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REVIEW 4 major objections 7 minor 79 references

Generalized Neural Operator for Parametric and Boundary-Value Problems

T0 review · 4 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper argues that explicitly feeding PDE parameters and boundary conditions to a neural operator removes the trade-off between generalization and speed.

desk verdict A plausible architecture with strong reported numbers, but the discrete parameter gate lacks a gradient path as written; needs a straight-through or relaxation and more transparency to be reproducible. read the letter →

arxiv 2607.21932 v1 pith:7BALYFCP submitted 2026-07-24 cs.CE cs.AI

Ruoyan Li , Yizhou Sun , Wei Wang This is my paper · ORCID
classification cs.CEcs.AI
keywords neuraloperatorspartialdifferentialequationsparametricPDEsboundaryconditionsmixtureofkernelsdistributionallyrobustoptimizationNavier-Stokesphysics-agnosticgeneralization
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 sets out to show that the standard trade-off in learned PDE solvers — between physics-agnostic deployment, physical fidelity, and inference speed — is avoidable if the solver is explicitly told the PDE parameters and boundary conditions instead of being left to infer them from data. It proposes a Generalized Neural Operator built from three parts: a parameter-gated mixture of kernels that picks one integration kernel per physical regime, a boundary transfer operator that converts arbitrary boundary conditions into a shared latent Dirichlet representation, and a group distributionally robust training objective that equalizes accuracy across parameter groups. The paper also argues formally that a single-valued neural operator mapping only initial conditions to solutions cannot represent a family of solution operators, because one initial condition can produce different trajectories under different configurations. On the paper's datasets the method reduces normalized mean-squared error to $5.23\times10^{-5}$ on incompressible Navier-Stokes, against $1.68\times10^{-2}$ for the strongest baseline, while running roughly 2–4x faster than conventional numerical solvers at matched accuracy. A sympathetic reader would take the point to be that rigorous conditioning is what unlocks both generalization and efficiency.

What carries the argument

The load-bearing mechanism is the parameter-gated mixture of kernels: rather than applying one static integral kernel at each layer, the network holds $K$ base kernels $\kappa_k$ and a lightweight score function $s_k(\theta)$, and a hard gate $g_k(\theta)\in\{0,1\}$ selects exactly one kernel per layer based on the PDE parameter $\theta$. This lets the operator assemble a different integration geometry for each physical regime — laminar versus turbulent, diffuse versus sharp — without activating the whole network. Around that core sit two supporting mechanisms: the generalized boundary transfer operator $T_\lambda(B,v^{(\ell)})$, which maps boundary type and boundary data together with the current latent state into a single latent Dirichlet vector $h^{(\ell)}$ used as a conditioning input, and the Group DRO objective, which after an MSE warm-up trains against the worst parameter group in each batch. The update rule is $$$v^{{(\ell+1)}}$(x)=\$\sigma$\Big(W\langle\tilde\$\theta$,$h^{{(\ell)}}$,$v^{{(\ell)}}$(x)\rangle+\int_{\$\Omega$\cup X_{\mathrm{ctx}}}\Big(\textstyle\sum_{k=1}^{K}g_k(\$\theta$)\kappa_k(x,y)\Big)\xi(y)\,d\mu(y)+b(x)\Big),$$ with $\xi(y)$ carrying either the latent state or the conditioning information. Proposition 4.1 supplies the theoretical job: it forces the domain to include the configuration that disambiguates trajectories, which is why the gating and boundary transfer are placed where they are.

What would settle it

Train the same architecture on a high-fidelity dataset whose boundary conditions include types outside the training set (for example, Robin or mixed-type conditions) and whose parameter values are drawn sparsely between the training grid points; if the normalized error on those cases is no better than a single-kernel baseline that receives the same explicit inputs, then the claim that the parameter-gated mixture and boundary transfer operator are the cause of the generalization would be falsified.

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Extended reading notes

Core claim

The paper's central claim is that the well-posedness conditions of the underlying PDE family should be mirrored inside the neural operator: the operator's domain must include the PDE parameters $\theta$ and the boundary operator $B$, not just the initial condition $u_0$. Proposition 4.1 states that a single-valued map $M_\phi: X_{\mathrm{init}}\to Y$ cannot universally represent the family $\mathcal{S}=\{S_{\theta,B}\}$, because a fixed initial condition can yield distinct trajectories under distinct configurations, and a single-valued operator would have to output both. The proposed Generalized Neural Operator therefore learns $M_\phi: X_{\mathrm{init}}\times\Theta\times\mathcal{B}\to Y$, with kernel integration re-routed per parameter value, boundary data entering through a learned transfer operator that approximates a latent Dirichlet value, and a worst-group training objective. Empirically, the paper reports normalized MSE well below all compared baselines across Heat, Advection, Burgers, and incompressible Navier-Stokes equations, with the largest gain on Navier-Stokes ($5.23\times10^{-5}$ versus $1.68\times10^{-2}$ for the best baseline), and an accuracy-matched runtime advantage of roughly 2–4x over conventional numerical solvers.

Load-bearing premise

The load-bearing premise is that the self-generated training data — with dense parameter grids and Gaussian-random initial fields — represents the variety of real PDE problems, and that the exact PDE parameters and boundary conditions are always known at inference time; if either condition fails, the reported margins could shrink or disappear.

