Thermodynamics-Informed Input Reparameterization for Neural Prediction of Real-Fluid Thermodynamic Properties in Supercritical Combustion Haoze Zhang,1 Han Li,1, 2, a) Ke Xiao,2 Yangchen Xu,1 Runze Mao,1 and Zhi X. Chen1, 2 1) State Key Laboratory of Turbulence and Complex Systems, College of Engineering, Peking University, Beijing 100871,
China 2) AI for Science Institute (AISI), Beijing 100080, China
arXiv:2607.19241v1 [cs.LG] 21 Jul 2026
(Dated: 22 July 2026)
Real-fluid thermodynamic property evaluation is a major computational cost in supercritical combustion simulations. In the enthalpy-based pressure-correction formulation considered here, the cell-local closure evaluates temperature T , density ρ , and the compressibility coefficient ψ from the solver-available state (h, p, Y) through enthalpy–temperature inversion and repeated real-fluid equation-of-state evaluations. Neural-network surrogates offer fixed-cost inference, but a direct mapping from (h, p, Y) to (T, ρ , ψ ) must represent both the caloric relation between enthalpy and temperature and the non-ideal equation-of-state response, resulting in a complex regression problem. This work introduces a thermodynamics-informed input reparameterization strategy, termed target-aligned input reparameterization (TAIR). TAIR replaces the raw enthalpy coordinate of each property network with a target-matched thermodynamic coordinate: the temperature network uses a temperature estimate obtained by inverting a constant-c p ideal-gas mixture enthalpy approximation, whereas the density and compressibility networks use an ideal-gas density estimate. These explicit algebraic transformations use only solver-available variables and species constants, guiding the networks to learn realfluid departures from ideal-gas-based baselines rather than reconstructing the full closure from raw enthalpy. The method is assessed using supercritical methane–oxygen counterflow flame data against a raw-input baseline and targetinconsistent cross-reparameterization controls. TAIR reduces held-out test-set root-mean-squared error (RMSE) by factors of approximately 1.5, 2.0, and 7.5 for T , ρ , and ψ , respectively. For an unseen-strain-rate flame within the augmented thermodynamic envelope, the corresponding factors are 3.6, 14.5, and 6.0. The target-inconsistent controls perform markedly worse, indicating that the gains arise from thermodynamically matched input design rather than generic preprocessing. Keywords: Supercritical combustion, Real-fluid thermodynamics, Thermodynamic closure, Neural-network surrogate, Input reparameterization I.
INTRODUCTION
Liquid-propellant rocket engines and related high-pressure propulsion systems commonly operate at pressures above the critical pressure of one or more propellants; consequently, propellant injection, mixing, and combustion are strongly influenced by real-fluid thermodynamics1–4 . Under these conditions, dense-fluid interactions cause pronounced departures from ideal-gas behavior, including strong nonlinearities in density and thermodynamic derivatives near critical or pseudo-critical states5–7 . In reacting-flow simulations, these effects are accounted for through the thermodynamic closure. For the enthalpy-based pressure-correction formulation considered here, the local closure evaluates temperature T , density ρ , and the compressibility coefficient ψ from the solveravailable state (h, p, Y), where h is the specific enthalpy and Y is the species mass-fraction vector. Temperature is required for chemical-kinetic and temperature-dependent property evaluations, density enters the conservation equations, and ψ enters the pressure-correction equation. Accurate and robust evaluation of this closure is therefore essential for supercritical combustion simulations. The computational expense of this closure arises from repeated real-fluid state evaluations. Given a local state
a) Electronic mail: [email protected]
(h, p, Y), temperature is recovered by solving the nonlinear relation h = h(T, p, Y) typically through Newton-type iteration8 . Each temperature iteration requires an equationof-state (EoS) evaluation; for a cubic EoS such as Peng– Robinson9 , this includes solving the corresponding cubic relation for the molar volume, from which density follows10 . In the reference closure considered here, ψ is evaluated by an isenthalpic finite-difference pressure perturbation, which requires an additional enthalpy inversion and EoS evaluation at the perturbed pressure. Because these operations are repeated in every computational cell and at every time step, real-fluid thermodynamic closure can constitute a major computational cost in supercritical combustion simulations11,12 . Several approaches have been developed to reduce this cost. Tabulation methods precompute thermodynamic states and replace online EoS evaluations with interpolation13,14 . Their storage requirement and interpolation complexity, however, increase rapidly with the dimensionality and resolution of the tabulated state space. Near critical or pseudo-critical regions, steep real-fluid property gradients additionally require dense local resolution to maintain accuracy15,16 . Correlated dynamic evaluation reduces repeated property calculations by reusing evaluations within dynamically constructed groups of thermodynamically similar states12,17 , but its efficiency and accuracy depend on the quality of the state-space partitioning. These difficulties become increasingly important for multicomponent reacting mixtures, motivating surrogate models
