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REVIEW 3 major objections 5 minor 45 references

Physics-Infused Reduced-Order Modeling for Analysis of Multi-Layered Hypersonic Thermal Protection Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A physics-plus-data reduced-order model predicts the transient temperature response of a multi-layered hypersonic thermal protection system with below 1 percent error on heat-flux and material-property conditions not seen in training, at…

desk verdict Solid PIROM application to TPS with plausible ~1% extrapolation on boundary conditions, but the material-property generalization rests on a thin D8 test set and should be verified before the headline claim is taken at face value. read the letter →

arxiv 2505.22890 v1 pith:VO43LDVM submitted 2025-05-28 physics.comp-ph cs.NAmath.NA

Carlos A. Vargas Venegas , Daning Huang , Patrick Blonigan , JohnTencer This is my paper
classification physics.comp-phcs.NAmath.NA
keywords physics-infusedreduced-ordermodelingthermalprotectionsystemsMori-Zwanzigformalismlumped-capacitancemodelhypersonicheatfluxoperatorinferenceneuralODEcoarse-graining
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tackles the 'impossible trinity' of reduced-order modeling - accuracy, generalizability, and efficiency - for transient heat conduction in multi-layered hypersonic thermal protection systems. It proposes PIROM, which combines a lumped-capacitance model as a physics backbone with a data-driven correction term derived from the Mori-Zwanzig formalism. The claim is that, trained only on thermocouple temperature histories from uniform constant heat flux and nominal material properties, PIROM keeps normalized root-mean-square error below 1 percent across space-time varying heat fluxes and temperature-dependent material perturbations never seen in training, while running about 100 times faster than the finite-element full-order model. Compared with Operator Inference and neural ODE baselines, the paper reports that PIROM generalizes best: NODE fails on new loading shapes, OpInf degrades moderately and cannot adapt to new materials, while PIROM retains its accuracy.

What carries the argument

The load-bearing object is the hidden-state extension of the lumped-capacitance model: $\bar{A}(\bar{u})\dot{\bar{u}} = \bar{B}(\bar{u})\bar{u} + \bar{f}(t) + P\boldsymbol{\beta}$, with $\dot{\boldsymbol{\beta}} = Q\bar{u} - \Lambda\boldsymbol{\beta} + R\bar{f}(t)$. The first equation is the physics backbone; $P$, $Q$, $\Lambda$, and $R$ are learned matrices, and the diagonal decay rates are replaced by a fully populated constant matrix. This construction converts a non-Markovian memory integral into Markovian hidden states, so the model keeps explicit parametric dependence on material properties through $\bar{A}(\bar{u})$ and $\bar{B}(\bar{u})$ while the hidden states correct the residual dynamics. Training uses adjoint-based trajectory optimization with a tolerance-sweeping schedule that gradually extends the time horizon and tightens the error tolerance.

What would settle it

Train PIROM on D1 and test on material-property perturbations far outside the sampled ranges, for example alpha values outside the box [-1/2,1/2] in both linear and sinusoidal amplitudes, or a material property curve with a sharp knee not representable by the linear-plus-sinusoid formula; if the NRMSE crosses the 1 percent threshold, the constant-coefficient exponential-memory closure is the failing link.

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Extended reading notes

Core claim

On its own terms, the paper's central result is that coarse-graining a discontinuous-Galerkin finite-element model onto component-averaged temperatures does not merely approximate the lumped-capacitance model; it produces exactly the LCM plus a residual term. The Mori-Zwanzig closure turns that residual into a memory integral, and approximating the memory kernel as a sum of exponentials gives a finite-dimensional extended system whose hidden states evolve linearly in the resolved temperatures and the heat-flux forcing. With a three-lump LCM backbone and six hidden states, PIROM reports errors around 0.29 percent on a challenging extrapolative test case and mean errors below 1 percent across the out-of-distribution datasets, including material-property perturbations. The model needs only thermocouple temperature histories for training, in contrast to OpInf, which requires full-field temperature snapshots.