Editorial extensions

If this is right

  • A single trained model can cover a whole family of PDE configurations — varying diffusivities, viscosities, Reynolds numbers, velocity fields, and boundary types — without instance-specific retraining or fine-tuning.
  • Because exactly one kernel is active per layer, the added generalization capacity costs almost nothing at inference, so the model can stay within a small constant factor of traditional numerical solvers.
  • Boundary conditions of different types can be treated as one modality after the transfer operator, which is why the model is not tied to a single boundary geometry or a fixed boundary type.
  • Training against the worst parameter group rather than the average removes the incentive to specialize on easy regimes, so accuracy becomes more uniform across the parameter range.
  • If the formal argument is right, purely implicit, data-only operator learning for parametric families is not just practically risky but underdetermined, and any foundational PDE model should include explicit physical configuration inputs.

Reading between the lines

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

  • The non-degeneracy argument generalizes beyond PDEs: any single-valued learned simulator whose output depends on an exogenous conditioning variable — a material property, a control input, an external forcing — is underdetermined if that variable is withheld, so the same explicit-conditioning design should apply to other scientific surrogate models.
  • The hard-gate scores could be repurposed as interpretable diagnostics: if the chosen kernel correlates with Reynolds number or diffusivity, the model is implicitly learning regime boundaries, and one could test whether those boundaries match physical transition thresholds.
  • The runtime comparison is sensitive to how the numerical baseline is calibrated; testing on a high-resolution three-dimensional turbulent problem, where the cheap numerical solver is much less accurate, would show whether the 2–4x speedup survives in the regime where neural solvers are most needed.
  • A natural robustness extension is to feed the transfer operator a posterior over parameters and boundary data rather than point values; if the model degrades gracefully under uncertainty, it would broaden the method beyond exactly-known configurations.
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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

4 major / 7 minor

Summary. This paper proposes a Generalized Neural Operator (GNO) that conditions a neural PDE solver explicitly on PDE parameters and boundary conditions. The architecture uses a parameter-gated mixture of kernels, a learned boundary transfer operator producing a latent Dirichlet state, and a Group DRO training objective. The paper gives a short proposition (Prop. 4.1) on the impossibility of a single-valued operator from initial conditions alone, and reports experiments on Heat, Advection, Burgers, and incompressible Navier-Stokes equations plus a public 2D Turbulent Radiative Layer dataset, with lower normalized MSE than several baselines and 2-4x speedups over accuracy-matched numerical solvers.

Significance. If the results held as reported, GNO would be a practical contribution to parametric and boundary-condition-aware neural operators. Strengths include the breadth of the evaluations (four PDE families, a public complex PDE, OOD and 3D tests), the ablation study in Appendix J.1, the OOD experiment in Appendix L, and the honest discussion of the missing community benchmark in Appendix B. The main weaknesses are that the core gating mechanism is not specified in a trainable form, the headline results lack uncertainty quantification, and the theoretical formalization is much weaker than the framing suggests. These are load-bearing for the paper's central claims, so the manuscript needs major revision.

major comments (4)
  1. [Section 4.2, Eqs. (5)-(6), Appendix E] The hard gate g_k(θ) is defined in Eq. (6) as the one-hot argmax of the softmax scores, and in Eq. (5) these discrete gates multiply the kernel sum inside the integral. Under standard backpropagation, the gradient of g_k with respect to the score parameters is zero almost everywhere, so the gating MLP cannot learn to partition parameter space. The manuscript does not mention a straight-through estimator, Gumbel-softmax, REINFORCE, or a soft-gating training phase; Appendix E only states that the gating function is a lightweight MLP with hidden size 32. Please specify the exact training scheme for the gates, or if a relaxation was used, state it explicitly and provide an ablation of the relaxed versus the hard formulation.
  2. [Tables 1-3, Section 5.1] The central empirical comparisons are reported as single-point nMSE values without error bars, number of seeds, or significance tests. The uncertainty issue matters most for Table 1, where the incompressible Navier-Stokes margin (5.23e-5 vs. 1.68e-2) is orders of magnitude larger than the gains on other equations and is attributed only to a hypothesis about 'modest parameter variation' in Section 5.1. Please report mean ± standard deviation over at least three seeds for all tables, and provide per-parameter error curves for the proposed model on the Navier-Stokes dataset, analogous to Figure 4, to support the claimed margin.
  3. [Appendix C and Appendix B] The boundary-condition and joint parameter-boundary experiments rest entirely on self-generated datasets; the public Turbulent Radiative Layer validation in Appendix N varies only PDE parameters with fixed boundaries, so it does not exercise the generalized boundary transfer operator. The authors acknowledge the absence of a community benchmark in Appendix B, but this is a direct limitation on the abstract's claim of generalization across boundary conditions. Please either validate on an independent dataset with boundary-condition variation or weaken the claims about boundary-condition generalization in the abstract and Section 5.2.
  4. [Proposition 4.1, Appendix O] The proof of Proposition 4.1 is a set-theoretic argument: under Assumption O.3, a single-valued map from u0 alone cannot reproduce two different solution trajectories for the same initial condition. This observation is correct, but it does not formalize well-posedness in the Hadamard sense (existence, uniqueness, continuous dependence), and it gives no approximation or generalization bound for the proposed architecture. The abstract and contribution (ii) therefore overstate what the theory establishes; please either supply a substantive theoretical analysis or revise the claims to say that explicit conditioning makes the learning problem well-posed rather than 'theoretically grounded' in the sense claimed.
minor comments (7)
  1. [Abstract] The manuscript contains two nearly identical abstracts, one at the top and one after the author list; please remove the duplicate.
  2. [Figure 1] Figure 1 contains garbled embedded text (for example '0QUJNBM' and 'EFFJDJFODZ'), indicating a rendering problem; please regenerate the figure.
  3. [Appendix C] In the incompressible Navier-Stokes paragraph, the text says the range is for the Reynolds number and then refers to 'each specific kinematic viscosity'; since the two are related by ν = uL/Re, please state unambiguously which parameter is varied.
  4. [Appendix E] The CAPE+Attn implementation description says the CAPE representation is 'then fed into the FNO', which appears to be a copy-paste error from CAPE+FNO; it should refer to the attention-based model.
  5. [Table 11] Table 11's method column is labeled 'Advection' although the table reports the 2D Turbulent Radiative Layer dataset; please correct the column header.
  6. [Section 6] The first sentence of the conclusion contains a stray 'B' ('BWe achieve'); please fix the typo.
  7. [Section 4.4, Eq. (10)] The DRO temperature τ and the number/size of parameter bins used in the grouping are not reported anywhere; please provide these hyperparameters for reproducibility.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-definitional flavor in Prop. 4.1; central empirical claims are independent, and self-citations are not load-bearing.

