Pith. sign in

REVIEW 2 major objections 4 minor 42 references

Execution-grounded search can discover entirely new PINN learning algorithms, and one discovered architecture, SLRC-PINN, retains its advantage under parameter-matched retraining.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-01 14:38 UTC pith:G4PRA5KF

load-bearing objection EvoPINN is a serious, honestly reported agentic search framework, but the headline 'rigorous parameter-matched' claim outruns the protocol: budget matches steps and parameter counts, not FLOPs or wall-clock, and the artifacts needed to reproduce the discoveries are not released. the 2 major comments →

arxiv 2607.26490 v1 pith:G4PRA5KF submitted 2026-07-29 cs.AI cs.LGcs.NE

EvoPINN: Agentic Discovery of Executable Algorithms for Physics-Informed Neural Networks

Peng Yin , Kai Li , Yifan Zhang , Jian Cheng This is my paper
classification cs.AI cs.LGcs.NE
keywords physics-informed neural networksautomated algorithm discoverylarge language modelsevolutionary searchSLRC-PINNPDE solvingexecution-grounded search
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that open-ended algorithm discovery for scientific computing is possible in practice when a large language model is tied to an execution loop: the agent proposes changes to a physics-informed neural network, runs them, reads diagnostics, and keeps only changes that survive structural checks and improve solution error under a shared budget. The central proposal is to split a PINN into a representation module and a training-program module, mutate one at a time, and let a diagnostic-guided memory steer the search, converting PINN design from handcrafting into a verifiable search problem. Across Poisson, Burgers, wave, and heat benchmarks, the frozen discovered algorithms reduce relative L2 error relative to the seed PINN and match or beat a pool of expert-designed variants on three of four tasks. The paper's flagship discovery is SLRC-PINN, a representation u(q)=u_G(q)+alpha(q)u_L(q) with a zero-initialized corrective branch and a learned bank of localized bases, and the paper argues its advantage is due to this topology, not capacity. If true, this means an automated agent can invent new, transferable computational mechanisms rather than merely select among known ones.

Core claim

On its own terms, the paper establishes that EvoPINN, an evolutionary search where an LLM proposes programmatic changes and an execution pipeline verifies them, can discover complete and reusable PINN learning algorithms. The algorithm is represented as a pair A=(M,T), with M the neural representation and T the training program; each generation mutates only one module, conditions the LLM on a compact diagnostic summary plus evolutionary memory, and accepts a candidate only if it passes structural abstract-syntax-tree verification and beats the parent's relative L2 error under a shared training budget. Frozen, freshly retrained versions of the discovered algorithms report lower mean relative

What carries the argument

The load-bearing mechanism is the global-local decomposition of SLRC-PINN: u(q)=u_G(q)+alpha(q)u_L(q), with u_G a persistent global decoder, u_L a corrective decoder whose final layer is initialized to zero, and alpha(q) a smooth gating field computed from a learned bank of localized basis functions with learned centers and input-dependent widths. Zero initialization guarantees the network starts as the global branch, so local enrichment is introduced as a continuous, function-preserving extension rather than as a competing subnetwork. Around it, the EvoPINN loop does the discovery work: one-module-at-a-time mutation of M versus T, UCB-style scheduling between the two modules, LLM proposals

Load-bearing premise

The quantitative claims rest on the fairness of the comparison protocol, which matches optimization steps and initial collocation counts but not wall-clock time or floating-point work per step; if discovered algorithms spend more compute per step, or if the expert baselines are weaker than their original published versions, the performance gap could shrink or disappear.

