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Generative Discovery of Partial Differential Equations by Learning from Math Handbooks

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A generative model trained on sentence-like encodings of handbook PDEs recovers governing equations from sparse, noisy data, including a previously unreported equation for surface gravity waves approaching breaking.

desk verdict EqGPT framework is a legitimate step for PDE discovery and deserves serious review, but the 'previously unreported wave-breaking PDE' is currently overclaimed. read the letter →

arxiv 2505.05869 v1 pith:5GARGVA3 submitted 2025-05-09 cs.LG cs.AIphysics.comp-ph

Hao Xu , Yuntian Chen , Rui Cao , Tianning Tang , Mengge Du , Jian Li , Adrian H. Callaghan , Dongxiao Zhang This is my paper · ORCID
classification cs.LGcs.AIphysics.comp-ph
keywords PDEdiscoverygenerativerepresentationofequationsscientificallyaugmentedtrainingknowledge-guidedneuralsurrogatemodelwavebreakingnonlineardynamicssymbolicregression
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

Data-driven PDE discovery usually faces a trade-off: sparse regression is fast but limited to a hand-built candidate library, while free-form symbolic search explores more but is slow and fragile. This paper proposes embedding prior knowledge instead of brute-force search: existing PDEs from a mathematical handbook are tokenized into operators and base terms, and an autoregressive generative model learns which term combinations are plausible. In a generation–evaluation–optimization loop, candidate equations are produced by the model, scored by how well they fit derivatives from a neural surrogate of the data, and the best candidates are used to fine-tune the generator. The paper reports that this recovers canonical PDEs with tens of seconds of compute and tolerates over 50 percent noise, handles unusual terms such as $u_{xt}$ and $u_x/x$, works on irregular and high-dimensional domains, and uncovers a previously unreported equation for strongly nonlinear surface gravity waves moving toward breaking from real wave-tank data.

What carries the argument

The load-bearing object is the generative representation of equations (GRE): a tokenization that writes a PDE as a sentence of operators and basic terms, so that a transformer language model can learn co-occurrence patterns among terms. The trained model, EqGPT, proposes candidate structures; a neural-network surrogate trained on the observations supplies values and derivatives everywhere, and a reward combining a sparsity penalty with the coefficient of determination $R^2$ of a least-squares regression scores each candidate. The top-scoring structures fine-tune the generator, and the loop drives the model toward equations that are both data-consistent and plausible under the handbook's prior.

What would settle it

Train the same pipeline on a synthetic PDE with a known solution, delete a subregion from the training data, and compare automatic-differentiation derivatives from the surrogate against exact derivatives in that subregion; if the reward ranks wrong structures above the true PDE whenever the derivative error is substantial, or if the discovered wave equation fails to time-march a new breaking-wave experiment it was not fitted to, the central claim would be refuted.

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Extended reading notes

Core claim

The paper's central claim is that the combinatorial difficulty of PDE discovery can be shifted from the optimizer to a learned prior over equation structure. Concretely, 221 handbook PDEs are rewritten as sentences over a 56-token vocabulary of operators and terms, augmented by term shuffling to 7,072 sentences, and used to train EqGPT, an autoregressive transformer that generates free-form PDEs by next-token sampling under syntactic and dimensional constraints. The generated structures are evaluated by least-squares fitting against meta-data produced by a neural-network surrogate, with a reward that combines the fit's $R^2$ with a sparsity penalty; the top ten structures fine-tune the generator, and the loop repeats. Against this machinery the paper reports rediscovery of eight canonical PDEs, including forms with the mixed derivative $u_{xt}$ and the fraction $u_x/x$, recovery of Poisson's equation on disk, smiley-face, letter-glyph, and three-dimensional shuttle geometries, and reconstruction of coupled oil–water flow equations in three dimensions. Its headline experimental result is a previously unreported PDE for nondimensional surface elevation $\eta^*$ of strongly nonlinear waves propagating toward breaking, with $\eta_t^*$ on the left and a combination of spatial derivative terms on the right whose three coefficients are re-fit per experiment; the paper argues the leading linear term agrees with classical potential-flow wave theory and that the newly identified nonlinear term can generate the higher-frequency content seen as breaking approaches.