  1. self definitional [Section 4.1, Proposition 4.1 and Appendix O, Assumption O.3 (Eq. 11)]
    "Assumption O.3 (Non-Degeneracy of PDE Configuration): ... If an initial condition u0 sufficiently excites the system, then there exist at least two distinct configurations c1, c2 ∈ C whose corresponding solution trajectories are not identical: ∃c1, c2 ∈ C, c1 ≠ c2 such that S_c1(u0) ≠ S_c2(u0). ... Proposition 4.1. ... a single-valued operator Mφ : X_init → Y defined solely on the initial condition space cannot universally represent the family S."

    The proposition does not derive the need for explicit conditioning from well-posedness; it restates Assumption O.3. Given O.3, the proof's contradiction (S_c1(u0) = y* = S_c2(u0)) follows immediately from the definition of a single-valued map. The non-degeneracy assumption already asserts that the same initial condition leads to different solutions under different (θ, B), which is precisely the content of the proposition. Thus the theoretical benefit claimed in the Abstract — that formalizing classical well-posedness conditions 'demonstrates the theoretical benefits of explicitly conditioning on PDE parameters and boundary conditions' — is an assumption repackaged as a theorem rather than an independent result.

full rationale

The paper's central empirical claims are obtained by a standard supervised-learning pipeline: a conditional operator Mφ(u0, θ, B) is trained with MSE and a Group DRO objective on self-generated datasets, then evaluated against external baselines. No reported nMSE is produced by inserting a fitted constant into the evaluation metric; the reported numbers are measured on held-out parameter grids and boundary setups. The boundary transfer operator is openly described as a learned surrogate for the ground-truth trace (Eq. 7 approximated by Eq. 8), which is a distillation pattern rather than a hidden identity. The hard-gating issue in Eqs. 5–6 (argmax gives no gradient path for the gating scores as written) is a reproducibility and correctness concern, not a circularity in the sense of Eq. X reducing to Eq. Y by construction. The self-citations (Wang et al. 2025a; Li et al. 2025b) appear in runtime methodology, kernel-choice justification, and benchmark motivation, but none of the paper's main generalization results is forced by those citations alone; the architecture choices remain testable against baselines. The only genuine definitional short-circuit is Proposition 4.1, whose content is essentially contained in Assumption O.3; since this proposition is not the source of the reported performance gains, the overall circularity is minor.

Assumptions & free parameters 3 free parameters · 4 assumptions · 1 invented entities

The method contributes a learned latent representation and a training scheme; the theory is a well-posedness argument that assumes the standard well-posedness of the underlying PDEs, not a first-principles derivation of the architecture choices.

free parameters (3)
  • Number of kernels K = 4 (3 for Navier-Stokes)
    Chosen by ablation in Appendix J.2; not derived from theory.
  • DRO switching epoch P = 70
    Selected from a validation sweep in Table 7 to balance training stability and robust optimization time.
  • DRO temperature tau = not reported
    Equation 10 includes an unspecified temperature hyperparameter; its value affects the shape of the loss and is not given.
assumptions (4)
  • domain assumption Assumption O.1: Existence of solutions for every initial condition
    States that every u0 has a realizable solution in Y; used in the proof of Prop. 4.1 (Appendix O).
  • domain assumption Assumption O.2: Uniqueness of solutions
    Assumes S_theta,B is single-valued; without it the operator learning problem is one-to-many (Appendix O).
  • domain assumption Assumption O.3: Non-degeneracy of PDE configuration, i.e., there exist c1 and c2 with different solution trajectories for the same initial condition
    This is the load-bearing premise of Prop. 4.1; if parameters and boundary conditions did not affect solutions, the theorem would be vacuous (Appendix O).
  • domain assumption Fixed structural form of the differential operator L; only parameters theta and boundary operator B vary
    Section 3 explicitly restricts to a fixed L with varying parameters and boundary operators; the framework does not generalize across PDE families.
invented entities (1)
  • Latent Dirichlet boundary value h
    purpose: Unifies arbitrary boundary conditions into a single latent Dirichlet representation so the kernel only handles one boundary type
    Defined via the trace of the ground truth solution (Eq. 7), then approximated by a learned transfer operator (Eq. 8); an internal representation with no external falsifiable handle.