Load-bearing premise

The residual difference between the true component-averaged temperature evolution and the lumped-capacitance backbone can be represented by a small fixed number of hidden states that evolve linearly in temperature and heat-flux inputs with constant coefficients and exponentially decaying memory.

Editorial extensions

If this is right

  • If the claimed accuracy holds, PIROM can serve as a many-query surrogate for thermal protection system design optimization and uncertainty quantification, cutting each transient evaluation from minutes to well under a second.
  • Thermocouple-only training means the method can be applied to flight-test or experimental data where full-field temperature snapshots are unavailable.
  • The Mori-Zwanzig derivation supplies a template: any reduced-physics model that is a coarse-graining of a fine discretization can be corrected by hidden states of the same form.
  • The reported speedup makes PIROM a candidate for real-time thermal state estimation or control during hypersonic flight.
  • The higher training cost is a one-time overhead, and the paper points to weak-form, adjoint-free training as a route to reduce that cost substantially.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The exponential-memory closure is the natural first thing to stress: testing the six-hidden-state model on material properties whose variation is discontinuous in temperature, or where radiation coupling matters, would map the limits of the linear-constant-coefficient hidden-state assumption.
  • Because the hidden-state dynamics are linear in temperature and forcing with constant coefficients, the current PIROM form may not capture state-dependent memory; a temperature-dependent or nonlinear closure would be needed if the residual dynamics are strongly regime-dependent.
  • The benchmark mixes two advantages: OpInf is trained on full-field snapshots while PIROM uses only three thermocouple signals, so the accuracy comparison reflects both model form and data efficiency.
  • The same architecture could be applied to other coarse-grainable physics, such as battery thermal management or convection-diffusion in porous media, whenever an engineering lumped model is available as a backbone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes PIROM, a reduced-order model for multi-layer hypersonic thermal protection systems that couples a lumped-capacitance backbone with a data-driven hidden-state correction derived from a Mori-Zwanzig coarse-graining argument. The model is trained only on thermocouple temperature histories from uniform, constant heat-flux cases with nominal material properties (dataset D1) and tested on in-distribution data (D2) and on extrapolative datasets with space-time varying heat fluxes (D3-D7) and perturbed temperature-dependent material properties (D8). The authors report NRMSE below 1% for boundary-condition extrapolation, approximately 1% for material-property perturbations, and online speedups of two orders of magnitude over the full-order FEM solver. The PIROM is benchmarked against Operator Inference and Neural ODEs, and a convergence study over the number of hidden states is used to select m=6.

Significance. If the reported robustness holds, the paper makes a useful contribution to hybrid physics/data-driven ROMs: the architecture retains explicit parametric dependence through the LCM while correcting residual dynamics from data, and it requires only observable trajectories rather than full-field snapshots. The numerical benchmark is thoughtfully structured: training on D1, in-distribution testing on D2, extrapolation tests D3-D8, hyperparameter selection limited to D3, and a hidden-state convergence study are all clear strengths. The comparison with OpInf and NODE is informative and the computational-cost reporting is transparent. The main significance hinges on whether the material-property extrapolation claim is supported by the evidence, because that is the regime where the fixed-size linear hidden-state correction is least justified by the derivation.