Load-bearing premise

The load-bearing premise is that the neural surrogate trained on the observations yields accurate values and derivatives everywhere in the domain—including the non-contiguous gaps in the wave-camera images—because every generated equation is judged by how well its terms fit those derivatives.

Editorial extensions

If this is right

  • The fixed candidate-library bottleneck disappears: terms like $u_{xt}$ and $u_x/x$ are generated and can be selected when the data call for them.
  • PDE discovery becomes feasible on irregular, discontinuous, and three-dimensional geometries, because the surrogate supplies predictions and derivatives at arbitrary observation points.
  • The loop can operate with as few as hundreds of data points under heavy noise in the tested cases, and each optimization cycle costs seconds, so the method is cheap enough to rerun for many candidate hypotheses.
  • Growing the equation corpus improves the success rate; with the current 221-equation handbook the reported rediscovery success rates exceed 0.7 even when the target equation is held out.
  • For multi-variable systems, the method recovers expanded forms of nested-operator equations, giving scientists a ranked set of structurally plausible laws rather than a single guess.

Reading between the lines

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

  • The scaling behaviour implied by the paper's ablation is that EqGPT's discovery power is bounded by the coverage of the handbook corpus; a sharper test would hold out entire families of terms and ask whether the model can compose them from parts it has seen.
  • Because the reward is computed from surrogate derivatives, the wave-equation result should be checked against a second, independent derivative estimator on the same data; a mismatch would indicate the discovered form may be an artifact of the surrogate.
  • The same loop could be repurposed for closure discovery by freezing the known operators of a system and letting EqGPT generate only the unknown coupling or source terms, which would turn the method into a hypothesis generator for unresolved physics.
  • A natural next experiment is to test whether the discovered wave equation predicts surface elevation for wave groups with spectral parameters outside the 12 calibration cases, especially closer to or past the onset of breaking.
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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 EqGPT, a generative approach to PDE discovery that pretrains an autoregressive transformer on 221 PDE structures extracted from a mathematical handbook, encoded as token sequences under the proposed generative representation of equations (GRE). Candidate equations are generated by sampling from this model, evaluated through a reward combining least-squares R2 on surrogate-reconstructed fields with a sparsity penalty, and used to fine-tune the generator in a generation-evaluation-optimization loop. The authors report rediscovery of eight canonical PDEs from sparse and noisy data, comparisons against eight existing PDE-discovery methods, extensions to irregular computational domains and high-dimensional systems, and a real-world case study in which the method yields a reported 'previously unreported' PDE for strongly nonlinear surface gravity waves propagating toward breaking, obtained by fitting a linear propagation term, a quadratic nonlinear dispersive term, and a linear dispersive term to wave-tank data.

Significance. If all claims are substantiated, this is a valuable contribution: the knowledge-guided generative prior is a simple and potentially effective way to expand the search space of PDE discovery beyond fixed candidate libraries, and the public code and data make the benchmark portion reproducible. The benchmark results on canonical PDEs, irregular domains, and high-dimensional systems are largely plausible and would be useful to the community. However, the central scientific-discovery claim for the wave-breaking equation depends critically on the accuracy of automatic-differentiation derivatives of ANN surrogates trained on gappy camera data, and on the stability of the fitted coefficients. As the evidence stands, that headline claim is not fully supported and requires additional validation before it can be accepted.

