pith. sign in

arxiv: 2606.09638 · v1 · pith:G7IKQR4Enew · submitted 2026-06-08 · 💻 cs.LG · cs.SC· math-ph· math.MP· physics.comp-ph· stat.AP

Data-driven discovery of governing differential equations across physical systems

Siyu Lou , Hao Xu , Wenguan Wang , Lu Lu , Hao Sun , Yang Liu , Linfeng Zhang , Dongxiao Zhang
show 1 more author
Yuntian Chen
This is my paper

Pith reviewed 2026-06-27 17:11 UTC · model grok-4.3

classification 💻 cs.LG cs.SCmath-phmath.MPphysics.comp-phstat.AP
keywords data-driven discoverydifferential equationsphase diagramREO frameworkscientific discoveryphysics applicationsequation learning
0
0 comments X

The pith

A phase diagram of structural and coefficient complexity plus the REO framework organizes data-driven differential equation discovery across methods.

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

The paper proposes a two-dimensional phase diagram that places discovery problems according to structural complexity and coefficient complexity to explain how the field has progressed from simple sparse equations to more complex laws. It introduces the representation-evaluation-optimization framework as an abstraction that isolates the persistent core problems shared by all algorithms, shifting attention from specific methods to the principles that govern what can be discovered. The authors apply these ideas to examples in physics and adjacent fields and conclude that the next stage requires using recovered equations to revise theories and form new concepts rather than stopping at recovery itself.

Core claim

The authors claim that the two-dimensional phase diagram of equation discoverability, with axes of structural complexity and coefficient complexity, accounts for why different methodological families succeed or fail in different settings, while the representation-evaluation-optimization framework abstracts the discovery process into three persistent components that determine what equations can be found from data across physical systems.

What carries the argument

The representation-evaluation-optimization (REO) framework, which decomposes any discovery algorithm into choosing a representation for candidate equations, evaluating their fit to data, and optimizing over that representation.

If this is right

  • Methods can be compared and improved by mapping them onto the regions of the phase diagram they can reliably address.
  • Discovery algorithms will need to advance along both axes simultaneously to handle richer structures and more flexible coefficients.
  • Recovered equations become inputs for revising existing physical theories rather than end products.

Where Pith is reading between the lines

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

  • The phase diagram could be used to design benchmark suites that systematically test coverage of complexity regions.
  • REO might serve as a template for analyzing related inverse problems such as discovering discrete rules or stochastic dynamics.

Load-bearing premise

That the two axes of structural complexity and coefficient complexity are sufficient to organize discoverability problems and explain method success or failure.

What would settle it

A data set whose governing equation lies in one region of the proposed phase diagram yet is recovered by a method predicted to fail there, or fails to be recovered by a method predicted to succeed.

Figures

Figures reproduced from arXiv: 2606.09638 by Dongxiao Zhang, Hao Sun, Hao Xu, Linfeng Zhang, Lu Lu, Siyu Lou, Wenguan Wang, Yang Liu, Yuntian Chen.

Figure 1
Figure 1. Figure 1: Overview of data-driven differential equation discovery. a Comparison between traditional first-principles and data-driven workflows for discovering differential equations. b Representative application domains in which differential equation discovery is used to model complex systems and uncover governing dynamics. c Major challenges in data-driven differential equation discovery. 2/35 [PITH_FULL_IMAGE:fig… view at source ↗
Figure 2
Figure 2. Figure 2: A phase diagram of equation discoverability based on structure and coefficient complexity. a Illustration of closed library, expandable library, and open-form library. b A two-dimensional taxonomy of existing differential equation discovery algorithms. The horizontal axis indicates increasing equation structure complexity (from closed-form libraries to open-form libraries), while the vertical axis indicate… view at source ↗
Figure 3
Figure 3. Figure 3: Overview of the REO framework for discovering governing equations. a Illustration of the REO pipeline, which integrates prior knowledge and observed data through representation, evaluation, and optimization to infer the governing equation. b The evaluation metrics for differential equation discovery, reflecting increasingly stringent criteria for scientific usability. c Representative optimization strategi… view at source ↗
Figure 4
Figure 4. Figure 4: Proof-of-concept validations reported across representative PDE discovery methods. A blue box indicates that the paper explicitly demonstrates recovery or identification of the equation (or standard variants). A pink box indicates that the paper does not demonstrate this explicitly. The purpose of this table is to showcase the canonical PDEs used in various studies, rather than to evaluate the capability o… view at source ↗
Figure 5
Figure 5. Figure 5: Outlook for differential equation discovery: from recovering equations to forming scientific concepts. The future development of differential equation discovery may be understood at two levels. At the methodological level, important challenges remain in dealing with noisy, partially observed, sparsely sampled, high-dimensional, and multiscale systems, as well as in establishing reliable evaluation protocol… view at source ↗
read the original abstract