major comments (3)
  1. [Sec. IV.C.2, Table 2 (dataset D8), Fig. 12] The material-property extrapolation claim rests on a single small test set of 10 cases with a fixed m=6 hidden-state correction, and the paper reports only a mean NRMSE of 'approximately 1%' with no per-case distribution and no hidden-state sweep in this regime. The convergence study in Sec. IV.B.3 varies the number of hidden states only for D2/D3, i.e., for heat-flux generalization. Because Eq. (32) assumes constant-coefficient, linear hidden dynamics trained on nominal materials, the residual induced by the perturbations in Eq. (41) could in principle require more states or state-dependent coefficients. Please report the per-case NRMSEs for all D8 cases, provide a hidden-state sweep or residual diagnostic on D8, or explicitly restrict the claimed 'wide range' generalization to the tested 10-case family.
  2. [Appendix A, Eqs. (47)-(48), Sec. III.A] The derivation that the resolved dynamics 'is exactly the LCM' invokes the 'mild assumption that the element sizes are the same within a component' (Appendix A, after Eq. (47)), but the actual FEM mesh is refined at the layer interfaces (Sec. IV.A.2, Fig. 3(a)). For the mesh used in the numerical study, the coarse-graining identity W Phi^+ B Phi = \bar B therefore does not hold exactly, and the discrepancy is absorbed by the same fixed hidden states that correct the residual physics. This conflates a discretization mismatch with the physical residual dynamics and weakens the theoretical claim that Eq. (22) is the exact resolved LCM. Please either extend the coarse-graining derivation to non-uniform meshes or quantify the magnitude of the mismatch for the meshes used.
  3. [Sec. III.B, Eqs. (28)-(32)] The representation of the memory kernel as a finite sum of exponentials and the resulting constant-coefficient linear hidden-state dynamics are introduced without error control or empirical justification. The leading-order expansion in Eq. (26) motivates the general structure, but it does not guarantee that m=6 scalar hidden states with coefficients trained only on D1 can represent the residual dynamics under material-property perturbations. Since this is the core representational assumption that enables the extrapolative claims, the paper would be strengthened by a diagnostic showing, for example, that the identified kernel decays as a small set of exponentials and that the residual lies in the span of the learned hidden states for the D8 cases.
minor comments (5)
  1. [Sec. IV.B.2] The time-horizon schedule T = {1500/38 * j}_{j=1}^{38} is stated without justification; please explain why 38 stages or whether this choice affects the convergence behavior.
  2. [Appendix A, Eq. (45)] In the derivation of term #1, the notation \rho c_{p,k} is used inside a sum over i in V_k before the index k is clearly identified as the component label; a sentence clarifying this would improve readability.
  3. [Fig. 10(b) and Sec. IV.C.3] The caption and text should be more explicit that the acceleration ratio is evaluated on D1 while the overhead ratio is evaluated on D2-D7, so that readers do not infer a single dataset for both ratios.
  4. [Appendix B.B] There is a typo, 'overfiting' for 'overfitting', and the regularization discussion would benefit from stating how the reported 54-minute training cost relates to the overhead ratio in Fig. 10(b).
  5. [References] Reference [15] contains a typo: 'Convoutional' should be 'Convolutional'.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: PIROM's extrapolative accuracy claims are validated on out-of-sample datasets D4-D8 that are not used for fitting, so the central derivation is not equivalent to its inputs.

full rationale

PIROM's target predictions (thermocouple histories on D2-D8) are not defined in terms of the quantities being predicted. The hidden-state parameters Theta={P,Q,E,R,D} are learned solely from D1 via the trajectory objective (36), and the extrapolation claims are measured on D4-D7 (grid heat-flux sweeps) and D8 (material-property perturbations) that are neither fitted nor used for hyperparameter selection; D3 is used only to choose m=6, which is a mild selection on one out-of-distribution family and does not force the D4-D8 results. The lumped-capacitance backbone enters Eq. (32a) with material-dependent A_bar, B_bar, and f_bar, so the D8 response to perturbed properties is carried by the physics term rather than by a parameter fit to D8. The exponential-kernel hidden-state model (28)-(32) is an explicit modeling ansatz, introduced as 'Consider the representation of the kernel as a finite sum of simpler functions, e.g., exponentials', and is validated empirically rather than being equivalent to the objective by construction. The only self-citation that is load-bearing in the derivation narrative is [29] for the leading-order Mori-Zwanzig approximation and hidden-state form, but it is not used to define the measured observables or to forbid alternatives; the paper's central accuracy claims are independently benchmarked against OpInf and NODE on out-of-sample data. Appendix A's equal-element-size assumption is a consistency gap between the derivation and the refined mesh, not a circularity. Overall, no prediction in the paper reduces by construction to its training input.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claim rests on a hybrid model whose physics backbone (LCM) is a deliberate coarse-grained approximation, and whose correction is a learned linear hidden-state system with hand-chosen architecture. The numerical generalization results are the main support; the theoretical derivation of the correction form depends on several unverified ansatze (exponential kernel, equal element sizes, penalty choice). No new physical entities are proposed; the hidden states are latent modeling variables.