2 that provide rapid local property evaluation without explicitly constructing a high-dimensional table18 . Machine-learning models have consequently been explored for accelerating thermodynamic calculations in both nonreacting and reacting real-fluid systems19 . In broader thermodynamic applications, neural networks and related regressors have been used to emulate equations of state and iterative flash calculations for complex fluids20,21 . For real-fluid flow simulations, Milan et al.22 developed deep-learning surrogates for property evaluation in complex reacting flowfields, demonstrating the feasibility of replacing repeated thermodynamic calculations with neural inference. Sahranavardfard et al.23 subsequently implemented a neural real-fluid property model in OpenFOAM for transcritical LOX/GH2 mixing, addressing practical issues including training-domain selection, output transformation, and solver integration. More recently, Cai et al.24 coupled a real-fluid property network with a chemical-ODE network to accelerate both thermodynamicproperty evaluation and chemistry integration in supercritical combustion. Collectively, these studies establish neural surrogates as a promising route for accelerating real-fluid reactingflow calculations. The available studies have primarily improved surrogate performance through network architecture, training-data coverage, output transformation, loss formulation, and solver coupling. By comparison, the thermodynamic input coordinates themselves have received relatively little systematic investigation, although they determine the regression map presented to the network. In an enthalpy-based formulation, a direct neural closure naturally inherits the solver interface, (h, p, Y) 7→ (T, ρ , ψ ). This input contains sufficient information to determine the target properties and is convenient for CFD coupling. However, the network must represent both the caloric relation between enthalpy and temperature and the non-ideal EoS response within the same learned mapping. This combination may increase the complexity of the regression problem, particularly for density and compressibility in thermodynamically sensitive high-pressure states25 . It therefore motivates the hypothesis examined in this work: a lowcost thermodynamic transformation of the input coordinates may improve the learnability and accuracy of neural real-fluid closures. To examine this hypothesis, we introduce a thermodynamics-informed input-coordinate design strategy, termed target-aligned input reparameterization (TAIR), for neural prediction of real-fluid thermodynamic properties. Instead of providing every property network with the same raw enthalpy coordinate, TAIR replaces h with a targetmatched thermodynamic input. The temperature network uses an estimated temperature T̃ obtained by inverting an ideal-gas mixture enthalpy approximation, whereas the density and compressibility networks use an ideal-gas-based density estimate ρ̃ = p/(Rm T̃ ). These quantities serve only as input coordinates; the network outputs remain the real-fluid properties T , ρ , and ψ . The transformed coordinates incorporate the leading ideal-gas dependence of the corresponding targets, leaving the networks to represent the remaining real-fluid departures from the ideal-gas baseline rather than
reconstructing the complete closure directly from the raw enthalpy coordinate. The role of input-coordinate design is isolated by comparing TAIR with a raw-input baseline and target-inconsistent cross-reparameterization controls under identical network architectures, training data, and optimization settings. The assessment considers training behavior, held-out test-set accuracy, transfer to a methane–oxygen counterflow flame at an unseen strain rate within the augmented thermodynamic envelope, and closure-level computational cost. The remainder of this paper is organized as follows. Section II presents the reference real-fluid closure, the TAIR construction, data generation, and neural-network training. Section III reports the model comparisons, and Section IV discusses the thermodynamic interpretation and implications of the proposed reparameterization, followed by conclusions in Section V. II.
METHODS
A.
Reference Real-Fluid Thermodynamic Closure
The present work considers the cell-local thermodynamic closure used in the enthalpy-based pressure-correction formulation of the DeepFlame dfLowMachFoam solver26 . At each thermodynamic update, the closure receives the local solver state xh = (h, p, Y),
(1)
where h is the mass-specific enthalpy variable selected in the present solver configuration, p is the local pressure, and Y = (Y1 , . . . ,YNs ) is the species mass-fraction vector. The same enthalpy definition and reference convention are used by the flow solver, the Cantera-based reference thermodynamic evaluator27 , and the input transformations introduced in Sec. II B. The reference real-fluid closure defines the state-toproperty mapping G : (h, p, Y) 7−→ (T, ρ , ψ ), where T is temperature, ρ is density, and ∂ρ ψ= ∂ p h,Y
(2)
(3)
is the compressibility coefficient used in the pressurecorrection formulation. For a prescribed state (h, p, Y), temperature is recovered by solving the nonlinear enthalpy relation f (T ) = h(T, p, Y) − h = 0,
(4)
where the real-fluid mixture enthalpy is expressed as h(T, p, Y) = hig (T, Y) + hdep (T, p, Y).