Differential equations play a critical role in scientific discovery because they provide a mathematical framework to describe the behaviour of physical phenomena. As a promising alternative to traditional first principles, data-driven differential equation discovery has attracted increasing attention for its ability to infer governing laws directly from experimental or simulated data, especially when the underlying physics is unclear. However, the field has expanded rapidly along diverse methodological directions, particularly with the emergence of AI-based approaches, and still lacks a clear organizing perspective. In this Review, we propose a problem-oriented perspective on data-driven differential equation discovery. We first introduce a two-dimensional phase diagram of equation discoverability, where discovery problems are organized according to structural complexity and coefficient complexity. This phase diagram shows how the field has moved from the discovery of sparse equations with simple coefficients toward more complex governing laws with richer structures and more flexible parameterizations. It also clarifies why different methodological families succeed or fail in different problem settings. We then present the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process. By identifying the core problems of equation discovery that persist across algorithmic variations, REO shifts the discussion from individual algorithms to the fundamental principles that determine discoverability. We connect these perspectives to applications across physics and adjacent sciences, and argue that the next challenge is not merely recovering equations, but using them to revise existing theories, distil mechanisms and form new scientific concepts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper is a review proposing a two-dimensional phase diagram that organizes data-driven differential equation discovery problems according to structural complexity and coefficient complexity. It introduces the representation-evaluation-optimization (REO) framework as a fundamental abstraction of the discovery process that persists across algorithmic variations. The review connects these ideas to applications in physics and adjacent sciences and argues that the next challenge is using discovered equations to revise theories and form new concepts.

Significance. If the proposed perspectives hold, the work offers a unifying problem-oriented view that synthesizes the rapidly expanding literature on data-driven equation discovery, shifting emphasis from individual algorithms to core principles of discoverability. This could help clarify why methodological families succeed or fail in different settings and guide future research toward mechanism distillation and theory revision.

major comments (1)
  1. [Phase diagram] Phase diagram (as described in the abstract): the axes of structural complexity and coefficient complexity are introduced conceptually but without operational definitions, scoring functions, or explicit procedures for mapping an arbitrary discovery task onto coordinates. This directly undermines the asserted explanatory power that the diagram clarifies method success/failure and enables prediction across the field.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the recognition of the paper's potential to provide a unifying perspective. We address the single major comment below.

read point-by-point responses

  1. Referee: [Phase diagram] Phase diagram (as described in the abstract): the axes of structural complexity and coefficient complexity are introduced conceptually but without operational definitions, scoring functions, or explicit procedures for mapping an arbitrary discovery task onto coordinates. This directly undermines the asserted explanatory power that the diagram clarifies method success/failure and enables prediction across the field.