free parameters (4)
  • PIROM learnable matrices P, Q, E, R, D = not reported (trained on D1)
    These matrices in Eq. (33) define the hidden-state correction dynamics, the central learned component whose values determine the extrapolation errors.
  • Number of hidden states m = 6 (LCM-3 with 6 hidden states, total 9 states)
    Selected via convergence study on D2/D3 (Sec. IV.B.3); this is a hand-chosen hyperparameter balancing accuracy and training cost.
  • LCM lumped parameters (thermal resistances, heat flux distribution) = determined from geometry and material properties; heat flux split among exposed lumped masses
    The LCM-3/LCM-5 models in Appendix B.A require a choice of which lumped masses receive the distributed heat flux; the paper describes this qualitatively but does not give the exact mapping formula.
  • Training schedule hyperparameters (learning rate, tolerances, time horizons) = lr=1e-4, E={1e-1,...,1e-5}, T={1500 j/38}, Nmax=300
    These are hand-chosen to stabilize the tolerance-sweeping training (Alg. 1); they affect training cost and final model quality but are not independently benchmarked.
assumptions (5)
  • domain assumption Mori-Zwanzig exact reformulation and its leading-order kernel approximation (Eqs. 25-26)
    The paper relies on the MZ formalism as an exact identity for the resolved dynamics, and then on the leading-order expansion from prior work [29] to approximate the memory kernel as Eq. (26). If this approximation is inaccurate, the hidden-state model form may be misspecified. The expansion is not re-derived in this paper.
  • ad hoc to paper Exponential-sum representation of the memory kernel (Eq. 28)
    The kernel is assumed to be a finite sum of exponentials with linear dependence on the resolved states and forcing. This is an ansatz introduced to enable Markovianization; no proof is given that the true memory kernel for the TPS has this form.
  • domain assumption Equal element sizes within each component in the coarse-graining (Appendix A)
    Equations (47) and (50) assume all elements inside a component have the same size to simplify the coarse-grained B and f. The paper calls this a mild assumption; for non-uniform meshes (as in Fig. 3a), it is only approximate.
  • ad hoc to paper DG penalty factor equals the inverse thermal resistance (Sec. III.A, Eq. 16, Appendix A)
    The equivalence between the LCM and the coarse-grained DG model requires choosing the IPG penalty as 1/R. This is a parameter choice that makes the derivation work; it is not derived from the PDE.
  • domain assumption Temperature-dependent material property curves and perturbation model (Eq. 41) are known and representative
    The FOM trajectories and the D8 generalization test rely on the nominal and perturbed property curves shown in Fig. 3. The perturbation form (linear plus sinusoidal) is an arbitrary test construction, not based on measured uncertainty data.
invented entities (1)
  • Hidden states beta
    purpose: Augmented latent variables that convert the non-Markovian memory integral into a Markovian system (Eqs. 29-32)
    The hidden states are unobserved and their values are only inferred through the learned matrices; they have no direct physical measurement or independent falsifiable prediction. They function as a black-box memory buffer.

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Cite this review

Pith. "Pith review of Physics-Infused Reduced-Order Modeling for Analysis of Multi-Layered Hypersonic Thermal Protection Systems." pith.science (2026). https://pith.science/paper/VO43LDVM

@misc{pith2026250522890,
  author       = {Pith},
  title        = {Pith review of: Physics-Infused Reduced-Order Modeling for Analysis of Multi-Layered Hypersonic Thermal Protection Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VO43LDVM}},
  note         = {Machine review of arXiv:2505.22890}
}
read the original abstract