(5)
Here, hig is the ideal-gas contribution obtained from the species thermochemical data, whereas hdep is the real-fluid departure contribution associated with the equation of state. The
3 enthalpy inversion is performed iteratively. In Newton form, the temperature update can be written as h(T k , p, Y) − h , c p (T k , p, Y)
where
c p (T, p, Y) =
∂h ∂T
(6)
Newton iteration: ℎ(𝑇 𝑘 , 𝑝, Y) − ℎ 𝑐 𝑝 (𝑇 𝑘 , 𝑝, Y)
𝑇 𝑘+1 = 𝑇 𝑘 −
(7)
(each iteration requires PR-EoS evaluation)
p,Y
is the mass-specific real-fluid heat capacity at constant pressure and composition. At each trial temperature T k , the real-fluid state at (T k , p, Y) is established using the Peng–Robinson equation of state. The ideal-gas contributions to h and c p are evaluated from the species thermochemistry at T k , while their departure contributions are evaluated from the corresponding Peng–Robinson state. Thus, h and c p are evaluated consistently from the same trial state and do not require separate EoS root solves. The Peng–Robinson equation of state is written as9 Ru T am p= − , V̄ − bm V̄ (V̄ + bm ) + bm (V̄ − bm )
where W̄ is the mixture molecular weight. In the reference implementation, the compressibility coefficient is evaluated by a first-order backward finite difference at fixed h and Y. Introducing the perturbed pressure p0 = p(1 − ε ),
(10)
gives
ρ (h, p, Y) − ρ (h, p0 , Y) , p − p0
Compute density 𝜌 Solve PR-EoS for 𝑉¯ (molar volume) ¯ 𝑉¯ 𝜌 = 𝑊/
Perturbed pressure state 𝑝 0 = 𝑝(1 − 𝜖) at fixed ℎ, Y
Repeat 𝑇-𝜌 closure Obtain 𝜌(ℎ, 𝑝 0, Y)
Compute compressibility coefficient 𝜓 𝜌(ℎ, 𝑝, Y) − 𝜌(ℎ, 𝑝 0, Y) 𝜓≈ 𝜖𝑝
(8)
where V̄ is the molar volume and am and bm are mixture parameters calculated using the classical one-fluid mixing rules. For each trial temperature, Eq. (8) is solved for the molar volume. Once the enthalpy inversion has converged, the density is retained from the final Peng–Robinson state as !−1 Ns W̄ Yi , (9) ρ= , W̄ = ∑ V̄ i=1 Wi
ψ≈
Recover temperature 𝑇 Solve ℎ(𝑇, 𝑝, Y) = ℎ
Cell-local thermodynamic closure (repeated calls)
T k+1 = T k −
Solver state ℎ, 𝑝, Y
Thermochemical evaluation and pressure correction use 𝑇, 𝜌, 𝜓
FIG. 1. Schematic of the reference solver-native thermodynamic closure in each computational cell.
The corresponding raw-input neural surrogate preserves the external state-to-property interface, (T, ρ , ψ ) = Fθ (h, p, Y),
(12)
where Fθ denotes the aggregate neural-surrogate mapping. This configuration serves as the raw-input baseline. The TAIR strategy introduced in Sec. II B retains p, Y, and the real-fluid output quantities, but reparameterizes the enthalpy coordinate separately for each target.
(11)
where ε > 0 is the relative pressure perturbation. Evaluation of the perturbed state requires a second enthalpy–temperature inversion and real-fluid EoS evaluation at fixed h and Y. Fig. 1 schematically summarizes this cell-local workflow. Temperature recovery and density evaluation are shown sequentially for clarity, although they are numerically coupled because each temperature iteration requires an EoS state evaluation. Evaluation of ψ further requires repeating the temperature–density closure at the perturbed pressure. The reference thermodynamic states underlying Eqs. (4)–(11) are evaluated using the Cantera Peng–Robinson implementation. These calculations provide the supervised targets used throughout this work.
B.
Target-Aligned Input Reparameterization
The raw-input baseline defined in Eq. (12) supplies the same solver-native enthalpy coordinate h to all property networks. Target-aligned input reparameterization (TAIR) modifies only this thermodynamic coordinate while retaining pressure, composition, network architecture, and output definitions. The construction follows two requirements. First, every transformed coordinate must be evaluated exclusively from the solver-available state (h, p, Y), without using the reference values of T , ρ , or ψ . Second, the transformation must involve only species-weighted sums and explicit algebraic operations, so that it does not reintroduce enthalpy inversion or real-fluid
4 EoS iteration. Its cost therefore scales linearly with the number of species and is fixed for a prescribed chemical mechanism. Because the energy variable used in the present solver configuration is the mass-specific absolute enthalpy, the ideal-gas mixture enthalpy under the same thermochemical reference convention can be written as Z T Ns ig (13) hig (T, Y) = ∑ Yi h◦f ,i + c p,i (τ ) dτ , T0
i=1
where T0 = 298.15 K, h◦f ,i is the mass-specific standard enig
thalpy of formation of species i at T0 , and c p,i is its massspecific ideal-gas heat capacity at constant pressure. All species enthalpies and heat capacities in this section are expressed on a mass basis before mass-fraction averaging. A low-cost temperature coordinate is obtained by treating the solver-provided real-fluid enthalpy as an ideal-gas mixture enthalpy and approximating the species heat capacities by their values at T0 : Ns hig (T, Y) ≈ e hig (T, Y) = ∑ Yi h◦f ,i + c◦p,i (T − T0 ) ,
TABLE I. First thermodynamic coordinate supplied to each property network. Pressure p and composition Y are retained in all configurations. Configuration T network ρ network ψ network Raw-input baseline h h h TAIR T̃ ρ̃ ρ̃ Cross-reparameterization ρ̃ T̃ T̃
The use of ρ̃ for the compressibility network follows from the same approximate thermodynamic relation. At fixed h and Y, Eq. (15) contains no explicit pressure dependence. Consequently, ∂ ρ̃ 1 ρ̃ ψ̃ ≡ = . = (18) ∂ p h,Y Rm T̃ p Thus, because p is retained as an independent input, (ρ̃ , p) contains the corresponding ideal-gas baseline for the isenthalpic density response. The density and compressibility networks use xTAIR = xTAIR = (ρ̃ , p, Y). ρ ψ
(14)
(19)
i=1
where c◦p,i = c p,i (T0 ). The thermodynamic coordinate temperature T̃ is defined by equating the solver-provided enthalpy to this approximate ideal-gas mixture enthalpy e hig (T̃ , Y) = h. Solving this equation gives ig
Ns
h − ∑ Yi h◦f ,i T̃ = T0 +
i=1 Ns
∑
.