What would settle it

Measure per-step FLOPs and wall-clock time of SLRC-PINN versus the parameter-matched MLP on Burgers1D under the reported 20,000-step budget, and retrain both with a hard equal-FLOPs envelope; alternatively rerun the expert baselines using their original published numerical protocols. If the relative-L2 gap vanishes or reverses, the claimed advantage would be refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Frozen discovered algorithms, retrained from scratch, improve over the seed PINN in all four benchmark regimes and beat the selected expert baseline on three of them.
  • SLRC-PINN outperforms a parameter-matched MLP and domain-decomposition variants under the same training budget, so the architecture's effect is separable from parameter count.
  • The zero-initialized local-correction principle means the model can begin from the smooth global solution and grow local capacity only where the physics residual demands it.
  • The discovered training policy for the Poisson problem, which spreads optimizer effort and refresh points based on run telemetry, outperforms the standard equal-split handoff, showing the search can invent training programs as well as representations.
  • Ablation results indicate that run-informed diagnostics and modular mutation are both needed: removing them degrades accuracy or stability, especially on the wave benchmark.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A strict test of the 'budget-matched' claim would cap total floating-point operations per candidate rather than optimization steps; if the gap shrinks, part of the reported gain may be uncharged computation—an extension the paper itself flags by noting the shared envelope does not imply equal wall-clock or FLOP counts.
  • The zero-init global-plus-local template could be extracted as a hand-crafted prior and tested on other shock-forming PDEs or multiscale learning tasks, which would tell whether the discovered topology is a general principle or a special fix for Burgers.
  • Varying the search-time reference set across independent runs and checking whether the same mechanisms emerge would sharpen the evidence that the search is discovering robust structures rather than overfitting to one evaluation set.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes EvoPINN, an agentic framework that treats PINN design as an execution-grounded search over a two-module algorithm space A=(M,T), where M is the neural representation and T is the training program. An LLM proposes one-module-at-a-time mutations guided by training diagnostics, evolutionary memory, and diversified search focuses; candidates are screened by structural AST comparison, executable smoke tests, and budget-matched PDE evaluation. The frozen winner is retrained from scratch on five fresh seeds and evaluated on a disjoint reporting set. Across four PDE benchmarks the discovered algorithms improve on the seed PINN and are competitive with or better than three of four expert baselines; the framework also discovers SLRC-PINN and TCROP. The paper includes ablations and zero-shot transfer experiments.

Significance. If the results are robust, EvoPINN is a meaningful step toward automated discovery of PINN learning mechanisms rather than hyperparameter configuration. The paper is unusually careful in several respects: frozen algorithms are retrained from scratch, search- and reporting-time point sets are separated, the TCROP control is executed as a stateless program, and the main protocol limitations are explicitly disclosed. The central claim is concrete and falsifiable. The main weakness is that the quantitative fairness controls are not fully quantified: the budget envelope matches steps and parameter counts but not FLOPs or wall-clock time, and the expert baselines are acknowledged to be reimplementations of published methods rather than exact reproductions. These gaps are disclosed, but they are load-bearing for the abstract's claim of 'rigorous parameter-matched comparisons.'

major comments (2)
  1. [Section 5.4 and Appendix D.1 (Eq. 38, Table 3)] The abstract's 'rigorous parameter-matched comparisons' and Table 3's 'identical training budgets' match parameter counts and optimization-step counts, but Appendix D.1 explicitly states that matching the envelope does not imply identical wall-clock time or floating-point operation count. SLRC-PINN evaluates six Gaussian basis responses, a gating field, and two decoders each forward pass; the width-230 global MLP does not. If SLRC-PINN has materially higher per-step FLOPs, the 55.1% error reduction over the MLP may reflect additional computational budget rather than the discovered topology. Since the claim of a 'genuinely new mechanism' depends on excluding this explanation, please add a FLOP-matched or wall-clock-matched control, or report per-step FLOP counts and show the advantage survives an equal-compute comparison.
  2. [Appendix C.4 and Tables 1/3] The expert baselines are 'architecture-faithful reimplementations... rather than claiming exact reproduction of the numerical protocols in their original studies.' If the reimplemented APINN, FBPINN, HyResPINNs, and other expert variants are less tuned than their published versions, the 'match-or-beat expert' results in Table 1 and the controlled comparison in Table 3 may be inflated. The paper reports that experts were screened under a unified pipeline but does not report whether the screened variants reach accuracy levels comparable to the original publications. Please provide screening scores, compare against published reference accuracies, or otherwise demonstrate that the expert pool is not systematically weaker than the source literature.
minor comments (4)
  1. [Appendix B.1/B.2] The diagnostic summary includes 'the log-scale gap between the final training loss and the search-time solution error,' which is derived from reference-solution values. The prompt protocol in B.2 says the prompt 'never contains reference-solution values.' This should be qualified to make clear that summary statistics derived from reference values are permitted; as written it is easy to read as contradictory.
  2. [Section 5.4 vs C.4] The main text presents SLRC-PINN as the discovered Burgers1D algorithm, but C.4 states that the SLRC-PINN controlled study is distinct from the frozen Burgers1D algorithm used in Table 1. This distinction should appear in the main text to prevent readers from comparing the two sets of numbers as if they were the same experiment.
  3. [Table 1] Formatting glitch: the Poisson2D expert-baseline entry '4.7025e-3±8.23e-44.1413e-3' has no separating whitespace between the expert and EvoPINN columns. Please correct the table rendering.
  4. [Section 5.3/Table 2] The comparison with LLM Best-of-56 uses 56 independent samples, matching EvoPINN's 56 proposals. Please state explicitly whether the Best-of-56 and PINNsAgent baselines also regenerate from seed with the same five reporting seeds and disjoint reporting points; the text implies this but it should be explicit in the protocol.