This work presents a physics-infused reduced-order modeling (PIROM) framework for efficient and accurate prediction of transient thermal behavior in multi-layered hypersonic thermal protection systems (TPS). The PIROM architecture integrates a reduced-physics backbone, based on the lumped-capacitance model (LCM), with data-driven correction dynamics formulated via a coarse-graining approach rooted in the Mori-Zwanzig formalism. While the LCM captures the dominant heat transfer mechanisms, the correction terms compensate for residual dynamics arising from higher-order non-linear interactions and heterogeneities across material layers. The proposed PIROM is benchmarked against two non-intrusive reduced-order models (ROMs): Operator Inference (OpInf) and Neural Ordinary Differential Equations (NODE). The PIROM consistently achieves errors below 1% for a wide range of extrapolative settings involving time- and space-dependent boundary conditions and temperature-varying material property perturbations. In contrast, OpInf exhibits moderate degradation, and NODE suffers substantial loss in accuracy due to its lack of embedded physics. Despite higher training costs, PIROM delivers online evaluations of two orders of magnitude faster than the full-order model. These results demonstrate that PIROM effectively reconciles the trade-offs between accuracy, generalizability, and efficiency, providing a robust framework for thermal modeling of TPS under diverse operating conditions.

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Reference graph

Works this paper leans on

45 extracted references · 28 canonical work pages

  1. [1]

    R.,Space Shuttle: The History of the National Space Transportation System, Voyageur Press, 2007

    Jenkins, D. R.,Space Shuttle: The History of the National Space Transportation System, Voyageur Press, 2007

    work page 2007

  2. [2]

    DesignReport-ThermalProtectionSystemX-15A-2,

    Price,A.B.,“DesignReport-ThermalProtectionSystemX-15A-2,”Tech.rep.,NationalAeronauticsandSpaceAdministration, 1968

    work page 1968

  3. [3]

    Development of X-43A Mach 10 Leading Edges,

    Ohlhosrt, C. W., Glass, D., Bruce, W. E., lindell, M. C., Vaughn, W. L., Smith, R. W., Dirling, R. B., Hogenson, P. A., Nichols, J., Risner, N. W., Thompson, D. R., Kowbel, W., Sullivan, B. J., Koenig, J. R., and Cuneo, J. C., “Development of X-43A Mach 10 Leading Edges,”56th International Astronautical Congress of the International Astronautical Federati...

    work page doi:10.2514/6.i 2005

  4. [4]

    Development of the Mars Science Laboratory Heatshield Thermal Protection System,

    Beck, R. A. S., Driver, D. M., Wright, M. J., and Hwang, H. H., “Development of the Mars Science Laboratory Heatshield Thermal Protection System,”Journal of Spacecraft and Rockets, Vol. 51, No. 4, 2014. https://doi.org/10.2514/1.A32635

    work page doi:10.2514/1.a32635 2014

  5. [5]

    E., and Williams, C

    Rasmussen, C. E., and Williams, C. K. I.,Gaussian Processes for Machine Learning, The MIT Press, 2005. https: //doi.org/10.7551/mitpress/3206.001.0001. 43

    doi:10.7551/mitpress/3206.001.0001 2005

  6. [6]

    Neural Ordinary Differential Equations,

    Chen, R. T. Q., Rubanova, Y., Bettencourt, J., and Duvenaud, D., “Neural Ordinary Differential Equations,”arXiv, 2018. https://doi.org/10.48550/arXiv.1806.07366

    arXiv doi:10.48550/arxiv.1806.07366 2018

  7. [7]

    Turbulence and the Dynamics of Coherent Structures Part I: Coherent Structures,

    Sirovich, L., “Turbulence and the Dynamics of Coherent Structures Part I: Coherent Structures,”Quarterly of Applied Mathematics, Vol. 45, No. 3, 1987, pp. 561–571

    work page 1987

  8. [8]

    Turbulence and the Dynamics of Coherent Structures Part II: Symmetries and Transformations,

    Sirovich, L., “Turbulence and the Dynamics of Coherent Structures Part II: Symmetries and Transformations,”Quarterly of Applied Mathematics, Vol. 45, No. 3, 1987, pp. 573–582

    work page 1987

Show all 45 references
  1. [9]

    Turbulence and the Dynamics of Coherent Structures Part III: Dynamics and Scaling,

    Sirovich, L., “Turbulence and the Dynamics of Coherent Structures Part III: Dynamics and Scaling,”Quarterly of Applied Mathematics, Vol. 45, No. 3, 1987, pp. 583–590

    1987

  2. [10]