    Authors: We agree that the phase diagram is presented at a conceptual level in the current manuscript. Structural complexity is defined qualitatively as the syntactic form of the governing equation (e.g., number and type of terms, degree of nonlinearity), while coefficient complexity refers to the parameterization of those terms (e.g., constant scalars versus spatially or temporally varying functions). The manuscript illustrates these axes through multiple examples drawn from the literature, showing how different methodological families align with regions of the diagram. However, we acknowledge that formal scoring functions and a step-by-step mapping procedure for arbitrary tasks are not provided. In the revised manuscript we will add a new subsection that supplies (i) explicit, computable proxies for each axis based on equation structure and parameter type, (ii) worked examples that assign coordinates to published discovery problems, and (iii) a discussion of the diagram's intended use as an organizing heuristic rather than a quantitative predictor. This addition will preserve the high-level synthesis while addressing the concern about operational clarity. revision: yes

Circularity Check

0 steps flagged

No circularity: review organizes literature via conceptual axes without self-referential reductions

full rationale

This is a review paper that proposes a two-dimensional phase diagram (structural complexity vs. coefficient complexity) and the REO framework as organizing perspectives on existing equation-discovery methods. No derivation chain, quantitative predictions, or fitted parameters are introduced that reduce to inputs defined within the paper itself. The abstract and described content draw from cited prior work to classify problems and explain method performance, without any self-definitional, fitted-input, or uniqueness-imported steps. This matches the default expectation for non-circular reviews; the phase-diagram axes are presented conceptually rather than as outputs of an internal fit or self-citation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; no new free parameters, axioms, or invented entities are introduced beyond conceptual organization of existing methods. The phase diagram and REO are proposed abstractions rather than fitted quantities or postulated physical entities.

pith-pipeline@v0.9.1-grok · 5812 in / 1054 out tokens · 17685 ms · 2026-06-27T17:11:09.972883+00:00 · methodology

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

140 extracted references · 6 canonical work pages · 2 internal anchors

  1. [1]

    & Ott, E

    Pathak, J., Hunt, B., Girvan, M., Lu, Z. & Ott, E. Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach.Phys. review letters120, 024102 (2018)

    2018

  2. [2]

    & Karniadakis, G

    Lu, L., Jin, P., Pang, G., Zhang, Z. & Karniadakis, G. E. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators.Nat. machine intelligence3, 218–229 (2021)

    2021

  3. [3]

    InInternational Conference on Learning Representations(2021)

    Li, Z.et al.Fourier neural operator for parametric partial differential equations. InInternational Conference on Learning Representations(2021)

    2021

  4. [4]

    & Lipson, H

    Bongard, J. & Lipson, H. Automated reverse engineering of nonlinear dynamical systems.Proc. Natl. Acad. Sci.104, 9943–9948 (2007). 11.Schmidt, M. & Lipson, H. Distilling free-form natural laws from experimental data.science324, 81–85 (2009)

    2007

  5. [5]

    L., Proctor, J

    Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems.Proc. national academy sciences113, 3932–3937 (2016). 21/35

    2016

  6. [6]

    & Nair, P

    Course, K. & Nair, P. B. State estimation of a physical system with unknown governing equations.Nature622, 261–267 (2023)

    2023

  7. [7]

    Supekar, R.et al.Learning hydrodynamic equations for active matter from particle simulations and experiments.Proc. Natl. Acad. Sci.120, e2206994120 (2023)

    2023

  8. [8]

    A., Kageorge, L

    Reinbold, P. A., Kageorge, L. M., Schatz, M. F. & Grigoriev, R. O. Robust learning from noisy, incomplete, high- dimensional experimental data via physically constrained symbolic regression.Nat. communications12, 3219 (2021)

    2021

  9. [9]

    Li, Z.et al.Bi-level identification of governing equations for nonlinear physical systems.Nat. Comput. Sci.5, 456–466 (2025)

    2025

  10. [10]

    & Capecelatro, J

    Beetham, S. & Capecelatro, J. Formulating turbulence closures using sparse regression with embedded form invariance. Phys. Rev. Fluids5, 084611 (2020)

    2020

  11. [11]

    Beetham, S., Fox, R. O. & Capecelatro, J. Sparse identification of multiphase turbulence closures for coupled fluid–particle flows.J. Fluid Mech.914, A11 (2021)

    2021

  12. [12]