(15)
Yi c◦p,i
i=1
The quantity T̃ is a thermodynamic input coordinate rather than a replacement for the real-fluid temperature. It retains the leading caloric relation among enthalpy, temperature, and composition, while omitting the temperature dependence of the species heat capacities and the real-fluid enthalpy departure. The temperature network therefore uses xTAIR = (T̃ , p, Y). T
(16)
Pressure is retained because the real-fluid correction to temperature at fixed enthalpy and composition remains pressure dependent. For the density-related targets, the thermodynamic coordinate temperature is further combined with the ideal-gas equation of state to obtain the ideal-gas density estimate as
ρ̃ =
p , Rm T̃
(17)
where Rm is the mixture-specific gas constant. This coordinate incorporates the leading pressure, temperature, and mixturemolecular-weight dependence of density, while omitting the non-ideal EoS contribution.
Equation (18) is used only to motivate the input design; the real-fluid compressibility coefficient ψ remains the network output. The resulting target-specific assignments are summarized in Table I. The raw-input baseline retains the solver-native enthalpy coordinate for all three targets. TAIR assigns each network the target-matched thermodynamic coordinate that represents the leading ideal-gas dependence of its target. The cross-reparameterization control deliberately exchanges these assignments: ρ̃ is used for temperature, whereas T̃ is used for density and compressibility. This control distinguishes the effect of target-consistent thermodynamic input design from the generic effect of replacing h with an analytically transformed variable. All three configurations predict the absolute real-fluid quantities T , ρ , and ψ ; no residual target is introduced in the loss function. Therefore, the interpretation that TAIR guides the networks to learn real-fluid departures from idealgas-based baselines refers to the structure of the input coordinates rather than to an explicit residual-output formulation. The resulting input–target relations are examined in Sec. III.
C.
Data Generation and Augmentation
The supervised database consists of input–target pairs oN n D = (h, p, Y)(n) , (T, ρ , ψ )(n) . n=1
(20)
It is worth noting that TAIR is applied only after these reference pairs have been generated, as a deterministic preprocessing transformation of the enthalpy coordinate. It therefore does not alter the sampled thermodynamic states, data augmentation, or reference targets. The raw-input, TAIR,
5 (a)
Temperature, T (K)
4000
Sampled Data Augmented Data
3000 2000 1000 0.0
0.2
0.4
0.6
Mixture Fraction, Z
0.8
1.0
Pressure, p (MPa)
10.40 (b)
FIG. 2. Schematic of the 2D counterflow diffusion flame configuration used for data generation.
10.20 10.00 9.80 9.60
and cross-reparameterization models use the same underlying database and differ only in the first thermodynamic coordinate supplied to each property network. The base thermochemical states are sampled from twodimensional laminar counterflow diffusion flames28,29 computed using the DeepFlame dfLowMachFoam solver. For each sampled state, the reference thermodynamic properties are evaluated using the Cantera real-fluid closure described in Sec. II A. The counterflow configuration is shown in Fig. 2. The computational domain is a 2 mm × 2 mm square discretized using a 400 × 100 mesh. Pure methane at 269 K and pure oxygen at 278 K are introduced through opposing boundaries with equal inlet-speed magnitudes. The nominal operating pressure is 100 bar, which exceeds the critical pressures of both pure inlet species. For equal inlet-speed magnitudes Uin , the nominal strain rate is defined as a = 2Uin /Lx , where Lx = 2 mm is the distance between the opposing inlets. Three flames are simulated at a = 1000, 5000, and 10000 s−1 , corresponding to inletspeed magnitudes of 1, 5, and 10 m s−1 , respectively. Chemical kinetics are represented using the 18-species, 44-reaction reduced mechanism for high-pressure methane–oxygen combustion developed by Monnier et al.30 . For each strain rate, thermochemical states are sampled from multiple snapshots during the transient evolution toward a quasi-steady flame, thereby including states from the inlet streams, mixing layers, reaction zone, and burned-gas region. The flame-derived states occupy a relatively narrow portion of the thermochemical state space. To broaden the thermochemical coverage of the flame-derived database, the physically constrained random-augmentation procedure following our previous work31 is adopted. The augmentation is performed in (T, p, Y) space rather than by directly perturbing enthalpy: temperature and pressure are randomly varied within prescribed bounds, while the species mass fractions are perturbed using a common power-law exponent and subsequently renormalized. Candidate states that vio-
0
1000
2000
3000
Temperature, T (K)
4000
FIG. 3. Thermochemical state-space coverage of sampled and augmented flame states. (a) mixture fraction-temperature space and (b) temperature-pressure space.