Circularity Check

0 steps flagged

No significant circularity: the reported results are separated from the search objective by fresh retraining, disjoint evaluation points, and frozen algorithms.

full rationale

EvoPINN's central claim is an empirical discovery claim, not a derivation fitted to its own output. The selection objective (Eq. 4) is evaluated on a search reference set, while final results are computed after freezing the algorithm, retraining from scratch on five new seeds, and evaluating on a disjoint reporting set (Sec. 5.1, App. B.4, App. D.1). The prompts explicitly exclude reference-solution values (App. B.2, App. D.3), and the LLM is not given reporting coordinates. SLRC-PINN's formulation (Eqs. 7-8) is a mathematical description of the discovered architecture, not a self-defined prediction; its comparison in Table 3 uses parameter counts and equal step budgets, and the paper's own Appendix D.1 concedes that this does not imply identical FLOPs or wall-clock time - a fairness limitation, not circularity. Likewise, the C.4 note that expert baselines are architecture-faithful reimplementations is a baseline-strength caveat, not an identity between input and output. There are no self-citations, no imported uniqueness theorems, and no fitted parameter renamed as a prediction. The only near-concerns (search focuses include 'global-local scale separation'; budget matching excludes FLOPs) are disclosed exploratory priors and evaluation-protocol caveats, respectively, and neither makes the reported result equivalent to the search objective by construction.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 2 invented entities

The central claim is empirical rather than mathematical, so the ledger lists the hand-chosen controls of the framework (β, reward weights, opportunity budget, temperature, diagnosis windows) and the domain assumptions that make the evaluation protocol fair and meaningful. The most load-bearing assumptions are the fairness of the budget envelope (which does not match wall-clock/FLOP cost) and the accuracy of the reference solutions. SLRC-PINN and TCROP are listed as invented entities but both carry independent falsifiable evidence, so the 'graviton problem' does not apply.

free parameters (5)
  • UCB exploration coefficient β = 0.35
    Hand-chosen (Table 9); controls how aggressively the scheduler alternates between representation and training-program modules; shapes the search trajectory that produced the reported winners.
  • Scheduler reward weights (ω_a, ω_δ, ω_v, ω_n) = (0.55, 0.20, 0.15, 0.10)
    Hand-chosen (Table 9, Appendix B.1); determine module-selection rewards and therefore which modules get evolved in later generations.
  • Opportunity budget: generations G and proposals per batch K = G=7, K=8 (max 56 full evaluations)
    Hand-chosen; bounds the search and is the comparison budget given to all competing search strategies in Table 2.
  • LLM sampling temperature = 0.7
    Hand-chosen (Table 9); affects proposal diversity and hence which mechanisms are discovered.
  • Diagnostic windows W_early/W_late and contraction/oscillation statistics = Not numerically specified
    Postulated in Appendix B.1 to convert training traces into LLM guidance; window boundaries are not given, so the diagnosis step is not exactly reproducible from the text.
axioms (6)
  • domain assumption Structural AST verification plus smoke tests are sufficient to ensure scientific validity of LLM-generated PINN code
    Section 4.4 / B.3. If the checks miss mathematically invalid or numerically unstable programs, the validity guarantee for discovered algorithms collapses.
  • domain assumption Budget-matched evaluation (equal optimization steps, initial collocation counts, shared seed) is a fair proxy for computational fairness across structurally different architectures
    Appendix D.1 explicitly disclaims identical wall-clock/FLOP cost; candidate programs may reallocate resources via adaptive sampling, making true compute cost unequal.
  • domain assumption Reference solutions (COMSOL finite-element field, numerical Burgers grid, analytic wave/heat solutions) are accurate enough to grade algorithms
    Appendix C.1; errors in the reference fields would bias all relative-L2 comparisons.
  • domain assumption Five evaluation seeds suffice to rank algorithms reliably without significance testing
    Section 5.1 / D.2; Table 1 error bars overlap between EvoPINN and expert baselines on 3 of 4 benchmarks, so the ranking is statistically fragile.
  • domain assumption The search-time reference set and the reporting set are disjoint such that selection does not leak into final scores
    Appendix D.1–D.2 asserts separation; the paper's headline results depend on this holding in the implementation.
  • domain assumption DeepXDE 1.15.0 / PyTorch 2.6.0 autodiff and solver behavior is numerically correct in the regimes tested
    Appendix D.4; all derivative computations and training loops go through these libraries.
invented entities (2)
  • SLRC-PINN (self-localizing residual-correction architecture) independent evidence
    purpose: Flagship discovered artifact: global base predictor u_G plus gated, zero-initialized local corrective branch u_L with a bank of K=6 learned localized bases (Eqs. 7–8), aimed at capturing Burgers steep-gradient structure.
    An output of the search, not an ad hoc postulate; exposed to refutation via parameter-matched controlled comparisons (Table 3), five-seed retraining, and transfer tests. Novelty claim rests on the authors' literature survey, not a systematic prior-art search.
  • TCROP (telemetry-conditioned resampling and optimizer policy) independent evidence
    purpose: Discovered training program for Poisson2D: diagnostic state maps to an optimizer split (17.2k Adam / 2.8k L-BFGS) and global collocation refresh every 1,000 steps.
    A stateless control executing the discovered schedule reproduces the accuracy (Appendix C.5, Table 7), giving the entity a falsifiable handle outside the search loop.