    Water Resour

    Song, W.et al.Deep learning identification of the governing equation for water flow in heterogeneous soils from data. Water Resour. Res.61, e2024WR037786 (2025)

    2025

  13. [13]

    Zhan, Y .et al.Physics-informed identification of pdes with lasso regression, examples of groundwater-related equations. J. Hydrol.638, 131504 (2024)

    2024

  14. [14]

    Xu, H., Wu, W., Chen, Y ., Zhang, D. & Mo, F. Explicit relation between thin film chromatography and column chromatography conditions from statistics and machine learning.Nat. Commun.16, 832 (2025). 22.Ying, J.et al.A neural symbolic model for space physics.Nat. Mach. Intell.1–16 (2025)

    2025

  15. [15]

    S., Wikle, C

    North, J. S., Wikle, C. K. & Schliep, E. M. A review of data-driven discovery for dynamic systems.Int. Stat. Rev.91, 464–492 (2023)

    2023

  16. [16]

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

    2024

  17. [17]

    & Wedelin, D

    Gennemark, P. & Wedelin, D. Benchmarks for identification of ordinary differential equations from time series data. Bioinformatics25, 780–786 (2009)

    2009

  18. [18]

    Shojaee, P., Meidani, K., Gupta, S., Farimani, A. B. & Reddy, C. K. Llm-sr: Scientific equation discovery via programming with large language models. InThe Thirteenth International Conference on Learning Representations(2025)

    2025

  19. [19]

    & Ahmed, S

    Sanderse, B., Stinis, P., Maulik, R. & Ahmed, S. E. Scientific machine learning for closure models in multiscale problems: A review.Foundations Data Sci.7, 298–337 (2025)

    2025

  20. [20]

    H., Brunton, S

    Rudy, S. H., Brunton, S. L., Proctor, J. L. & Kutz, J. N. Data-driven discovery of partial differential equations.Sci. advances3, e1602614 (2017)

    2017

  21. [21]

    Messenger, D. A. & Bortz, D. M. Weak sindy: Galerkin-based data-driven model selection.Multiscale Model. & Simul. 19, 1474–1497 (2021). 30.Messenger, D. A. & Bortz, D. M. Weak sindy for partial differential equations.J. Comput. Phys.443, 110525 (2021)

    2021

  22. [22]

    & Garikipati, K

    Wang, Z., Huan, X. & Garikipati, K. Variational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise.Comput. Methods Appl. Mech. Eng.356, 44–74 (2019)

    2019

  23. [23]

    & Garikipati, K

    Wang, Z., Huan, X. & Garikipati, K. Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data.Comput. Methods Appl. Mech. Eng.377, 113706 (2021). 33.Loiseau, J.-C. & Brunton, S. L. Constrained sparse galerkin regression.J. Fluid Mech.838, 42–67 (2018)

    2021

  24. [24]

    Kaiser, E., Kutz, J. N. & Brunton, S. L. Sparse identification of nonlinear dynamics for model predictive control in the low-data limit.Proc. Royal Soc. A474, 20180335 (2018)

    2018

  25. [25]

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

    2019

  26. [26]

    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.Proc. Royal Soc. A478, 20210904 (2022). 22/35

    2022

  27. [27]

    L., Sbalzarini, I

    Maddu, S., Cheeseman, B. L., Sbalzarini, I. F. & Müller, C. L. Stability selection enables robust learning of differential equations from limited noisy data.Proc. Royal Soc. A478, 20210916 (2022)

    2022

  28. [28]

    Learning partial differential equations via data discovery and sparse optimization.Proc

    Schaeffer, H. Learning partial differential equations via data discovery and sparse optimization.Proc. Royal Soc. A: Math. Phys. Eng. Sci.473, 20160446 (2017)

    2017

  29. [29]

    Kaheman, K., Kutz, J. N. & Brunton, S. L. Sindy-pi: a robust algorithm for parallel implicit sparse identification of nonlinear dynamics.Proc. Royal Soc. A476, 20200279 (2020)