    Balanced Model Reduction via the Proper Orthogonal Decomposition,

    Willcox, K., and Peraire, J., “Balanced Model Reduction via the Proper Orthogonal Decomposition,”AIAA Journal, Vol. 40, No. 11, 2012, pp. 2323–2330. https://doi.org/10.2514/2.1570

    2012 doi

  3. [11]

    Operator Inference for Non-Intrusive Model Reduction of Systems with Non-Polynomial Nonlinear Terms,

    Benner, P., Goyal, P., Kramer, B., Peherstorfer, B., and Willcox, K., “Operator Inference for Non-Intrusive Model Reduction of Systems with Non-Polynomial Nonlinear Terms,”Computer Methods in Applied Mechanics and Engineering, Vol. 372, 2020, p. 113433. https://doi.org/10.1016...

    2020

  4. [12]

    Projection-Based Model Reduction for Finite-Element Simulations of Thermal Protection Systems,

    Arienti, M., Blonigan, P. J., Rizzi, F., Tencer, J., and Howard, M., “Projection-Based Model Reduction for Finite-Element Simulations of Thermal Protection Systems,”AIAA SciTech Forum, Virtual, 2021. https://doi.org/10.2514/6.2021-1717

    2021 doi

  5. [13]

    Projection-based Reduced-Order Models with Hyperreduction for Finite Element Simulations of Thermal Protection Systems,

    Blonigan, P. J., Tencer, J. T., and Rizzi, F., “Projection-based Reduced-Order Models with Hyperreduction for Finite Element Simulations of Thermal Protection Systems,”AIAA AVIATION 2023 Forum, San Diego, California, 2023. https://doi.org/10.2514/6.2023-3913

    2023 doi

  6. [14]

    Physics-Informed Machine Learning,

    Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., and Yang, L., “Physics-Informed Machine Learning,”ResearchGate, 2021, pp. 1–19. https://doi.org/10.1038/s42254-021-00314-5

    2021 doi

  7. [15]

    Convoutional Neural Networks for the Construction of Surrogate Models for Fluid Flows,

    Lui, H. F. S., and Wolf, W. R., “Convoutional Neural Networks for the Construction of Surrogate Models for Fluid Flows,”

  8. [16]

    Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications,

    Hao, Z., Liu, S., Zhang, Y., Ying, C., Feng, Y., Su, H., and Zhu, J., “Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications,”arXiv, 2022. https://doi.org/10.48550/arXiv.2211.08064

    2022 doi arXiv

  9. [17]

    A metalearning approach for Physics-Informed Neural Networks (PINNs): Application to parameterized PDEs,

    Penwarden, M., Zhe, S., Narayan, A., and Kirby, R. M., “A metalearning approach for Physics-Informed Neural Networks (PINNs): Application to parameterized PDEs,”Journal of Computational Physics, Vol. 477, 2023, p. 111912. https: //doi.org/10.1016/j.jcp.2023.111912

    2023

  10. [18]

    Physics-Informed Neural Networks for Enhanced Critical Heat Flux Prediction in Hypersonic Flows,

    Yasar, H. A., and Sevinc, O. K., “Physics-Informed Neural Networks for Enhanced Critical Heat Flux Prediction in Hypersonic Flows,”AIAA Aviation Forum and ASCEND Conference, Las Vegas, Nevada, 2024. https://doi.org/10.2514/6.2024-4203

    2024 doi

  11. [19]

    Physics-Guided Deep Learning for Dynamical Systems: A Survey,

    Wang, R., and Yu, R., “Physics-Guided Deep Learning for Dynamical Systems: A Survey,”arXiv, 2021. https://doi.org/10. 48550/arXiv.2107.01272. 44

    2021 doi arXiv

  12. [20]

    DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators,

    Mao, Z., Lu, L., Marxen, O., Zaki, T. A., and Karniadakis, G. E., “DeepM&Mnet for hypersonics: Predicting the coupled flow and finite-rate chemistry behind a normal shock using neural-network approximation of operators,”Journal of Computational Physics, Vol. 447, 2021, p. 1106...