pith-pipeline@v1.3.0-daily-deepseek · 19562 in / 21115 out tokens · 181084 ms · 2026-08-01T14:38:39.667551+00:00 · methodology

0 comments
read the original abstract

Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs), yet their performance heavily relies on the manual, trial-and-error engineering of neural representations, loss formulations, and optimization dynamics. While Large Language Models (LLMs) offer a promising avenue for automated design, unconstrained code generation often yields mathematically invalid or numerically unstable solutions under strict scientific computing constraints. To bridge this gap, we propose \textbf{EvoPINN}, an agentic framework that reformulates PINN development from labor-intensive manual design into a rigorous, execution-grounded algorithm discovery problem. EvoPINN navigates a modular search space by decoupling neural representations from training programs, utilizing an LLM agent to iteratively propose memory-conditioned programmatic modifications. To ensure scientific validity, all candidates undergo strict structural verification and budget-matched PDE evaluation. Extensive experiments across diverse PDE regimes (oscillatory, elliptic, dissipative, and nonlinear transport) demonstrate that EvoPINN discovers PDE-specialized learning algorithms that significantly reduce relative $L_{2}$ error compared to baselines. Crucially, EvoPINN autonomously invented SLRC-PINN, a novel architecture whose performance gains persist under rigorous parameter-matched comparisons, establishing the viability of execution-grounded agents for discovering genuinely new scientific computing mechanisms.

Figures

Figures reproduced from arXiv: 2607.26490 by Jian Cheng, Kai Li, Peng Yin, Yifan Zhang.

Figure 1
Figure 1. Figure 1: Overview of EvoPINN. A PINN algorithm is represented as [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

42 extracted references · 6 linked inside Pith

  1. [1]

    Journal of Computational Physics , volume =

    Raissi, Maziar and Perdikaris, Paris and Karniadakis, George Em , title =. Journal of Computational Physics , volume =. 2019 , doi =

    2019

  2. [2]

    SIAM Review , volume =

    Lu, Lu and Meng, Xuhui and Mao, Zhiping and Karniadakis, George Em , title =. SIAM Review , volume =. 2021 , doi =

    2021

  3. [3]

    Advances in Neural Information Processing Systems , year =

    Hao, Zhongkai and Yao, Jiachen and Su, Chang and Su, Hang and Wang, Ziao and Lu, Fanzhi and Xia, Zeyu and Zhang, Yichi and Liu, Songming and Lu, Lu and Zhu, Jun , title =. Advances in Neural Information Processing Systems , year =

  4. [4]

    SIAM Journal on Scientific Computing , volume =

    Wang, Sifan and Teng, Yujun and Perdikaris, Paris , title =. SIAM Journal on Scientific Computing , volume =. 2021 , doi =

    2021

  5. [5]

    Journal of Computational Physics , volume =

    Wang, Sifan and Yu, Xinling and Perdikaris, Paris , title =. Journal of Computational Physics , volume =. 2022 , doi =

    2022

  6. [6]

    Computer Methods in Applied Mechanics and Engineering , volume =

    Wu, Chenxi and Zhu, Min and Tan, Qinyang and Kartha, Yadhu and Lu, Lu , title =. Computer Methods in Applied Mechanics and Engineering , volume =. 2023 , doi =

    2023

  7. [7]