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Pith. "Pith review of Generalized Neural Operator for Parametric and Boundary-Value Problems." pith.science (2026). https://pith.science/paper/7BALYFCP

@misc{pith2026260721932,
  author       = {Pith},
  title        = {Pith review of: Generalized Neural Operator for Parametric and Boundary-Value Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7BALYFCP}},
  note         = {Machine review of arXiv:2607.21932}
}
read the original abstract

Developing foundational neural simulators for Partial Differential Equations (PDEs) requires robust generalization across diverse physical parameters and boundary conditions. However, current deep learning approaches largely face a structural trade-off between condition-agnostic deployment and physical fidelity. Purely data-driven operators infer the underlying physics implicitly and thus lack the explicit constraints needed to ensure physically valid solutions across varying domains, rendering the learning problem ill-posed. On the other hand, Physics-Informed Neural Networks (PINNs) enforce rigorous physical constraints but necessitate costly, instance-specific optimization. Furthermore, the massive scale of emerging foundational operators has severely degraded their inference speeds, making them computationally uncompetitive with traditional numerical solvers. To address this bottleneck between condition-agnostic deployment, physical rigor, and inference efficiency, we propose a \textit{Generalized Neural Operator}. By formalizing the classical conditions for well-posedness within neural operators, our framework demonstrates the theoretical benefits of explicitly conditioning on PDE parameters and boundary conditions. To implement this synthesis without compromising computational speed, we introduce three novel architectural components: a parameter-gated mixture of kernels for efficient parameter generalization, a generalized boundary transfer operator that projects arbitrary boundary constraints into a unified latent Dirichlet representation, and a specialized training objective to ensure stability. Extensive experiments demonstrate that our theoretically grounded approach achieves superior generalization across heterogeneous physical regimes while maintaining strict inference efficiency comparable to conventional numerical baselines.

Figures

Figures reproduced from arXiv: 2607.21932 by the authors.

Figure 1
Figure 1. Trilemma of existing PDE solvers. Tra￾ditional numerical solvers guarantee mathematical rigor but are computationally expensive and de￾mand instance-specific design. PINNs maintain this rigor and offer efficient inference, yet still re￾quire instance-specific retraining. Neural operators achieve favorable accuracy-efficiency tradeoffs and physics-agnostic deployment, but often yield ill￾posed predictions. Existing f… view at source ↗
Figure 2
Figure 2. (a) Lack of physical inputs renders the problem ill-posed, causing the model to converge toward an averaged solution. (b) We propose a Generalized Neural Operator, which incorporates a parameter-gated mixture of kernels and a robust optimization training objective to improve gen￾eralization. (c) To enhance boundary generalization, we introduce a generalized boundary transfer operator that maps arbitrary boundary con… view at source ↗
Figure 3
Figure 3. Runtime comparison between our GNO and numerical solvers. The ground truth, operating at a 128×128 resolution, is denoted as Solver-128. For our accuracy-matched baselines, we utilize Solver-70 and Solver-100, which operate at 70×70 and 100 × 100 resolutions, respectively [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Error Trends vs. PDE Parameters on Incompressible Navier–Stokes dataset for [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Standard deviation of model performance on the Advection equation in generalization [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

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Works this paper leans on

79 extracted references · 43 canonical work pages

  1. [1]

    Hyperfno: Improving the generalization behavior of fourier neural operators

    Alesiani, F., Takamoto, M., and Niepert, M. Hyperfno: Improving the generalization behavior of fourier neural operators. In Workshop on Machine Learning and the Physical Sciences (ML4PS) at the 36th Conference on Neural Information Processing Systems (NeurIPS), 2022. URL https://ml4physicalsciences.github.io/2022/files/NeurIPS_ML4PS_2022_89.pdf

    work page 2022

  2. [2]

    Universal physics transformers

    Alkin, B., Fürst, A., Schmid, S., Gruber, L., Holzleitner, M., and Brandstetter, J. Universal physics transformers. arXiv preprint arXiv:2402.12365, 2024

    arXiv 2024

  3. [3]

    Arendt, W., ter Elst, A. F. M., Kennedy, J. B., and Sauter, M. The dirichlet-to-neumann operator via hidden compactness. Journal of Functional Analysis, 266: 0 1757--1786, 2014

    work page 2014

  4. [4]

    Flow matching meets pdes: A unified framework for physics-constrained generation, 2025

    Baldan, G., Liu, Q., Guardone, A., and Thuerey, N. Flow matching meets pdes: A unified framework for physics-constrained generation, 2025. URL https://arxiv.org/abs/2506.08604

    arXiv 2025

  5. [5]

    T., Duvenaud, D., and Jacobsen, J.-H

    Behrmann, J., Grathwohl, W., Chen, R. T., Duvenaud, D., and Jacobsen, J.-H. Invertible residual networks. In International Conference on Machine Learning, pp.\ 573--582. PMLR, 2019

    work page 2019

  6. [6]

    Distributionally robust optimization and robust statistics, 2024

    Blanchet, J., Li, J., Lin, S., and Zhang, X. Distributionally robust optimization and robust statistics, 2024. URL https://arxiv.org/abs/2401.14655

    arXiv 2024

  7. [7]

    The 'scalar' poincar \'e --steklov operator and the 'vector' one: algebraic structures which underlie their duality

    Bossavit, A. The 'scalar' poincar \'e --steklov operator and the 'vector' one: algebraic structures which underlie their duality. In Glowinski, R. (ed.), Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, pp.\ 19--26, Philadelphia, PA, 1991. SIAM. ISBN 978-0-89871-278-0

    work page 1991

  8. [8]