major comments (3)
  1. [Supplementary Information S1.1; Materials and Methods ('the surrogate model can ...] The wave-breaking result rests on automatic-differentiation derivatives of per-experiment ANN surrogates trained on three non-contiguous spatial subdomains, yet the paper provides no validation that these surrogates, and their spatial derivatives up to third order, are accurate in the gaps or near the breaking onset. Since Eq. (1) computes the R2 reward on these surrogate derivatives and Eq. (15) is selected by that reward, any bias in the surrogate derivatives can select a structure that fits surrogate artifacts rather than the physics. Please provide held-out derivative tests, such as comparison with finite differences on observed frames, train/test splits, or ensemble variability across surrogate initializations, and report how the selected equation changes when the surrogate architecture or training seed is varied.
  2. [Table S3; Results, 'Discovery of new PDEs from real-world experimental data'] Table S3 shows that the coefficients of Eq. (15) vary substantially across the 12 experiments: c2 ranges from 4.819e-4 to 2.376e-3 and c3 changes sign (negative in cases 1-11, positive in case 12). The statement that the coefficients 'may be slightly different' is therefore inaccurate, and the sign change in the linear dispersive term is physically significant. The paper should report per-experiment R2 values or other goodness-of-fit measures, uncertainties for c1-c3, and a sensitivity analysis with respect to random seeds and surrogate training; without these, the claim that this PDE 'performs well across all experiments' and the physical interpretation of the new nonlinear term are not established.
  3. [Results, 'Discovery of new PDEs from real-world experimental data'; Fig. 6e] The evidence that Eq. (15) is a 'previously unreported' governing equation is incomplete. The paper does not report quantitative out-of-sample skill, does not compare Eq. (15) with classical weakly nonlinear models such as KdV, NLS, or Boussinesq-type equations, and does not show that the discovered equation outperforms plausible alternatives on data withheld from structure selection. The time-marching posterior prediction in Fig. 6e needs a quantitative error metric per experiment; otherwise the central claim of scientific discovery is not falsifiable from the reported material.
minor comments (5)
  1. [Eq. (1)] Please make the components of Eq. (1) explicit in the typeset version; in the provided text the expression is difficult to read, and the definitions of alpha0, Nterm, and R2 should appear in one place.
  2. [Eq. (15)] Please typeset Eq. (15) with unambiguous subscripts and superscripts; the current display is difficult to parse and could mislead readers about which terms are multiplied.
  3. [Fig. 2; Supplementary S1.2] The main-text benchmark figures appear to show single representative trials; consider reporting success rates and standard errors in the main text, as already computed in Supplementary S1.2, rather than only in the supplement.
  4. [Materials and Methods, 'The experimental settings'] The sparsity penalty alpha0 is set to different values in different experiment groups (0.2, 1, and 0.02), but no sensitivity analysis is given; a brief report on how the selected structure depends on alpha0 would strengthen the robustness claims.
  5. [Discussion, paragraph on compactons] Please cite the specific handbook chapter or equation number for the term (eta*)^2 eta_xxx so that readers can verify that Eq. (15) itself is not simply memorized from the handbook, rather than being a genuinely new recombination.

Circularity Check

1 steps flagged · score 4.0 of 10

The wave-breaking PDE is selected and validated with the same fitted R2 objective on surrogate derivatives, so the real-world 'discovery' is partially a fitted-input-called-prediction; the controlled deletion benchmarks retain independent content.

  1. fitted input called prediction [Results, 'Discovery of new PDEs from real-world experimental data'; Materials and Methods, 'The optimization cycle for PDE discovery'; Eq. (1), Eq. (15), Fig. 6d]
    "where the reward is composed of two components: the penalty term α and the determination coefficient R2. ... the latter is employed to measure the consistency with observation data through the R2 of the regression. ... the coefficients are obtained from the least squares regression via meta-data. ... It is discovered that the left-hand side term (ηt*) and the regressed right-hand side terms of the PDE align well at the time of wave breaking onset (Fig. 6d)."

    By Eq. (1), the reward is a function of the R2 of a least-squares regression of the generated LHS term on the RHS terms over surrogate meta-data. The final wave equation (15) is selected as the highest-reward structure, and its coefficients C1-C3 (Table S3) are obtained by least-squares regression on the same meta-data. Thus the Fig. 6d agreement between ηt* and the regressed RHS is the very objective being maximized, not an independent confirmation. The subsequent 'time-marching posterior prediction' is likewise in-sample if the observed data were used to fit the coefficients. The automatic-differentiation derivatives of the ANN trained on non-contiguous camera observations are the unverified input, and the R2-based reward cannot distinguish physical terms from surrogate artifacts.