    2020

  30. [30]

    Open Source Softw.7, 3994 (2022)

    Kaptanoglu, A.et al.Pysindy: A comprehensive python package for robust sparse system identification.J. Open Source Softw.7, 3994 (2022)

    2022

  31. [31]

    & Brenner, M

    Bar-Sinai, Y ., Hoyer, S., Hickey, J. & Brenner, M. P. Learning data-driven discretizations for partial differential equations. Proc. Natl. Acad. Sci.116, 15344–15349 (2019)

    2019

  32. [32]

    & Kusters, R

    Both, G.-J., Choudhury, S., Sens, P. & Kusters, R. Deepmod: Deep learning for model discovery in noisy data.J. Comput. Phys.428, 109985 (2021)

    2021

  33. [33]

    & Sun, H

    Chen, Z., Liu, Y . & Sun, H. Physics-informed learning of governing equations from scarce data.Nat. communications12, 6136 (2021)

    2021

  34. [34]

    & Fukui, K.-i

    Thanasutives, P., Morita, T., Numao, M. & Fukui, K.-i. Noise-aware physics-informed machine learning for robust pde discovery.Mach. Learn. Sci. Technol.4, 015009 (2023)

    2023

  35. [35]

    Yuan, Y .et al.Machine discovery of partial differential equations from spatiotemporal data: A sparse bayesian learning framework.Chaos: An Interdiscip. J. Nonlinear Sci.33(2023)

    2023

  36. [36]

    & Earls, C

    Stephany, R. & Earls, C. Pde-read: Human-readable partial differential equation discovery using deep learning.Neural Networks154, 360–382 (2022)

    2022

  37. [37]

    & Townsend, A

    Boullé, N., Nakatsukasa, Y . & Townsend, A. Rational neural networks.Adv. neural information processing systems33, 14243–14253 (2020)

    2020

  38. [38]

    & Earls, C

    Stephany, R. & Earls, C. Pde-learn: Using deep learning to discover partial differential equations from noisy, limited data. Neural Networks174, 106242 (2024)

    2024

  39. [39]

    & Yin, W

    Chartrand, R. & Yin, W. Iteratively reweighted algorithms for compressive sensing. In2008 IEEE international conference on acoustics, speech and signal processing, 3869–3872 (IEEE, 2008)

    2008

  40. [40]

    & 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)

    2024

  41. [41]

    InInternational conference on learning representations, vol

    Liu, Z.et al.Kan: Kolmogorov–arnold networks. InInternational conference on learning representations, vol. 2025, 70367–70413 (2025)

    2025

  42. [42]

    C., Kim, S

    Koenig, B. C., Kim, S. & Deng, S. Kan-odes: Kolmogorov–arnold network ordinary differential equations for learning dynamical systems and hidden physics.Comput. Methods Appl. Mech. Eng.432, 117397 (2024)

    2024

  43. [43]

    & Tegmark, M

    Kantamneni, S., Liu, Z. & Tegmark, M. Optpde: Discovering novel integrable systems via ai-human collaboration.arXiv preprint arXiv:2405.04484(2024)

    work page arXiv 2024

  44. [44]

    & Dong, B

    Long, Z., Lu, Y ., Ma, X. & Dong, B. Pde-net: Learning pdes from data. InInternational conference on machine learning, 3208–3216 (PMLR, 2018). 55.So, C. C., Li, T. O., Wu, C. & Yung, S. P. Differential spectral normalization (dsn) for pde discovery. InProceedings of the AAAI Conference on Artificial Intelligence, vol. 35, 9675–9684 (2021)

    2018

  45. [45]

    & Sun, H

    Rao, C., Ren, P., Liu, Y . & Sun, H. Discovering nonlinear pdes from scarce data with physics-encoded learning.Int. Conf. on Learn. Represent.(2022)

    2022

  46. [46]

    & Kalyuzhnaya, A

    Maslyaev, M., Hvatov, A. & Kalyuzhnaya, A. Data-driven partial derivative equations discovery with evolutionary approach. InInternational Conference on Computational Science, 635–641 (Springer, 2019)