    L., de Bezenac, E., Serrano, L., Regueiro-Espino, R

    Boudec, L. L., de Bezenac, E., Serrano, L., Regueiro-Espino, R. D., Yin, Y., and Gallinari, P. Learning a neural solver for parametric PDE s to enhance physics-informed methods. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=jqVj8vCQsT

    work page 2025

Show all 79 references
  1. [9]

    Message passing neural pde solvers

    Brandstetter, J., Worrall, D., and Welling, M. Message passing neural pde solvers. arXiv preprint arXiv:2202.03376, 2022

    2022 arXiv

  2. [10]

    Omniarch: Building foundation model for scientific computing

    Chen, T., Zhou, H., Li, Y., Wang, H., Gao, C., Shi, R., Zhang, S., and Li, J. Omniarch: Building foundation model for scientific computing. In Forty-second International Conference on Machine Learning, 2025. URL https://openreview.net/forum?id=UlprLwWYKP

    2025

  3. [11]

    A., and Patel, R

    Deighan, D., Actor, J. A., and Patel, R. G. Mixture of neural operator experts for learning boundary conditions and model selection, 2025. URL https://arxiv.org/abs/2502.04562

    2025 arXiv

  4. [12]

    An image is worth 16x16 words: Transformers for image recognition at scale, 2021

    Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn, D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M., Heigold, G., Gelly, S., Uszkoreit, J., and Houlsby, N. An image is worth 16x16 words: Transformers for image recognition at scale, 2021. URL https://arxiv.org/ab...

    2021 arXiv

  5. [13]

    Drivaernet++: A large-scale multimodal car dataset with computational fluid dynamics simulations and deep learning benchmarks

    Elrefaie, M., Morar, F., Dai, A., and Ahmed, F. Drivaernet++: A large-scale multimodal car dataset with computational fluid dynamics simulations and deep learning benchmarks. In Globerson, A., Mackey, L., Belgrave, D., Fan, A., Paquet, U., Tomczak, J., and Zhang, C. (eds.), Ad...

    2024

  6. [14]

    Drivaernet: A parametric car dataset for data-driven aerodynamic design and prediction

    Elrefaie, M., Dai, A., and Ahmed, F. Drivaernet: A parametric car dataset for data-driven aerodynamic design and prediction. Journal of Mechanical Design, 147 0 (4), 2025

    2025

  7. [15]

    L., Yanıkoglu, I., and den Hertog, D

    Gorissen, B. L., Yanıkoglu, I., and den Hertog, D. A practical guide to robust optimization. Omega, 53: 0 124–137, June 2015. ISSN 0305-0483. doi:10.1016/j.omega.2014.12.006. URL http://dx.doi.org/10.1016/j.omega.2014.12.006

    2015 doi

  8. [16]

    Learning to optimize multigrid PDE solvers

    Greenfeld, D., Galun, M., Basri, R., Yavneh, I., and Kimmel, R. Learning to optimize multigrid PDE solvers. In International Conference on Machine Learning, pp.\ 2415--2423. PMLR, 2019

    2019

  9. [17]

    Dpot: Auto-regressive denoising operator transformer for large-scale pde pre-training

    Hao, Z., Su, C., Liu, S., Berner, J., Ying, C., Su, H., Anandkumar, A., Song, J., and Zhu, J. Dpot: Auto-regressive denoising operator transformer for large-scale pde pre-training. arXiv preprint arXiv:2403.03542, 2024

    2024 arXiv

  10. [18]

    Flow completion network: Inferring the fluid dynamics from incomplete flow information using graph neural networks

    He, X., Wang, Y., and Li, J. Flow completion network: Inferring the fluid dynamics from incomplete flow information using graph neural networks. Physics of Fluids, 34 0 (8), August 2022. ISSN 1089-7666. doi:10.1063/5.0097688. URL http://dx.doi.org/10.1063/5.0097688

    2022 doi

  11. [19]

    Poseidon: Efficient foundation models for pdes, 2024

    Herde, M., Raonić, B., Rohner, T., Käppeli, R., Molinaro, R., de Bézenac, E., and Mishra, S. Poseidon: Efficient foundation models for pdes, 2024

    2024

  12. [20]

    Huang, J., Yang, G., Wang, Z., and Park, J. J. Diffusionpde: Generative pde-solving under partial observation, 2024. URL https://arxiv.org/abs/2406.17763

    2024 arXiv

  13. [21]

    An airflow velocity field reconstruction method with sparse or incomplete data using physics-informed neural network

    Jing, G., Wang, H., Li, X., Wang, G., and Yang, Y. An airflow velocity field reconstruction method with sparse or incomplete data using physics-informed neural network. Journal of Building Engineering, 88: 0 109231, July 2024. Published: 1 July 2024

    2024

  14. [22]

    Knockaert, L. F. and De Zutter, D. On the complex symmetry of the poincar \'e -steklov operator. Progress in Electromagnetics Research B, 7: 0 145--157, 2008. doi:10.2528/PIERB08022102

    2008 doi

  15. [23]

    K., Boudec, L

    Koupa \" , A. K., Boudec, L. L., Serrano, L., and Gallinari, P. ENMA : Tokenwise autoregression for continuous neural PDE operators. In The Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025. URL https://openreview.net/forum?id=3CYXSMFv55

    2025

  16. [24]

    K., Benet, J

    Koupaï, A. K., Benet, J. M., Yin, Y., Vittaut, J.-N., and Gallinari, P. Geps: Boosting generalization in parametric pde neural solvers through adaptive conditioning, 2024. URL https://arxiv.org/abs/2410.23889