late the prescribed temperature, pressure, mixture-fraction, or elemental-composition bounds are discarded. For every accepted augmented state, the mass-specific enthalpy, density, and compressibility coefficient are recomputed using the same Cantera-based real-fluid closure as that used for the base flame states. Consequently, the augmented input–target pairs remain thermodynamically consistent. This offline data-generation step is common to all model configurations and is independent of the subsequent TAIR preprocessing. Fig. 3 shows the state-space coverage before and after augmentation. In the displayed projections, the base flame states form relatively narrow bands, whereas the augmented samples populate a broader neighborhood in both the mixture-fraction– temperature and temperature–pressure spaces. A total of 1.2 × 106 accepted samples is retained. The augmented database is randomly partitioned into 1.0 × 106 training samples, 1.0 × 105 validation samples, and 1.0 × 105 held-out test samples. The resulting test set therefore evaluates prediction accuracy within the augmented-state distribution. Transfer across operating conditions is examined separately using a counterflow flame at the unseen strain rate a = 3000 s−1 , which is not included in the training, validation, or test sets but remains within the augmented thermodynamic envelope.
D.
Neural-Network Architecture and Training
The raw-input, TAIR, and cross-reparameterization models are implemented using the same neural-network architecture
6 MLPT
T (Temperature) Thermodynamic Approximation Solver State Raw Inputs h, p, Y
T̃ = T0 +
h − ∑i Yih0f ,i
ρ̃ =
∑i Yic0p,i p RmT̃
MLPρ
Target-Aligned Coordinates T̃ , ρ̃, p, Y T̃ → T,
ρ̃ → ρ, ψ
Z-Score Norm x−µ x̂ = σ
ρ (Density)
MLPψ
ψ (Compressibility)
FIG. 4. Neural-surrogate framework taking the TAIR models as an example.
and training procedure. The three configurations differ only in the thermodynamic coordinate supplied as the first input to each property network, as summarized in Table I. This controlled design isolates the effect of input reparameterization from those of model capacity, training data, and optimization. As illustrated in Fig. 4, temperature, density, and compressibility are predicted using three independent multilayer perceptrons (MLPs). Each network receives a 20-dimensional input vector comprising one thermodynamic coordinate, pressure, and 18 species mass fractions, and returns one scalar real-fluid property. The thermodynamic coordinate is selected according to the input strategy: the raw-input model uses h for all three networks; TAIR uses T̃ for the temperature network and ρ̃ for the density and compressibility networks as presented in Fig. 4; and the cross-reparameterization control interchanges these assignments. The target-matched thermodynamic coordinates are therefore used only as network inputs, while the outputs remain the real-fluid quantities T , ρ , and ψ . Each MLP contains three fully connected hidden layers with 64, 32, and 16 neurons, respectively. Gaussian error linear unit (GELU) activation functions32 are applied after the hidden layers, and the output layer is linear. The three property networks are trained independently and do not share parameters. In particular, density and compressibility are predicted by separate networks rather than deriving ψ from the density surrogate, thereby retaining a direct, fixed-cost evaluation of all three closure quantities. All input and output variables are standardized using means and standard deviations calculated from the training set. The same normalization parameters are subsequently applied to the validation set, held-out test set, and unseen-strain-rate flame. Each property network is trained using an L1 loss in the standardized output space, the Adam optimizer33 , and a cosine-annealing learning-rate schedule34 . Training is performed with a batch size of 1024 for 1500 epochs. Network predictions are transformed back to physical units before the accuracy metrics reported in Sec. III are calculated.
III.
RESULTS
The effect of thermodynamic input-coordinate design is examined from five complementary perspectives: projected input–target relations, training convergence, held-out testset accuracy, transfer to an unseen strain-rate condition, and closure-level computational cost. All comparisons use the same thermochemical database, property-network architectures, and training procedure; the only difference among the raw-input, TAIR, and cross-reparameterization configurations is the thermodynamic coordinate supplied as the first network input.
A.