full rationale

The paper's benchmark claims are substantially self-contained: in the canonical-PDE experiments the target equation is deleted from the EqGPT training set ('the target PDE is deleted from the dataset to ensure that the EqGPT model has never seen the target PDE'), and comparisons against PDE-FIND, GGA, DISCOVER, W-SINDy and others are run on the same surrogate meta-data, providing an external yardstick. Training the generative model on handbook PDEs is an explicit knowledge prior, not a hidden circularity; the 'new' wave term (η^2)_xxx being traceable to the handbook is openly acknowledged and is the intended mechanism of knowledge guidance. Self-citations appear mainly as baseline methods or implementation details (e.g., DLGA-PDE, SGA-PDE, DISCOVER, DL-PDE) and are not load-bearing; one physical-plausibility citation includes a co-author but only supports the well-known ηx leading-order term. The genuine circularity concern is confined to the real-world wave case: the reward (Eq. 1) is an R2 of a least-squares fit on surrogate automatic-differentiation derivatives, and the final Eq. (15) plus Table S3 coefficients are fit to and evaluated on that same surrogate field. The close LHS/RHS agreement in Fig. 6d is therefore a restatement of the selection objective, and calling the time-marching match a 'posterior prediction' presents an in-sample fit as independent evidence. This is partial circularity in the headline application, not a wholesale collapse of the method, so a score of 4 is appropriate.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The method introduces no new physical entities. It does introduce a new empirical PDE with fitted coefficients, which are listed as free parameters. The key unproven premises are the sufficiency of the handbook vocabulary and the reliability of the surrogate's derivatives, both load-bearing for the claimed discoveries.

free parameters (2)
  • c1, c2, c3 in the discovered wave equation = Per-experiment values vary, e.g., case 1: c1=1.461, c2=1.936e-3, c3=-7.38e-5; see Table S3
    The headline PDE eta_t = c1 eta_x + c2 (eta_x)^2 + c3 eta_xxx is fit to each of 12 experiments. Coefficients are not fixed across experiments, so the claimed governing equation is really a structure with fitted coefficients.
  • Sparsity penalty alpha0 = 0.2, 1, 0.1, 0.02 depending on the experiment
    Chosen by hand per experimental setting. This penalty directly controls the parsimony of the selected structure through the reward in Eq. (1).
assumptions (3)
  • domain assumption The true governing PDE for the wave data is expressible as a finite combination of the 56 tokens derived from 221 handbook PDEs.
    The EqGPT generator can only emit structures composed of the handbook-derived vocabulary. Any term outside this vocabulary is undiscoverable. Introduced in 'The construction of the PDE dataset from the math handbook' and implicitly used in the wave experiment.
  • domain assumption The ANN surrogate provides accurate values and derivatives, via automatic differentiation, on the full domain including the non-contiguous camera gaps.
    In the wave experiment, three camera views are spatially separated and no interpolation is applied, yet the surrogate is trained on these points and its derivatives are used for the reward. Stated in the wave experiment section and in Materials and Methods.
  • domain assumption Least squares regression of coefficients on surrogate meta-data yields coefficients that represent the true physical process.
    The reward uses R2 of this regression. If the surrogate meta-data are biased, both the selected structure and the coefficients are biased. No error bars are provided for the coefficients in Table S3.

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Pith. "Pith review of Generative Discovery of Partial Differential Equations by Learning from Math Handbooks." pith.science (2026). https://pith.science/paper/5GARGVA3

@misc{pith2026250505869,
  author       = {Pith},
  title        = {Pith review of: Generative Discovery of Partial Differential Equations by Learning from Math Handbooks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5GARGVA3}},
  note         = {Machine review of arXiv:2505.05869}
}
read the original abstract

Data driven discovery of partial differential equations (PDEs) is a promising approach for uncovering the underlying laws governing complex systems. However, purely data driven techniques face the dilemma of balancing search space with optimization efficiency. This study introduces a knowledge guided approach that incorporates existing PDEs documented in a mathematical handbook to facilitate the discovery process. These PDEs are encoded as sentence like structures composed of operators and basic terms, and used to train a generative model, called EqGPT, which enables the generation of free form PDEs. A loop of generation evaluation optimization is constructed to autonomously identify the most suitable PDE. Experimental results demonstrate that this framework can recover a variety of PDE forms with high accuracy and computational efficiency, particularly in cases involving complex temporal derivatives or intricate spatial terms, which are often beyond the reach of conventional methods. The approach also exhibits generalizability to irregular spatial domains and higher dimensional settings. Notably, it succeeds in discovering a previously unreported PDE governing strongly nonlinear surface gravity waves propagating toward breaking, based on real world experimental data, highlighting its applicability to practical scenarios and its potential to support scientific discovery.