    2019

  47. [47]

    & 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

  48. [48]

    & 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. physics378, 686–707 (2019)

    2019

  49. [49]

    & Zhang, D

    Xu, H. & Zhang, D. Robust discovery of partial differential equations in complex situations.Phys. Rev. Res.3, 033270 (2021). 23/35

    2021

  50. [50]

    & Dong, B

    Long, Z., Lu, Y . & Dong, B. Pde-net 2.0: Learning pdes from data with a numeric-symbolic hybrid deep network.J. Comput. Phys.399, 108925 (2019)

    2019

  51. [51]

    xiangnan et al.Nonlinear Dyn.1–21 (2025)

    Yu, X.et al.A data-driven framework for discovering fractional differential equations in complex systems: Y . xiangnan et al.Nonlinear Dyn.1–21 (2025). 63.Virgolin, M. & Pissis, S. P. Symbolic regression is np-hard.Transactions on Mach. Learn. Res.(2022)

    2025

  52. [52]

    & V oigt, H

    Kahlmeyer, P., Giesen, J., Habeck, M. & V oigt, H. Scaling up unbiased search-based symbolic regression. InProceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, 4264–4272 (2024)

    2024

  53. [53]

    & Sun, H

    Sun, F., Liu, Y ., Wang, J.-X. & Sun, H. Symbolic physics learner: Discovering governing equations via monte carlo tree search. InThe Eleventh International Conference on Learning Representations(2023). 66.Ruan, K.et al.Discovering physical laws with parallel symbolic enumeration.Nat. Comput. Sci.6, 53–66 (2026). 67.Dub ˇcáková, R. Eureqa: software review (2011)

    2023

  54. [54]

    & Tonda, A

    Gaucel, S., Keijzer, M., Lutton, E. & Tonda, A. Learning dynamical systems using standard symbolic regression. In European Conference on Genetic Programming, 25–36 (Springer, 2014)

    2014

  55. [55]

    & Spector, L

    La Cava, W., Danai, K. & Spector, L. Inference of compact nonlinear dynamic models by epigenetic local search.Eng. Appl. Artif. Intell.55, 292–306 (2016)

    2016

  56. [56]

    & 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, 023174 (2022)

    2022

  57. [57]

    G., Beykal, B

    Cohen, B. G., Beykal, B. & Bollas, G. M. Physics-informed genetic programming for discovery of partial differential equations from scarce and noisy data.J. Comput. Phys.514, 113261 (2024)

    2024

  58. [58]

    & Tegmark, M

    Udrescu, S.-M. & Tegmark, M. Ai feynman: A physics-inspired method for symbolic regression.Sci. advances6, eaay2631 (2020)

    2020

  59. [59]

    neural information processing systems33, 17429–17442 (2020)

    Cranmer, M.et al.Discovering symbolic models from deep learning with inductive biases.Adv. neural information processing systems33, 17429–17442 (2020)

    2020

  60. [60]

    Y ., Ariño Bernad, J

    Lu, P. Y ., Ariño Bernad, J. & Soljaˇci´c, M. Discovering sparse interpretable dynamics from partial observations.Commun. Phys.5, 206 (2022)

    2022

  61. [61]

    & Locatello, F

    Pervez, A., Gavves, E. & Locatello, F. Mechanistic pde networks for discovery of governing equations. InForty-second International Conference on Machine Learning(2025)

    2025

  62. [62]

    K.et al.Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients

    Petersen, B. K.et al.Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. InInternational Conference on Learning Representations(2021)

    2021

  63. [63]

    & 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, 013182 (2024)

    2024

  64. [64]

    & Zhang, D

    Du, M., Chen, Y ., Nie, L., Lou, S. & Zhang, D. Physics-constrained robust learning of open-form partial differential equations from limited and noisy data.Phys. Fluids36(2024)

    2024

  65. [65]