Projected Input–Target Relations
Fig. 5 compares two-dimensional projections of each target property against the first thermodynamic coordinate used by the three input configurations. Pressure and the full composition vector remain network inputs in all cases; consequently, these projections do not represent the complete highdimensional regression mapping. They are used only to examine how the choice of the leading scalar coordinate organizes the sampled thermodynamic states. For the raw-input configuration, the projections of ρ , T , and ψ against h are strongly multivalued. States with similar enthalpy can exhibit substantially different target values because pressure, composition, and real-fluid departures are not represented by the enthalpy coordinate alone. The methanemass-fraction coloring further shows that much of this projected spread is associated with composition. With TAIR, the target-matched coordinates produce substantially tighter and more monotonic projected relations. The mappings ρ̃ → ρ and ρ̃ → ψ are close to linear over most of the sampled range, while the mapping T̃ → T forms a comparatively narrow monotonic band. These observations indicate that the target-matched thermodynamic coordinates capture a large part of the leading ideal-gas dependence of their corresponding targets, although residual spread remains because the outputs retain their full real-fluid dependence. The cross-reparameterization projections remain organized by analytical coordinates but exhibit markedly less favorable
7
1.5 1.0 −2
Enthalpy h (
10
−4
−2
Enthalpy h (
10
1.0 0
2
4
2 2
125 100 75 50 0
1e3
Estimated temperature T̃ (K)
0
50
100
1.0 0
50
100
Estimated density ρ̃ (kg/m ) 3
0.4 0
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150 3
1e−5
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25
0.5
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Estimated density ρ̃ (kg/m )
Target ψ (s /m )
3
Target ρ (kg/m )
1.5
J/kg)
0.75
0.00
150
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2.0
0 6
0.25
175
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1.00
Estimated density ρ̃ (kg/m )
3.5
−2
Enthalpy h (
1.25
3
1e3
3.0
−4
1e−5
25
0.5
0.8
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Target ψ (s /m )
3
Target ρ (kg/m )
1.5
0.50
J/kg)
150
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0.25
J/kg)
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Target T (K)
2
50 0
0 6
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TAIR
75 25
0.5
1e3
Target T (K)
100
1.25
CH 4
2.0
125
2
3
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Target ψ (s /m )
150
3.0
1.0
1.50
Y
3.5
−4
Cross-reparameterization
1e−5
175
Target ρ (kg/m )
Target T (K)
Raw Input Model
1e3
0
2
4
1e3
Estimated temperature T̃ (K)
0.00
0
2
4
1e3
0.0
Estimated temperature T̃ (K)
FIG. 5. Projected input–target distributions for temperature, density, and compressibility under the raw-input, TAIR, and crossreparameterization strategies. Points are colored by the methane mass fraction (YCH4 ).
relations. In particular, using ρ̃ for temperature produces an inverse, strongly curved relation, whereas using T̃ for density and compressibility retains broad and nonlinear variations. The comparison therefore suggests that the benefit is associated with matching the analytical coordinate to the target thermodynamic dependence, rather than with analytical input transformation alone.
B.
Training Convergence
Fig. 6 compares the standardized L1 training losses of the three input configurations. Because each target is standardized independently, loss magnitudes can be compared among input configurations for the same property, but not across different properties. For density and compressibility, TAIR reaches lower losses from the early stages of training and maintains this advantage throughout the optimization. The final density loss is approxi-
mately half that obtained with the raw-input coordinate, while the reduction is more pronounced for compressibility. For temperature, the raw-input and TAIR models show similar behavior during the initial training stage, but TAIR converges to the lowest final loss. By contrast, the cross-reparameterization model, which uses the density-like coordinate ρ̃ for temperature, converges more slowly and remains at a substantially higher loss. The ordering of the converged losses is consistent with the projected relations in Fig. 5. Target-matched coordinates are associated with tighter projected relations and lower attainable training losses under the same network capacity and optimization procedure. The cross-reparameterization results further show that replacing enthalpy by an analytical variable does not by itself guarantee improved optimization; the relation between the transformed coordinate and the target property is essential.
8
FIG. 6. Standardized L1 training-loss histories for the raw-input, TAIR, and cross-reparameterization configurations: (a) temperature T , (b) density ρ , and (c) compressibility coefficient ψ .
FIG. 7. Predicted-versus-reference scatter plots for the held-out test set. Rows correspond to the target properties T , ρ , and ψ , whereas columns correspond to the raw-input, TAIR, and cross-reparameterization models. The dashed lines indicate exact agreement. Marker colors and sizes represent absolute errors. Error color scales are normalized separately for each target property.
C.
Held-Out Test-Set Accuracy
Fig. 7 compares the network predictions with the reference values on the held-out test subset of the augmented database.
All three configurations produce predictions close to the line of exact agreement, but their errors differ substantially when
9 quantified by the root-mean-squared error (RMSE) in physical units. For temperature, the RMSE decreases from 0.371 K for the raw-input model to 0.250 K for TAIR, an improvement factor of approximately 1.5. The target-inconsistent ρ̃ -to-T pairing instead increases the RMSE to 0.909 K, approximately 3.6 times the TAIR value and 2.5 times the raw-input value. For density, the raw-input model yields an RMSE of 0.0746 kg m−3 , whereas TAIR reduces it to 0.0368 kg m−3 , corresponding to an improvement factor of approximately 2.0. Cross-reparameterization gives 0.0886 kg m−3 , which is less accurate than both TAIR and the raw-input baseline. The strongest improvement is obtained for compressibility. TAIR reduces the RMSE from 3.19 × 10−8 s2 m−2 for the raw-input model to 4.24 × 10−9 s2 m−2 , corresponding to a reduction factor of approximately 7.5. The crossreparameterization result, 3.03 × 10−8 s2 m−2 , remains close to the raw-input error and provides no comparable improvement. Thus, TAIR gives the lowest held-out error for all three targets. The cross-reparameterization control does not provide a systematic benefit: it performs similarly to the raw-input model for compressibility and degrades the density and temperature predictions. This behavior supports the interpretation that the improvement originates from target-matched thermodynamic input design rather than from analytical preprocessing in general.
D.