    Proceedings of the 40th International Conference on Machine Learning , series =

    Yao, Jiachen and Su, Chang and Hao, Zhongkai and Liu, Songming and Su, Hang and Zhu, Jun , title =. Proceedings of the 40th International Conference on Machine Learning , series =. 2023 , publisher =

    2023

  8. [8]

    Computer Methods in Applied Mechanics and Engineering , volume =

    Yu, Jeremy and Lu, Lu and Meng, Xuhui and Karniadakis, George Em , title =. Computer Methods in Applied Mechanics and Engineering , volume =. 2022 , doi =

    2022

  9. [9]

    and Kawaguchi, Kenji and Karniadakis, George Em , title =

    Jagtap, Ameya D. and Kawaguchi, Kenji and Karniadakis, George Em , title =. Journal of Computational Physics , volume =. 2020 , doi =

    2020

  10. [10]

    and Kawaguchi, Kenji and Karniadakis, George Em , title =

    Jagtap, Ameya D. and Kawaguchi, Kenji and Karniadakis, George Em , title =. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , volume =. 2020 , doi =

    2020

  11. [11]

    Computer Methods in Applied Mechanics and Engineering , volume =

    Wang, Sifan and Wang, Hanwen and Perdikaris, Paris , title =. Computer Methods in Applied Mechanics and Engineering , volume =. 2021 , doi =

    2021

  12. [12]

    Sitzmann, Vincent and Martel, Julien N. P. and Bergman, Alexander W. and Lindell, David B. and Wetzstein, Gordon , title =. Advances in Neural Information Processing Systems , volume =

  13. [13]

    and Kharazmi, Ehsan and Karniadakis, George Em , title =

    Jagtap, Ameya D. and Kharazmi, Ehsan and Karniadakis, George Em , title =. Computer Methods in Applied Mechanics and Engineering , volume =. 2020 , doi =

    2020

  14. [14]

    and Karniadakis, George Em , title =

    Jagtap, Ameya D. and Karniadakis, George Em , title =. Communications in Computational Physics , volume =. 2020 , doi =

    2020

  15. [15]

    Advances in Computational Mathematics , volume =

    Moseley, Ben and Markham, Andrew and Nissen-Meyer, Tarje , title =. Advances in Computational Mathematics , volume =. 2023 , doi =

    2023

  16. [16]

    and Karniadakis, George Em , title =

    Shukla, Khemraj and Jagtap, Ameya D. and Karniadakis, George Em , title =. Journal of Computational Physics , volume =. 2021 , doi =

    2021

  17. [17]

    , title =

    Zoph, Barret and Le, Quoc V. , title =. International Conference on Learning Representations , year =

  18. [18]

    Journal of Machine Learning Research , volume =

    Elsken, Thomas and Metzen, Jan Hendrik and Hutter, Frank , title =. Journal of Machine Learning Research , volume =

  19. [19]

    , title =

    Real, Esteban and Aggarwal, Alok and Huang, Yanping and Le, Quoc V. , title =. Proceedings of the AAAI Conference on Artificial Intelligence , volume =. 2019 , doi =

    2019

  20. [20]

    , title =

    Bello, Irwan and Zoph, Barret and Vasudevan, Vijay and Le, Quoc V. , title =. Proceedings of the 34th International Conference on Machine Learning , series =. 2017 , publisher =

    2017

  21. [21]

    2021 , archivePrefix =

    Austin, Jacob and Odena, Augustus and Nye, Maxwell and Bosma, Maarten and Michalewski, Henryk and Dohan, David and Jiang, Ellen and Cai, Carrie and Terry, Michael and Le, Quoc and Sutton, Charles , title =. 2021 , archivePrefix =. 2108.07732 , primaryClass =

    Pith/arXiv arXiv 2021

  22. [22]

    Advances in Neural Information Processing Systems , volume =

    Shinn, Noah and Cassano, Federico and Gopinath, Ashwin and Narasimhan, Karthik and Yao, Shunyu , title =. Advances in Neural Information Processing Systems , volume =

  23. [23]

    2023 , archivePrefix =

    Wang, Guanzhi and Xie, Yuqi and Jiang, Yunfan and Mandlekar, Ajay and Xiao, Chaowei and Zhu, Yuke and Fan, Linxi and Anandkumar, Anima , title =. 2023 , archivePrefix =. 2305.16291 , primaryClass =

    Pith/arXiv arXiv 2023

  24. [24]