    2024 arXiv

  17. [25]

    Neural operator: Learning maps between function spaces with applications to pdes

    Kovachki, N., Li, Z., Liu, B., Azizzadenesheli, K., Bhattacharya, K., Stuart, A., and Anandkumar, A. Neural operator: Learning maps between function spaces with applications to pdes. JMLR, 24 0 (1), 2023

    2023

  18. [26]

    Benchmarking invertible architectures on inverse problems

    Kruse, J., Ardizzone, L., Rother, C., and K \"o the, U. Benchmarking invertible architectures on inverse problems. arXiv preprint, 2021

    2021

  19. [27]

    Distributionally robust optimization, 2025

    Kuhn, D., Shafiee, S., and Wiesemann, W. Distributionally robust optimization, 2025. URL https://arxiv.org/abs/2411.02549

    2025 arXiv

  20. [28]

    Li, E., Wang, Z., Huang, J., and Park, J. J. Video PDE : Unified generative PDE solving via video inpainting diffusion models, 2025 a . URL https://openreview.net/forum?id=CtBKITrZgp

    2025

  21. [29]

    Self-guided diffusion model for accelerating computational fluid dynamics, 2025 b

    Li, R., Huang, Z., Wang, H., Wan, G., Sun, Y., and Wang, W. Self-guided diffusion model for accelerating computational fluid dynamics, 2025 b . URL https://arxiv.org/abs/2504.04375

    2025

  22. [30]

    Flow field reconstruction with sensor placement policy learning

    Li, R., Wan, G., Huang, Z., Liu, Z., Wang, H., Luo, X., Wang, W., and Sun, Y. Flow field reconstruction with sensor placement policy learning. In The Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025 c . URL https://openreview.net/forum?id=1esFEjMUBS

    2025

  23. [31]

    B., Azizzadenesheli, K., liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A

    Li, Z., Kovachki, N. B., Azizzadenesheli, K., liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A. Fourier neural operator for parametric partial differential equations. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=c8P9NQVtmnO

    2021

  24. [32]

    B., Choy, C., Li, B., Kossaifi, J., Otta, S

    Li, Z., Kovachki, N. B., Choy, C., Li, B., Kossaifi, J., Otta, S. P., Nabian, M. A., Stadler, M., Hundt, C., Azizzadenesheli, K., and Anandkumar, A. Geometry-informed neural operator for large-scale 3d pdes, 2023 a . URL https://arxiv.org/abs/2309.00583

    2023 arXiv

  25. [33]

    Physics-informed neural operator for learning partial differential equations, 2023 b

    Li, Z., Zheng, H., Kovachki, N., Jin, D., Chen, H., Liu, B., Azizzadenesheli, K., and Anandkumar, A. Physics-informed neural operator for learning partial differential equations, 2023 b . URL https://arxiv.org/abs/2111.03794

    2023 arXiv

  26. [34]

    Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimisation horizons

    List, B., Chen, L.-W., and Thuerey, N. Learned turbulence modelling with differentiable fluid solvers: physics-based loss functions and optimisation horizons. Journal of Fluid Mechanics, 949: 0 A25, 2022

    2022

  27. [35]

    Transolver++: An accurate neural solver for PDE s on million-scale geometries

    Luo, H., Wu, H., Zhou, H., Xing, L., Di, Y., Wang, J., and Long, M. Transolver++: An accurate neural solver for PDE s on million-scale geometries. In Forty-second International Conference on Machine Learning, 2025. URL https://openreview.net/forum?id=AM7iAh0krx

    2025

  28. [36]

    R.-S., Parker, L

    McCabe, M., Blancard, B. R.-S., Parker, L. H., Ohana, R., Cranmer, M., Bietti, A., Eickenberg, M., Golkar, S., Krawezik, G., Lanusse, F., Pettee, M., Tesileanu, T., Cho, K., and Ho, S. Multiple physics pretraining for spatiotemporal surrogate models. In The Thirty-eighth Annua...

    2024

  29. [37]

    and Hakim, A

    McGreivy, N. and Hakim, A. Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations. Nature Machine Intelligence, 6 0 (10): 0 1256–1269, September 2024. ISSN 2522-5839. doi:10.1038/s42256-024-00897-5. URL htt...

    2024 doi

  30. [38]

    and Magri, L

    Mo, Y. and Magri, L. Reconstructing unsteady flows from sparse, noisy measurements with a physics-constrained convolutional neural network, 2024. URL https://arxiv.org/abs/2409.00260

    2024 arXiv

  31. [39]

    D., Barton, D

    Nzoyem, R. D., Barton, D. A., and Deakin, T. Neural context flows for meta-learning of dynamical systems. In The Thirteenth International Conference on Learning Representations, 2025. URL https://openreview.net/forum?id=8vzMLo8LDN

    2025

  32. [40]

    A., Malhotra, M., and Pinsky, P

    Oberai, A. A., Malhotra, M., and Pinsky, P. M. On the implementation of the dirichlet-to-neumann radiation condition for iterative solution of the helmholtz equation. Applied Numerical Mathematics, 27 0 (4): 0 443--464, 1998. doi:10.1016/S0168-9274(98)00024-5

    1998 doi

  33. [41]

    The well: a large-scale collection of diverse physics simulations for machine learning

    Ohana, R., McCabe, M., Meyer, L., Morel, R., Agocs, F., Beneitez, M., Berger, M., Burkhart, B., Dalziel, S., Fielding, D., et al. The well: a large-scale collection of diverse physics simulations for machine learning. Advances in Neural Information Processing Systems, 37: 0 44...