Figures

Figures reproduced from arXiv: 2505.05869 by the authors.

Figure 1
Figure 1. Overview of the proposed knowledge-guided generative framework for PDE discovery. (a) The construction of knowledge guidance, including extracting existing PDEs from a math handbook, which is tokenized into 56 vocabularies (basic terms) and converted to sentences through generative representation of equations [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The performance of the proposed generative framework in discovering [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. The comparison between the proposed method and existing methods on the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Discovery of Poisson’s equation within complex computed regions. (a) The dataset on a disk region (left) and the rewards of the top 10 equations generated from the EqGPT model in each optimization epoch (right). (b) The dataset on a “smiley face region” (left) and the …
Figure 5
Figure 5. Figure 5: Discovery of high-dimensional PDEs via the proposed generative framework. (a) For the 2-dimensional (2D) Burgers’ equation, the surface of u(x,y) when t=0 and t=1.8, the true PDE, the discovered PDE, and the rewards of the top 10 [PITH_FULL_IMAGE:figures/full_fig_p015…
Figure 6
Figure 6. Figure 6: Discovery of previously undisclosed governing equation [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

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Reference graph

Works this paper leans on

50 extracted references · 40 canonical work pages

  1. [1]

    Brunton, S. L. & Kutz, J. N. Methods for data -driven multiscale model discovery for materials. Journal of Physics: Materials 2, 044002 (2019)

    work page 2019

  2. [2]

    & Bolton, T

    Zanna, L. & Bolton, T. Data -driven equation discovery of ocean mesoscale closures. Geophys Res Lett 47, (2020)

    work page 2020

  3. [3]

    Beetham, S., Fox, R. O. & Capecelatro, J. Sparse identification of multiphase turbulence closures for coupled fluid-particle flows. J Fluid Mech 914, (2021)

    work page 2021

  4. [4]

    Brunton, S. L. & Kutz, J. N. Promising directions of machine learning for partial differential equations. Nat Comput Sci 4, 483–494 (2024)

    work page 2024

  5. [5]

    L., Proctor, J

    Brunton, S. L., Proctor, J. L., Kutz, J. N. & Bialek, W. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc Natl Acad Sci U S A 113, 3932–3937 (2016)

    work page 2016

  6. [6]

    H., Brunton, S

    Rudy, S. H., Brunton, S. L., Proctor, J. L. & Kutz, J. N. Data -driven discovery of partial differential equations. Sci Adv 3, 1–7 (2017)

    work page 2017

  7. [7]

    & Earls, C

    Stephany, R. & Earls, C. Weak-PDE-LEARN: A weak form based approach to discovering PDEs from noisy, limited data. J Comput Phys 506, 112950 (2024)

    work page 2024

  8. [8]

    Messenger, D. A. & Bortz, D. M. Weak SINDy for partial differential equations. J Comput Phys 443, 110525 (2021)

    work page 2021

Show all 50 references
  1. [9]

    & Kang, S

    Tang, M., Liao, W., Kuske, R. & Kang, S. H. WeakIdent: Weak formulation for identifying differential equation using narrow-fit and trimming. J Comput Phys 483, 112069 (2023)

    2023

  2. [10]

    & Zhang, D

    Xu, H., Chang, H. & Zhang, D. DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm. J Comput Phys 418, 109584 (2020)

    2020

  3. [11]

    & Zhang, D

    Chen, Y ., Luo, Y ., Liu, Q., Xu, H. & Zhang, D. Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE). Phys Rev Res 4, (2022)

    2022

  4. [12]