Transfer to an Unseen Strain Rate
To assess transfer across operating conditions, an additional counterflow flame is generated at the unseen strain rate a = 3000 s−1 . The assessment is performed a priori: the three trained models are queried on the thermochemical states generated by the reference dfLowMachFoam calculation, without replacing the real-fluid closure during time integration. The comparison therefore evaluates property-prediction transfer to a new flame structure rather than online solver stability or thermodynamic extrapolation. Fig. 8 shows the reference fields and the corresponding absolute-error maps at t = 7 × 10−4 s. For temperature, the TAIR RMSE is 0.332 K, compared with 1.18 K for the raw-input model, giving an improvement factor of approximately 3.6. Cross-reparameterization again performs poorly for this target, with an RMSE of 6.48 K. For density, TAIR yields an RMSE of 0.0295 kg m−3 , compared with 0.429 kg m−3 for the raw-input model and 0.0532 kg m−3 for cross-reparameterization. The raw-input-to-TAIR reduction factor is therefore approximately 14.5. For compressibility, TAIR reduces the RMSE from 1.81 × 10−8 s2 m−2 for the raw-input model to 3.03 × 10−9 s2 m−2 , corresponding to an improvement factor of approximately 6.0. The crossreparameterization RMSE is 9.47 × 10−9 s2 m−2 , approximately 3.1 times the TAIR value. For all three properties, the largest errors are confined primarily to the thin mixing and reaction layer, where the reference thermodynamic fields exhibit their strongest spatial gra-
TABLE II. Thermodynamic closure cost using 105 held-out thermochemical states. Method Reference (Cantera) Raw-input NN TAIR
Mean time (s) Speed-up 15.429 0.159 0.173
1.0 × 97.1 × 89.4 ×
dients. Errors in the bulk fuel and oxidizer streams remain small. The preservation of the TAIR accuracy advantage at this separately generated flame demonstrates transfer across strain-rate conditions within the augmented thermodynamic envelope. It does not, by itself, establish extrapolation beyond that envelope or stable inline coupling of the surrogate with the pressure-correction solver. E.
Closure-Level Computational Cost
The computational cost of the local thermodynamic closure is assessed using the 105 states in the held-out test subset. The reported wall-clock time corresponds to completing the evaluation of the full test set for each method. For every run, timing starts immediately before loading the test subset and ends after the ρ , T , and ψ outputs are materialized. All timings are obtained using one core of an Intel Xeon Gold 6330 CPU at 2.00 GHz. Each method is measured 12 times using a cyclically balanced execution order. After removing the largest and smallest values for each method, the reported result is the mean of the remaining ten measurements. As shown in Table II, the reference real-fluid closure requires 15.429 s for the full test set. The raw-input neural model reduces this cost to 0.159 s, corresponding to a speedup of 97.1×. TAIR requires 0.173 s and provides a speed-up of 89.4×. The TAIR workflow therefore requires an additional 0.014 s over the full test set relative to the raw-input neural model, corresponding to an observed overhead of approximately 8.8%. Nevertheless, the complete TAIR evaluation remains only about 1.1% of the reference closure cost. This overhead is therefore small relative to the iterative thermodynamic calculation that the surrogate replaces. These measurements quantify closure-level evaluation on the held-out test set only. They should not be interpreted as an equivalent speed-up of the complete CFD simulation, whose wall-clock cost also includes transport equations, chemical kinetics, pressure correction, communication, and other solver operations. IV.
DISCUSSION
A.
Thermodynamic Interpretation of TAIR
The target-matched thermodynamic coordinates introduced by TAIR are not approximate substitutes for the real-fluid
10
FIG. 8. Reference thermodynamic fields and absolute-error contours for the independently generated counterflow flame at the unseen strain rate a = 3000 s−1 and t = 7 × 10−4 s.
closure. The estimated temperature T̃ is constructed by neglecting the real-fluid enthalpy departure and approximating the ideal-gas species heat capacities as constants at the reference temperature. The estimated density ρ̃ further applies the ideal-gas equation of state to T̃ . These variables therefore retain the leading caloric or volumetric dependence of the corresponding targets, while the real-fluid departures remain to be represented by the neural networks.
B.
Target Matching and Local Sensitivity
TAIR does not introduce additional thermodynamic-state information. At fixed p and Y, the enthalpy h and T̃ are related by an invertible affine transformation. Because p and Y are retained, ρ̃ = p/(Rm T̃ ) is likewise an invertible reparameterization over the admissible positive-temperature domain. The raw-input, TAIR, and cross-reparameterization models therefore contain the same underlying state information and differ only in how that information is presented to the finite-capacity networks.
The target-dependent improvements are consistent with the structure of the thermodynamic closure. At fixed p and Y, enthalpy is already directly related to temperature through the caloric equation of state. The raw enthalpy is therefore a comparatively favorable coordinate for T , and the transformation to T̃ mainly accounts for the dominant formation-enthalpy offset and mixture heat-capacity scaling before regression. This is consistent with the moderate improvement obtained for temperature. In contrast, ρ̃ combines the leading dependence of density on pressure, caloric state, composition, and mixture molecular weight. Together with the retained pressure input, it also provides the ideal-gas baseline for the isenthalpic density response represented by ψ , as shown in Eq. (18). This is consistent with the larger improvements obtained for density and compressibility.
TAIR is thus best interpreted as a deterministic thermodynamic preconditioning of the input coordinates. The networks continue to predict the absolute real-fluid quantities T , ρ , and ψ , rather than explicit residuals. Nevertheless, the targetmatched coordinates expose the leading ideal-gas-based dependence of each target, leaving a more departure-like input– output relation for neural regression.