    2024

  34. [42]

    Pytorch: An imperative style, high-performance deep learning library, 2019

    Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Köpf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., and Chintala, S. Pytorch: An impera...

    2019 arXiv

  35. [43]

    Film: Visual reasoning with a general conditioning layer, 2017

    Perez, E., Strub, F., de Vries, H., Dumoulin, V., and Courville, A. Film: Visual reasoning with a general conditioning layer, 2017. URL https://arxiv.org/abs/1709.07871

    2017 arXiv

  36. [44]

    Learning mesh-based simulation with graph networks

    Pfaff, T., Fortunato, M., Sanchez-Gonzalez, A., and Battaglia, P. Learning mesh-based simulation with graph networks. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=roNqYL0_XP

    2021

  37. [45]

    Guaranteed conservation of momentum for learning particle-based fluid dynamics

    Prantl, L., Ummenhofer, B., Koltun, V., and Thuerey, N. Guaranteed conservation of momentum for learning particle-based fluid dynamics. In Conference on Neural Information Processing Systems, 2022

    2022

  38. [46]

    and Valli, A

    Quarteroni, A. and Valli, A. Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, 05 1999. ISBN 9780198501787. doi:10.1093/oso/9780198501787.001.0001. URL https://doi.org/10.1093/oso/9780198501787.001.0001

    1999

  39. [47]

    and Mehrotra, S

    Rahimian, H. and Mehrotra, S. Frameworks and results in distributionally robust optimization. Open Journal of Mathematical Optimization, 3: 0 1–85, July 2022. ISSN 2777-5860. doi:10.5802/ojmo.15. URL http://dx.doi.org/10.5802/ojmo.15

    2022 doi

  40. [48]

    A., Ross, Z

    Rahman, M. A., Ross, Z. E., and Azizzadenesheli, K. U- NO : U-shaped neural operators. Transactions on Machine Learning Research, 2023. ISSN 2835-8856. URL https://openreview.net/forum?id=j3oQF9coJd

    2023

  41. [49]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Raissi, M., Perdikaris, P., and Karniadakis, G. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378: 0 686--707, 2019. ISSN 0021-9991. doi...

    2019 doi

  42. [50]

    U-net: Convolutional networks for biomedical image segmentation, 2015

    Ronneberger, O., Fischer, P., and Brox, T. U-net: Convolutional networks for biomedical image segmentation, 2015. URL https://arxiv.org/abs/1505.04597

    2015 arXiv

  43. [51]

    Saad, N., Gupta, G., Alizadeh, S., and Maddix, D. C. Guiding continuous operator learning through physics-based boundary constraints. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=gfWNItGOES6

    2023

  44. [52]

    Sanchez-Gonzalez, A., Godwin, J., Pfaff, T., Ying, R., Leskovec, J., and Battaglia, P. W. Learning to simulate complex physics with graph networks. In International Conference on Machine Learning, 2020

    2020

  45. [53]

    Deep learning of preconditioners for conjugate gradient solvers in urban water related problems

    Sappl, J., Seiler, L., Harders, M., and Rauch, W. Deep learning of preconditioners for conjugate gradient solvers in urban water related problems. arXiv preprint, 2019

    2019

  46. [54]

    Sauter, M. S. Degenerate Elliptic Operators with Boundary Conditions via Form Methods. PhD thesis, The University of Auckland, 2013. URL http://hdl.handle.net/2292/21318

    2013

  47. [55]

    UPS : Efficiently building foundation models for PDE solving via cross-modal adaptation

    Shen, J., Marwah, T., and Talwalkar, A. UPS : Efficiently building foundation models for PDE solving via cross-modal adaptation. Transactions on Machine Learning Research, 2024. ISSN 2835-8856. URL https://openreview.net/forum?id=0r9mhjRv1E

    2024

  48. [56]

    W., and Gholami, A

    Subramanian, S., Harrington, P., Keutzer, K., Bhimji, W., Morozov, D., Mahoney, M. W., and Gholami, A. Towards foundation models for scientific machine learning: Characterizing scaling and transfer behavior. In Thirty-seventh Conference on Neural Information Processing Systems...

    2023

  49. [57]

    Towards a foundation model for partial differential equations: Multi-operator learning and extrapolation, 2025

    Sun, J., Liu, Y., Zhang, Z., and Schaeffer, H. Towards a foundation model for partial differential equations: Multi-operator learning and extrapolation, 2025. URL https://arxiv.org/abs/2404.12355

    2025 arXiv

  50. [58]

    A neural PDE solver with temporal stencil modeling

    Sun, Z., Yang, Y., and Yoo, S. A neural PDE solver with temporal stencil modeling. arXiv preprint, 2023

    2023

  51. [59]

    Learning neural pde solvers with parameter-guided channel attention, 2023

    Takamoto, M., Alesiani, F., and Niepert, M. Learning neural pde solvers with parameter-guided channel attention, 2023. URL https://arxiv.org/abs/2304.14118

    2023 arXiv

  52. [60]

    Pdebench: An extensive benchmark for scientific machine learning, 2024

    Takamoto, M., Praditia, T., Leiteritz, R., MacKinlay, D., Alesiani, F., Pflüger, D., and Niepert, M. Pdebench: An extensive benchmark for scientific machine learning, 2024. URL https://arxiv.org/abs/2210.07182

    2024 arXiv

  53. [61]

    and Choromanska, A

    Teng, Y. and Choromanska, A. Invertible autoencoder for domain adaptation. Computation, 7 0 (2): 0 20, 2019