    & Zhang, D

    Du, M., Chen, Y . & Zhang, D. DISCOVER: Deep identification of symbolically concise open-form partial differential equations via enhanced reinforcement learning. Phys Rev Res 6, (2024)

    2024

  5. [13]

    & Sun, H

    Sun, F., Liu, Y ., Wang, J.-X. & Sun, H. Symbolic Physics Learner: discovering governing equations via Monte Carlo tree search. arXiv preprint arXiv:2205.13134 (2022)

    2022 arXiv

  6. [14]

    & Virgolin, M

    Kamienny, P.-A., Lample, G., Lamprier, S. & Virgolin, M. Deep generative symbolic regression with monte-carlo-tree-search. in International Conference on Machine Learning 15655–15668 (PMLR, 2023)

    2023

  7. [15]

    & Chawla, S

    Makke, N. & Chawla, S. Interpretable scientific discovery with symbolic regression: a review. Artif Intell Rev 57, (2024)

    2024

  8. [16]

    Attention is all you need

    Vaswani, A. Attention is all you need. Adv Neural Inf Process Syst (2017)

    2017

  9. [17]

    & Zhang, D

    Du, M., Chen, Y ., Wang, Z., Nie, L. & Zhang, D. Large language models for automatic equation discovery of nonlinear dynamics. Physics of Fluids, 36(9), (2024)

    2024

  10. [18]

    Shojaee, P., Meidani, K., Gupta, S., Farimani, A. B. & Reddy, C. K. Llm -sr: Scientific equation discovery via programming with large language models. arXiv preprint arXiv:2404.18400 (2024)

    2024 arXiv

  11. [19]

    Li, Y . et al. Mllm-sr: Conversational symbolic regression base multi-modal large language models. arXiv preprint arXiv:2406.05410 (2024)

    2024 arXiv

  12. [20]

    Ye, Z. et al. Pdeformer: Towards a foundation model for one-dimensional partial differential 26 equations. arXiv preprint arXiv:2402.12652 (2024)

    2024 arXiv

  13. [21]

    & Schaeffer, H

    Sun, J., Liu, Y ., Zhang, Z. & Schaeffer, H. Towards a foundation model for partial differential equations: multi-operator learning and extrapolation. Physical Review E , 111(3), 035304, (2024)

    2024

  14. [22]

    Chen, J. et al. Symbol: Generating flexible black-box optimizers through symbolic equation learning. arXiv preprint arXiv:2402.02355. (2024)

    2024 arXiv

  15. [23]

    Mundhenk, T. N. et al. Symbolic regression via neural -guided genetic programming population seeding. In Proceedings of the 35th International Conference on Neural Information Processing Systems, 24912–24923 (2021)

    2021

  16. [24]

    Landajuela, M. et al. A unified framework for deep symbolic regression. In Proceedings of the 36th International Conference on Neural Information Processing Systems, 33985–33998 (2022)

    2022

  17. [25]

    & Ghodsi, A

    Valipour, M., You, B., Panju, M. & Ghodsi, A. SymbolicGPT: a generative transformer model for symbolic regression. arXiv preprint arXiv:2106.14131. (2021)

    2021 arXiv

  18. [26]

    Nonlinear Partial Differential Equations for Scientists and Engineers

    Debnath, L. Nonlinear Partial Differential Equations for Scientists and Engineers. (Springer, 2005)

    2005

  19. [27]

    Radford, A. et al. Language models are unsupervised multitask learners. OpenAI blog 1, 9 (2019)

    2019

  20. [28]

    & Zhang, D

    Xu, H., Chang, H. & Zhang, D. Dl-pde: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data. Commun Comput Phys 29, 698–728 (2021)

    2021

  21. [29]

    & Earls, C

    Stephany, R. & Earls, C. PDE -READ: Human -readable partial differential equation discovery using deep learning. Neural Networks 154, 360–382 (2022)

    2022

  22. [30]

    & Lin, G

    Zhang, S. & Lin, G. Robust data -driven discovery of governing physical laws with error bars. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, (2018)

    2018

  23. [31]

    Kaheman, K., Kutz, J. N. & Brunton, S. L. SINDy -PI: A robust algorithm for parallel implicit sparse identification of nonlinear dynamics: SINDy -PI. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, (2020)