    2021

  13. [21]

    Fusion DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic Flows on Arbitrary Grids,

    Peyvan, A., and Kumar, V., “Fusion DeepONet: A Data-Efficient Neural Operator for Geometry-Dependent Hypersonic Flows on Arbitrary Grids,”arXiv:2501.01934, 2025

    2025 arXiv

  14. [22]

    Instability-wave prediction in hypersonic boundary layers with physics-informed neural operators,

    Hao, Y., Clark Di Leoni, P., Marxen, O., Meneveau, C., Karniadakis, G. E., and Zaki, T. A., “Instability-wave prediction in hypersonic boundary layers with physics-informed neural operators,”Journal of Computational Science, Vol. 73, 2023, p. 102120. https://doi.org/10.1016/j....

    2023

  15. [23]

    Data-Driven Reduced-Order Models Via Regularized Operator Inference for a Single-Injector Combustion Process,

    McQuarrie, S., C., S. H., and Willcox, K., “Data-Driven Reduced-Order Models Via Regularized Operator Inference for a Single-Injector Combustion Process,”Journal of the Royal Society of New Zealand, 2021, pp. 194–211. https: //doi.org/10.1080/03036758.2020.1863237

    2021

  16. [24]

    OperatorInference-BasedModelOrderReductionofThermalProtection SystemFiniteElementSimulations,

    Blonigan,P.J.,Tencer,J.,Babiniec,S.,andMurray,J.,“OperatorInference-BasedModelOrderReductionofThermalProtection SystemFiniteElementSimulations,” AIAASCITECH2025Forum ,Orlando,Florida,2025. https://doi.org/10.2514/6.2025-2133

    2025 doi

  17. [25]

    Expedient Hypersonic Aerothermal Prediction for Aerothermoelastic Analysis Via Field Inversion and Machine Learning,

    Venegas, C. V., and Huang, D., “Expedient Hypersonic Aerothermal Prediction for Aerothermoelastic Analysis Via Field Inversion and Machine Learning,” Virtual Event, 2021. https://doi.org/10.2514/6.2021-1707

    2021 doi

  18. [26]

    Physics-Infused Reduced Order Modeling of Hypersonic Aerothermal Loads for Aerothermoe- lastic Analysis,

    Venegas, C. V., and Huang, D., “Physics-Infused Reduced Order Modeling of Hypersonic Aerothermal Loads for Aerothermoe- lastic Analysis,” San Diego and Virtual, 2022. https://doi.org/10.2514/6.2022-0989

    2022 doi

  19. [27]

    Physics-Infused Reduced-Order Modeling of Aerothermal Loads for Hypersonic Aerothermoe- lastic Analysis,

    Venegas, C. V., and Huang, D., “Physics-Infused Reduced-Order Modeling of Aerothermal Loads for Hypersonic Aerothermoe- lastic Analysis,”AIAA Journal, Vol. 61, No. 3, 2023. https://doi.org/10.2514/1.J062214

    2023 doi

  20. [28]

    Development of Weak-Form Physics-Infused Reduced Order Modeling With Applications,

    Venegas, C. V., and Huang, D., “Development of Weak-Form Physics-Infused Reduced Order Modeling With Applications,” Orlando, FL, 2024. https://doi.org/10.2514/6.2024-0782

    2024 doi

  21. [29]

    Learning Coarse-Grained Dynamics on Graph,

    Yu, Y., Harlim, J., Huang, D., and Li, Y., “Learning Coarse-Grained Dynamics on Graph,”arXiv preprint arXiv:2405.09324,

    arXiv

  22. [30]

    A Unified Framework for Multiscale Modeling using the Mori-Zwanzig Formalism and the Variational Multiscale Method,

    Parish, E., and Duraisamy, K., “A Unified Framework for Multiscale Modeling using the Mori-Zwanzig Formalism and the Variational Multiscale Method,” 2017. https://doi.org/10.48550/arXiv.1712.09669

    2017 doi arXiv

  23. [31]

    Non-Markovian Closure Models for Large Eddy Simulations Using the Mori-Zwanzig Formalism,