    2019

  54. [62]

    Lagrangebench: A lagrangian fluid mechanics benchmarking suite

    Toshev, A., Galletti, G., Fritz, F., Adami, S., and Adams, N. Lagrangebench: A lagrangian fluid mechanics benchmarking suite. Advances in Neural Information Processing Systems, 36, 2024 a

    2024

  55. [63]

    P., Erbesdobler, J

    Toshev, A. P., Erbesdobler, J. A., Adams, N. A., and Brandstetter, J. Neural sph: Improved neural modeling of lagrangian fluid dynamics, 2024 b . URL https://arxiv.org/abs/2402.06275

    2024 arXiv

  56. [64]

    Tran, A., Mathews, A., Xie, L., and Ong, C. S. Factorized fourier neural operators. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=tmIiMPl4IPa

    2023

  57. [65]

    N., Kaiser, L., and Polosukhin, I

    Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A. N., Kaiser, L., and Polosukhin, I. Attention is all you need, 2023. URL https://arxiv.org/abs/1706.03762

    2023 arXiv

  58. [66]

    BENO : Boundary-embedded neural operators for elliptic PDE s

    Wang, H., Jiaxin, L., Dwivedi, A., Hara, K., and Wu, T. BENO : Boundary-embedded neural operators for elliptic PDE s. In The Twelfth International Conference on Learning Representations, 2024. URL https://openreview.net/forum?id=ZZTkLDRmkg

    2024

  59. [67]

    Fd-bench: A modular and fair benchmark for data-driven fluid simulation, 2025 a

    Wang, H., Li, R., Xu, F., Sun, F., Han, K., Huang, Z., Wan, G., Chang, C., Luo, X., Wang, W., and Sun, Y. Fd-bench: A modular and fair benchmark for data-driven fluid simulation, 2025 a . URL https://arxiv.org/abs/2505.20349

    2025 arXiv

  60. [68]

    Mixture-of-experts operator transformer for large-scale pde pre-training, 2025 b

    Wang, H., Xin, H., Wang, J., Yang, X., Zha, F., Dong, H., and Jiang, Y. Mixture-of-experts operator transformer for large-scale pde pre-training, 2025 b . URL https://arxiv.org/abs/2510.25803

    2025

  61. [69]

    Solving high-dimensional pdes with latent spectral models

    Wu, H., Hu, T., Luo, H., Wang, J., and Long, M. Solving high-dimensional pdes with latent spectral models. In International Conference on Machine Learning, 2023

    2023

  62. [70]

    Transolver: A fast transformer solver for pdes on general geometries

    Wu, H., Luo, H., Wang, H., Wang, J., and Long, M. Transolver: A fast transformer solver for pdes on general geometries. In International Conference on Machine Learning, 2024

    2024

  63. [71]

    Rf-pinns: Reactive flow physics-informed neural networks for field reconstruction of laminar and turbulent flames using sparse data

    Yadav, V., Casel, M., and Ghani, A. Rf-pinns: Reactive flow physics-informed neural networks for field reconstruction of laminar and turbulent flames using sparse data. Journal of Computational Physics, 524: 0 113698, March 2025. Published: 1 March 2025

    2025

  64. [72]

    and Osher, S

    Yang, L. and Osher, S. J. Pde generalization of in-context operator networks: A study on 1d scalar nonlinear conservation laws, 2024. URL https://arxiv.org/abs/2401.07364

    2024 arXiv

  65. [73]

    Yang, L., Liu, S., Meng, T., and Osher, S. J. In-context operator learning with data prompts for differential equation problems. Proceedings of the National Academy of Sciences, 120 0 (39), September 2023. ISSN 1091-6490. doi:10.1073/pnas.2310142120. URL http://dx.doi.org/10.1...

    2023 doi

  66. [74]

    PDE former: Towards a foundation model for one-dimensional partial differential equations

    Ye, Z., Huang, X., Chen, L., Liu, H., Wang, Z., and Dong, B. PDE former: Towards a foundation model for one-dimensional partial differential equations. In ICLR 2024 Workshop on AI4DifferentialEquations In Science, 2024. URL https://openreview.net/forum?id=GLDMCwdhTK

    2024

  67. [75]

    Leads: Learning dynamical systems that generalize across environments, 2022

    Yin, Y., Ayed, I., de Bézenac, E., Baskiotis, N., and Gallinari, P. Leads: Learning dynamical systems that generalize across environments, 2022. URL https://arxiv.org/abs/2106.04546

    2022 arXiv

  68. [76]

    C., Li, M., and Smola, A

    Zhang, A., Lipton, Z. C., Li, M., and Smola, A. J. Dive into Deep Learning. Cambridge University Press, 2023. https://D2L.ai

    2023

  69. [77]

    Sparse sensor reconstruction of vortex-impinged airfoil wake with machine learning

    Zhong, Y., Fukami, K., An, B., et al. Sparse sensor reconstruction of vortex-impinged airfoil wake with machine learning. Theor. Comput. Fluid Dyn., 37: 0 269--287, 2023. doi:10.1007/s00162-023-00657-y

    2023 doi

  70. [78]

    Unisolver: PDE -conditional transformers are universal PDE solvers, 2025

    Zhou, H., Ma, Y., Wu, H., Wang, H., and Long, M. Unisolver: PDE -conditional transformers are universal PDE solvers, 2025. URL https://openreview.net/forum?id=f3xXPDCh8Q

    2025

  71. [79]

    write newline

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.