-hop

8 8 5 5

0.883 0.924 0.989 0.903

0.117 0.076 0.011 0.097

B4

Method

Direct

Thr.

Direct

Thr.

LS GR (Ours) GS BR

0.31 0.42 0.11 0.29

0.64 0.97 0.13 0.47

0.40 0.40 0.12 0.31

0.88 0.96 0.17 0.70

Table 10: Nominal maxflow and direct ratio (mean). Direct ratio = fraction of carried traffic allocated to direct (1-hop) tunnels; norm. throughput = delivered vs. demand (no-failure).

ADDITIONAL ABLATION DIAGNOSTICS

Variant

SIGCOMM’26, Colorado, USA

H

TRAINING COST

Table 11 reports the wall-clock training time for each topology on a single NVIDIA H200 GPU. Small topologies (B4, IBM) converge in under 10 minutes; GEANT finishes within 15 minutes; the largest topology GERMANY50 (with scenario chunking at chunk size 20 and gradient checkpointing both enabled) converges in under 4 hours. Figure 16 shows the convergence curve for this worst-case configuration: the relative error drops from ∼0.16 to ∼0.04 in the first 20 minutes, with steady refinement thereafter; the training objective and validation error track closely, indicating no overfitting. Topology B4 IBM GEANT GERMANY50

Training time < 10 min < 10 min ∼ 15 min ∼4h

Note

chunk = 20, checkpointing

Table 11: One-time training cost per topology on a single NVIDIA H200 GPU. Training is offline and amortized over all subsequent demand-scenario instances.

Table 9: Route structure in TeaVaR ablation on B4 (mean). Direct ratio = fraction of carried traffic allocated to direct (1-hop) tunnels; multi-hop ratio = 1 − direct ratio. Nominal (no-failure) throughput. Table 10 evaluates the same methods on a standard maxflow objective without failure scenarios. GR achieves the highest normalized throughput on both topologies (0.97 on GEANT, 0.96 on B4), demonstrating that its feasibility-by-construction design is effective for general TE, not only for risk-aware objectives. GS performs poorly (0.13 on GEANT, 0.17 on B4) because global scaling over-penalizes all tunnels when any single edge is congested (§3). LS and BR achieve intermediate throughput, with LS outperforming BR on B4 and GEANT. Notably, GR and LS have similar direct ratios in the nominal setting (0.40– 0.42 on B4), yet GR achieves higher throughput, suggesting that GR’s advantage stems from better allocation within tunnels rather than tunnel selection alone. 19

SIGCOMM’26, Colorado, USA

Mao et al.

Validation Relative Error

0.16 0.14 0.12 0.10 0.08 0.06 0.04 0.02

0.0

0.5

1.0

1.5

2.0

2.5

Training Time (Hours)

3.0

3.5

Figure 16: Training convergence on GERMANY50. Validation relative error (blue, left axis) and training objective (red, right axis) versus wall-clock time. Scenario chunking (chunk size 20) and gradient checkpointing are both enabled. Training converges within ∼4 hours on a single GPU.

20