    2020

  24. [32]

    N., Brunton, B

    Fasel, U., Kutz, J. N., Brunton, B. W. & Brunton, S. L. Ensemble -SINDy: Robust sparse model discovery in the low -data, high -noise limit, with active learning and control. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 478, (2022)

    2022

  25. [33]

    & Zhang, D

    Xu, H., Zeng, J. & Zhang, D. Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion. Research 6, 1–13 (2023)

    2023

  26. [34]

    & Zhang, D

    Xu, H. & Zhang, D. Robust discovery of partial differential equations in complex situations. Phys Rev Res 3, (2021)

    2021

  27. [35]

    Rudy, S., Alla, A., Brunton, S. L. & Kutz, J. N. Data -driven identification of parametric partial differential equations. SIAM J Appl Dyn Syst 18, 643–660 (2019)

    2019

  28. [36]

    Petersen, B. K. et al. Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. arXiv preprint arXiv:1912.04871 (2019)

    2019 arXiv

  29. [37]

    Udrescu, S. M. & Tegmark, M. AI Feynman: A physics -inspired method for symbolic regression. Sci Adv 6, (2020)

    2020

  30. [38]

    The Mathematica Book

    Wolfram, S. The Mathematica Book. Wolfram Media (2003). 27

    2003

  31. [39]

    Li, X., Zhang, D. & Li, S. A multi -continuum multiple flow mechanism simulator for unconventional oil and gas recovery. J Nat Gas Sci Eng 26, 652–669 (2015)

    2015

  32. [40]

    Cao, R., Padilla, E. M. & Callaghan, A. H. The influence of bandwidth on the energetics of intermediate to deep water laboratory breaking waves. J Fluid Mech 971, A11 (2023)

    2023

  33. [41]

    Rapp, R. J. & Melville, W. K. Laboratory measurements of deep -water breaking waves. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 331, 735–800 (1990)

    1990

  34. [42]

    M., Fang, Y

    Cao, R., Padilla, E. M., Fang, Y . & Callaghan, A. H. Identification of the free surface for unidirectional nonbreaking water waves from side -view digital images. IEEE Journal of Oceanic Engineering (2024)

    2024

  35. [43]

    H., Tang, T., Adcock, T

    Taylor, P. H., Tang, T., Adcock, T. A. A. & Zang, J. Transformed-FNV: Wave forces on a vertical cylinder — A free-surface formulation. Coastal Engineering 189, (2024)

    2024

  36. [44]

    Liu, P. C. Wavelet spectrum analysis and ocean wind waves. in Wavelet analysis and its applications vol. 4 151–166 (Elsevier, 1994)

    1994

  37. [45]

    & Enbo, W

    Jinshan, X., Jiwei, T. & Enbo, W. The application of wavelet transform to wave breaking. Acta Mechanica Sinica 14, 306–318 (1998)

    1998

  38. [46]

    T., Landajuela, M

    Kim, J. T., Landajuela, M. & Petersen, B. K. Distilling Wikipedia mathematical knowledge into neural network models. arXiv preprint arXiv:2104.05930. (2021)

    2021 arXiv

  39. [47]

    & Osindero, S

    Bornschein, J., Visin, F. & Osindero, S. Small data, big decisions: Model selection in the small-data regime. In International Conference on Machine Learning 1035–1044 (PMLR, 2020)

    2020

  40. [48]

    Kingma, D. P. & Ba, J. L. Adam: A method for stochastic optimization. in 3rd International Conference on Learning Representations, ICLR 2015 - Conference Track Proceedings (2015)

    2015

  41. [49]

    & Karniadakis, G

    Raissi, M., Perdikaris, P. & Karniadakis, G. E. Physics -informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378, 686–707 (2019). 28 Supplementary Information for the Gen...

    2019

  42. [50]

    smiley face region

    Supplementary text 1.1 Information on the dataset utilized in the experiments In the discovery of canonical PDEs, eight PDEs from different fields are employed to demonstrate the performance of the proposed generative framework. The information of these PDEs and their respecti...

Pith tools

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