    Parish, E., and Duraisamy, K., “Non-Markovian Closure Models for Large Eddy Simulations Using the Mori-Zwanzig Formalism,”Phys. Rev. Fluids, Vol. 2, 2017, p. 014604. https://doi.org/10.1103/PhysRevFluids.2.014604

    2017 doi

  24. [32]

    Higher-Order Mori-Zwanzig Models for the Euler Equations

    Stinis, P., “Higher-Order Mori-Zwanzig Models for the Euler Equations.”Multi-Scale Modeling and Simulation, Vol. 6, No. 3, 2007, pp. 741–760. https://doi.org/10.48550/arXiv.math/0607108. 45

    2007 doi arXiv

  25. [33]

    Data-Driven Learning for the Mori–Zwanzig Formalism: A Generalization of the Koopman Learning Framework,

    Lin, Y. T., Tian, Y., Livescu, D., and Anghel, M., “Data-Driven Learning for the Mori–Zwanzig Formalism: A Generalization of the Koopman Learning Framework,”SIAM Journal on Applied Dynamical Systems, Vol. 20, No. 4, 2021, pp. 2558–2601. https://doi.org/10.1137/21M1401759

    2021 doi

  26. [34]

    https://doi.org/10.1007/978-94-017-7761-2

    Cohen, G., and Pernet, S.,Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations, Springer Dordrecht, 2018. https://doi.org/10.1007/978-94-017-7761-2

    2018 doi

  27. [35]

    P., DeWitt, D

    Incropera, F. P., DeWitt, D. P., Bergman, T. L., and Lavine, A. S.,Fundamentals of Heat and Mass Transfer, 7th ed., Wiley, Hoboken, NJ, 2011

    2011

  28. [36]

    Parish, E., and Duraisamy, K.,Coarse-Graining Turbulence Using the Mori–Zwanzig Formalism, Cambridge University Press, 2025

    2025

  29. [37]

    Survey of Numerical Methods for Trajectory Optimization,

    Betts, J. T., “Survey of Numerical Methods for Trajectory Optimization,”Journal of Guidance Control Dynamics, Vol. 21, No. 2, 1998, pp. 193–207. https://doi.org/10.2514/2.4231

    1998 doi

  30. [38]

    Nonlinear Thermal Reduced-Order Modeling for Hypersonic Vehicles,

    Klock, R. J., and Cesnik, C. E. S., “Nonlinear Thermal Reduced-Order Modeling for Hypersonic Vehicles,”AIAA Journal, Vol. 55, No. 7, 2017, pp. 2358–2368. https://doi.org/10.2514/1.J055499

    2017 doi

  31. [39]

    SIERRA Multimechanics Module: Aria Verification Manual-Version 5.22,

    Clausen,J.,Brunini,V.,Collins,L.,Knaus,R.C.,Kucala,A.,Lin,S.,Moser,D.R.,Phillips,M.,Ransegnola,T.M.,Subia,S.R., et al., “SIERRA Multimechanics Module: Aria Verification Manual-Version 5.22,” Tech. rep., Sandia National Lab.(SNL-NM), Albuquerque, NM (United States), 2024

    2024

  32. [40]

    Automatic Differentiation in PyTorch,

    Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Antiga, L., and Lerer, A., “Automatic Differentiation in PyTorch,” Tech. rep., October 2017

    2017

  33. [41]

    Adam: AMethodforStochasticOptimization,

    Kingma,D.,andBa,J.,“Adam: AMethodforStochasticOptimization,” InternationalConferenceonLearningRepresentations ,

  34. [42]

    Learning Networked Dynamical System Models with Weak Form and Graph Neural Networks,

    Yu, Y., Huang, D., Park, S., and Pangborn, H., “Learning Networked Dynamical System Models with Weak Form and Graph Neural Networks,”arXiv preprint arXiv:2407.16779, 2024. https://doi.org/10.48550/arXiv.2407.16779. 46

    2024 doi arXiv

  35. [2014]

    https://doi.org/10.48550/arXiv.1412.6980

    doi arXiv

  36. [2021]

    https://doi.org/10.2514/6.2021-1675

    2021 doi

  37. [2024]

    https://doi.org/10.48550/arXiv.2405.09324

    doi arXiv

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.