    International Conference on Learning Representations , year =

    Ma, Yecheng Jason and Liang, William and Wang, Guanzhi and Huang, De-An and Bastani, Osbert and Jayaraman, Dinesh and Zhu, Yuke and Fan, Linxi and Anandkumar, Anima , title =. International Conference on Learning Representations , year =

  25. [25]

    Proceedings of the 45th International Conference on Software Engineering , pages =

    Xia, Chunqiu Steven and Wei, Yuxiang and Zhang, Lingming , title =. Proceedings of the 45th International Conference on Software Engineering , pages =. 2023 , doi =

    2023

  26. [26]

    Nature , author =

    Mathematical discoveries from program search with large language models , volume =. Nature , author =. 2024 , pages =. doi:10.1038/s41586-023-06924-6 , number =

    doi:10.1038/s41586-023-06924-6 2024

  27. [27]

    arXiv preprint arXiv:2401.02051 , year=

    Evolution of Heuristics: Towards Efficient Automatic Algorithm Design Using Large Language Model , author=. arXiv preprint arXiv:2401.02051 , year=

    Pith/arXiv arXiv

  28. [28]

    Advances in Neural Information Processing Systems , year=

    ReEvo: Large Language Models as Hyper-Heuristics with Reflective Evolution , author=. Advances in Neural Information Processing Systems , year=

  29. [29]

    Knowledge-Based Systems , volume =

    He, Xin and Zhao, Kaiyong and Chu, Xiaowen , title =. Knowledge-Based Systems , volume =. 2021 , doi =

    2021

  30. [30]

    arXiv preprint arXiv:2501.12053 , year =

    Wuwu, Qingpo and Gao, Chonghan and Chen, Tianyu and Huang, Yihang and Zhang, Yuekai and Wang, Jianing and Li, Jianxin and Zhou, Haoyi and Zhang, Shanghang , title =. arXiv preprint arXiv:2501.12053 , year =

    Pith/arXiv arXiv

  31. [31]

    arXiv preprint arXiv:2510.05158 , year =

    He, Xin and You, Liangliang and Tian, Hongduan and Han, Bo and Tsang, Ivor and Ong, Yew-Soon , title =. arXiv preprint arXiv:2510.05158 , year =

    arXiv

  32. [32]

    arXiv preprint arXiv:2505.08783 , year =

    Li, Shanda and Marwah, Tanya and Shen, Junhong and Sun, Weiwei and Risteski, Andrej and Yang, Yiming and Talwalkar, Ameet , title =. arXiv preprint arXiv:2505.08783 , year =

    arXiv

  33. [33]

    arXiv preprint arXiv:2511.00183 , year =

    Fazliani, Shaghayegh and Udell, Madeleine , title =. arXiv preprint arXiv:2511.00183 , year =

    arXiv

  34. [34]

    and Karniadakis, George Em and Kawaguchi, Kenji , title =

    Hu, Zheyuan and Jagtap, Ameya D. and Karniadakis, George Em and Kawaguchi, Kenji , title =. Engineering Applications of Artificial Intelligence , volume =. 2023 , doi =

    2023

  35. [35]

    , title =

    Zeng, Chengxi and Burghardt, Tilo and Gambaruto, Alberto M. , title =. arXiv preprint arXiv:2402.08367 , year =

    Pith/arXiv arXiv

  36. [36]

    and Zhe, Shandian and Shankar, Varun , title =

    Cooley, Madison and Kirby, Robert M. and Zhe, Shandian and Shankar, Varun , title =. arXiv preprint arXiv:2410.03573 , year =

    arXiv

  37. [37]

    2023 , eprint=

    Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture , author=. 2023 , eprint=

    2023

  38. [38]

    2024 , doi =

    Wang, Yifan and Zhong, Linlin , journal =. 2024 , doi =

    2024

  39. [39]

    Swarm and Evolutionary Computation , volume=

    Discovering physics-informed neural networks model for solving partial differential equations through evolutionary computation , author=. Swarm and Evolutionary Computation , volume=. 2024 , publisher=

    2024

  40. [40]

    Advances in neural information processing systems , volume=

    Fourier features let networks learn high frequency functions in low dimensional domains , author=. Advances in neural information processing systems , volume=

  41. [41]

    arXiv preprint arXiv:2207.02338 , year=

    Rethinking the importance of sampling in physics-informed neural networks , author=. arXiv preprint arXiv:2207.02338 , year=

    Pith/arXiv arXiv

  42. [42]

    Advances in neural information processing systems , volume=

    Characterizing possible failure modes in physics-informed neural networks , author=. Advances in neural information processing systems , volume=