The cross-reparameterization controls show that the improvement does not arise from analytical preprocessing alone. Using the density-like coordinate ρ̃ for temperature produces a strongly curved inverse relation, whereas using T̃ for density and compressibility omits the leading pressure and mixturemolecular-weight dependence contained in ρ̃ . The degraded performance of these target-inconsistent models therefore demonstrates the importance of matching the transformed co-
11
FIG. 9. Normalized local first-coordinate sensitivities of the reference thermodynamic mapping. One representative database state is selected in each temperature bin as the sample closest to the bin-median temperature.
ordinate to the thermodynamic dependence of the target. Fig. 5 and 9 describe complementary aspects of this coordinate effect. Fig. 5 provides a qualitative two-dimensional projection of how each first coordinate organizes the target states, whereas Fig. 9 quantifies the corresponding local slope at representative thermochemical states. For each temperature bin, one representative database state is selected, and the candidate first coordinate is locally perturbed while p and Y are held fixed. The normalized sensitivity is defined as Sy,x = |∂ y∗ /∂ x∗ | p,Y , where starred quantities denote the standardized variables used during training. For a monotonic onedimensional relation, a constant signed derivative corresponds to an affine mapping. Accordingly, when interpreted together with the monotonic projections in Fig. 5, a narrower variation of Sy,x indicates a more uniform local input–target scale and a relation that is closer to affine along the selected coordinate. Such a representation is generally more favorable for a finite-capacity multilayer perceptron, consistent with approximation results for smoother target functions and the spectral preference of standard neural networks for more slowly varying function components35–38 . The more uniform sensitivities of the target-matched coordinates are therefore consistent with their tighter projected relations, lower training losses, and lower prediction errors. This analysis characterizes only one representative state in each temperature bin and only the selected first input coordinate. The sensitivity magnitude does not capture derivativesign changes, cross-coordinate coupling, sample density, or the complete multivariate mapping. Fig. 9 should therefore be interpreted as a quantitative complement to the projected relations in Fig. 5, rather than as a complete measure of mapping nonlinearity or neural-network learnability.
C.
Generality and Limitations
The quantitative results reported in this work are obtained for supercritical methane–oxygen counterflow flames near a nominal pressure of 100 bar, using the Peng–Robinson equation of state and the thermodynamic envelope represented
by the present database. The specific error-reduction factors and computational speed-ups should therefore be interpreted within this validated domain. The unseen-strain-rate case demonstrates transfer to a different flame structure and operating condition within the augmented thermodynamic envelope, rather than unrestricted extrapolation to arbitrary thermodynamic states. The input-design principle underlying TAIR, however, is not intrinsically restricted to the present fuel–oxidizer system or to the three properties considered here. More generally, when a target quantity admits an inexpensive analytical or reduced-order approximation based on solver-available variables, that approximation may be used as a transformed input coordinate to expose the leading physical dependence of the target before neural regression. The network is then required to represent the remaining departure from the physical baseline rather than reconstructing the complete input–target relation directly from the native solver variables. This principle may provide a useful design strategy for other thermodynamic properties, transport quantities, or data-driven closure terms, provided that an appropriate target-matched baseline can be identified. Density and compressibility are predicted using separate networks. This choice provides direct fixed-cost inference, but does not enforce the differential consistency condition. Joint training or thermodynamic-consistency regularization may be examined in future work, together with the associated computational trade-offs. Finally, the flame-field assessment reported here is a priori: the surrogates are queried using states generated by the reference CFD solution. Stable online coupling with the enthalpy equation and pressure-correction procedure has not yet been demonstrated. Such a test must assess error accumulation, pressure–density feedback, conservation, and excursions outside the training envelope. Extension of the same inputdesign principle to transport properties or other closure terms will similarly require an inexpensive analytical baseline that is physically matched to the corresponding target.
12 V.
CONCLUSIONS
This work introduced target-aligned input reparameterization (TAIR) for neural prediction of real-fluid thermodynamic properties in supercritical combustion. TAIR replaces the raw enthalpy coordinate of each property network with a targetmatched thermodynamic coordinate: an estimated temperature derived from a constant-c p ideal-gas enthalpy approximation for temperature prediction, and an ideal-gas density estimate for density and compressibility prediction. The transformations are explicit, preserve the solver-available state information, and leave the training database, network architecture, and real-fluid output definitions unchanged. For supercritical methane–oxygen counterflow flames, TAIR reduced the held-out test-set RMSE by factors of approximately 1.5, 2.0, and 7.5 for T , ρ , and ψ , respectively, relative to the raw-input baseline. At an unseen strain rate within the augmented thermodynamic envelope, the corresponding reduction factors were approximately 3.6, 14.5, and 6.0. The degraded performance of the target-inconsistent cross-reparameterization controls shows that the improvement arises from thermodynamically matched input design rather than generic analytical preprocessing. TAIR also achieved a closure-level single-core speed-up of approximately 89.4× relative to the iterative Cantera reference. These results demonstrate that low-cost physical approximations can provide effective input coordinates for neural real-fluid closures. Future work will assess stable inline coupling with the pressure-based reacting-flow solver and thermodynamic consistency between the independently predicted density and compressibility.
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