    & Kilbertus, N

    d’Ascoli, S., Becker, S., Schwaller, P., Mathis, A. & Kilbertus, N. Odeformer: Symbolic regression of dynamical systems with transformers. InInternational Conference on Learning Representations(2024)

    2024

  66. [66]

    communications 16, 10255 (2025)

    Xu, H.et al.Generative discovery of partial differential equations by learning from math handbooks.Nat. communications 16, 10255 (2025)

    2025

  67. [67]

    Meidani, K., Shojaee, P., Reddy, C. K. & Farimani, A. B. Snip: Bridging mathematical symbolic and numeric realms with unified pre-training. InThe Twelfth International Conference on Learning Representations(2024)

    2024

  68. [68]

    & Chen, Y

    Li, Q., Hu, Y ., Liu, J. & Chen, Y . Gensr: Symbolic regression based in equation generative space. InInternational Conference on Learning Representations(2026)

    2026

  69. [69]

    & Zhang, D

    Du, M., Chen, Y ., Wang, Z., Nie, L. & Zhang, D. Large language models for automatic equation discovery of nonlinear dynamics.Phys. Fluids36(2024)

    2024

  70. [70]

    In Proceedings of the 41st International Conference on Machine Learning, 33940–33962 (2024)

    Ma, P.et al.Llm and simulation as bilevel optimizers: a new paradigm to advance physical scientific discovery. In Proceedings of the 41st International Conference on Machine Learning, 33940–33962 (2024)

    2024

  71. [71]

    & Chaudhuri, S

    Grayeli, A., Sehgal, A., Costilla Reyes, O., Cranmer, M. & Chaudhuri, S. Symbolic regression with a learned concept library.Adv. Neural Inf. Process. Syst.37, 44678–44709 (2024). 24/35

    2024

  72. [72]

    & Lehman, L.-w

    Li, J., Sun, G., Zhao, G. & Lehman, L.-w. H. Robust low-rank discovery of data-driven partial differential equations. In Proceedings of the AAAI Conference on Artificial Intelligence, vol. 34, 767–774 (2020)

    2020

  73. [73]

    & Zeng, J

    Xu, H., Zhang, D. & Zeng, J. Deep-learning of parametric partial differential equations from sparse and noisy data.Phys. Fluids33(2021)

    2021

  74. [74]

    Zhang, M., Kim, S., Lu, P. Y . & Soljaˇci´c, M. Deep learning and symbolic regression for discovering parametric equations. IEEE Transactions on Neural Networks Learn. Syst.(2023)

    2023

  75. [75]

    & Zhang, D

    Xu, H., Chen, Y . & Zhang, D. Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discovery.arXiv preprint arXiv:2511.09906(2025)

    work page arXiv 2025

  76. [76]

    & Phoon, K

    Huang, S., Quek, S. & Phoon, K. Convergence study of the truncated karhunen–loeve expansion for simulation of stochastic processes.Int. journal for numerical methods engineering52, 1029–1043 (2001)

    2001

  77. [77]

    Zhang, D. & Lu, Z. An efficient, high-order perturbation approach for flow in random porous media via karhunen–loeve and polynomial expansions.J. Comput. Phys.194, 773–794 (2004)

    2004

  78. [78]

    C., Tripura, T., Nayek, R

    Mathpati, Y . C., Tripura, T., Nayek, R. & Chakraborty, S. Discovering stochastic partial differential equations from limited data using variational bayes inference.Comput. Methods Appl. Mech. Eng.418, 116512 (2024)

    2024

  79. [79]

    InProceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 1595–1607 (2023)

    Luo, Y .et al.Physics-guided discovery of highly nonlinear parametric partial differential equations. InProceedings of the 29th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, 1595–1607 (2023)

    2023

  80. [80]

    & Parascandolo, G

    Biggio, L., Bendinelli, T., Lucchi, A. & Parascandolo, G. A seq2seq approach to symbolic regression. InLearning meets combinatorial algorithms at NeurIPS2020(2020)

    2020

Showing first 80 references.