pith. sign in

arxiv: 2606.20467 · v1 · pith:INNBXSWRnew · submitted 2026-06-18 · 💻 cs.LG · cs.NA· math.NA· physics.comp-ph

Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks

Zongmin Yu , Liu Yang This is my paper

Pith reviewed 2026-06-26 17:53 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NAphysics.comp-ph
keywords Agentic Symbolic SearchPDE characterizationsymbolic regressionevolutionary searchAllen-Cahn equationKeller-Segel modelinterpretable representationsgradient optimization
0
0 comments X

The pith

Agentic Symbolic Search turns PDE theory into evolved symbolic programs that yield closed-form approximations for previously intractable solution behaviors.

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

The paper proposes Agentic Symbolic Search (ASYS) to automate the creation of mathematical structures for PDE solutions. An agent converts known theory and constraints into differentiable symbolic expressions, which evolutionary search refines while gradient descent tunes continuous parameters. Experiments on five problems demonstrate recovery of known forms and discovery of new interpretable expressions, such as a geometric interface formula for two-dimensional Allen-Cahn dynamics and a nine-parameter contraction law for Keller-Segel blow-up. A reader would care because this approach supplies analytical insight directly rather than relying on numerical tables or opaque neural approximations.

Core claim

ASYS is a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. Across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up in settings where no closed-form description was previously available.

What carries the argument

Agentic Symbolic Search (ASYS), a framework that converts PDE theory into evolutionary-refined differentiable symbolic programs whose parameters are optimized by gradients.

If this is right

  • For problems with known analytical forms, ASYS recovers these forms naturally.
  • For problems without known forms, ASYS constructs analytical approximations that can guide further mathematical analysis.
  • ASYS offers a new paradigm for characterizing PDE solutions beyond hand-crafted expressions, meshes, and neural networks.

Where Pith is reading between the lines

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

  • The discovered symbolic expressions could serve as starting points for proving stability or existence results that were previously inaccessible.
  • Applying the same agentic loop to families of related PDEs might expose shared structural patterns across equation classes.
  • Coupling ASYS outputs with automated theorem provers could turn the generated formulas into machine-checked statements about long-time behavior.

Load-bearing premise

An agent can reliably convert PDE theory and constraints into symbolic programs whose evolutionary refinement will capture the essential solution structure.

What would settle it

Apply ASYS to a PDE whose exact closed-form solution is known and check whether the method recovers that exact form; or compare the discovered Allen-Cahn interface formula and Keller-Segel contraction law against high-resolution numerical simulations of those equations.

Figures

Figures reproduced from arXiv: 2606.20467 by Liu Yang, Zongmin Yu.

Figure 1
Figure 1. Figure 1: ASYS alternates between an outer loop over mathematical structure and an inner loop over continuous parameters. The agent reads problem guidance, previous high-scoring representations, and diagnostic summaries, then revises the representation, the training objective, and the optimizer configuration. The evaluator fits the resulting ansatz by quasi-Newton optimization and scores it using PDE-residual and pu… view at source ↗
Figure 2
Figure 2. Figure 2: ASYS uses EvE as its outer-loop search mechanism. EvE maintains the pool of representations, samples stronger ones more often as later context, and drives selection; ASYS specializes this framework to differentiable symbolic programs for PDE solution characterization. Physics residual The physics dimension measures equation satisfaction through the PDE residual evaluated on held-out collocation points. The… view at source ↗
Figure 3
Figure 3. Figure 3: NLS: breather profile snapshots for the 𝐿 2 -best symbolic candidate. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: NLS (continued): residual scoring reaches a purely symbolic soliton-form candidate, and offline validation confirms the resulting profile. While NLS is a case of analytic recall, Allen–Cahn 2D requires geometric construction: there is no classical formula for the evolving interface. Allen–Cahn 2D The two-dimensional Allen–Cahn equation (Allen and Cahn, 1979), 𝑢𝑡 = 𝜀 2Δ𝑢 + 𝑢 − 𝑢 3 , (𝑥, 𝑦) ∈ [−1, 1) 2 , 𝑡 ∈… view at source ↗
Figure 4
Figure 4. Figure 4: Allen–Cahn 2D: the 𝐿 2 -best symbolic candidate tracks the peanut-to-oval geometry. 0 1 2 3 4 5 outer-loop iteration 10 −4 10 −3 10 −2 physics loss (raw MSE) Allen-Cahn 2D 10 −2 10 −1 best-so-far validation L 2 physics loss (raw MSE) best-so-far validation L 2 L 2 -best candidate (b) Dual-axis trajectory: best-so-far validation 𝐿 2 drops early and the raw PDE residual continues to decrease through iteratio… view at source ↗
Figure 5
Figure 5. Figure 5: Radial profiles for the 𝐿 2 -best representation at iteration 8. Solid curves show the best representation found; dashed curves show the finite-volume reference. contrast: while SS-PINN requires 49,921 learned weights to fit the transformed profile, ASYS condenses the underlying dynamic mechanism into nine interpretable trainable scalars within an explicit contraction law. The best candidate blends an earl… view at source ↗
Figure 6
Figure 6. Figure 6: Keller–Segel: the 𝐿 2 -best candidate appears at iteration 8 with relative 𝐿 2 = 0.188 and nine parameters; the trajectory makes the residual-versus-validation distinction explicit. 3.3 Graveleau PME Focusing Keller–Segel shows progressive structural enrichment: the search builds its representation piece by piece across iterations. Graveleau tests a different mode: can ASYS instantiate a known mathematical… view at source ↗
Figure 7
Figure 7. Figure 7: should be read as a scaling-law check, not only as a profile overlay. The log–log radius analysis is the standard way to expose a second-kind exponent, and here it gives 𝛽model = 0.928 against the reference fit 𝛽ref = 0.877. The 6% gap reflects the indirect determination: the evaluator scores PDE satisfaction and public constraints, not the free-boundary radius. Despite this gap, the low validation 𝐿 2 = 0… view at source ↗
Figure 8
Figure 8. Figure 8: Graveleau residual and offline validation trajectory. The lowest raw residual occurs at the initial candidate, but the best profile match appears later at iteration 4, after the second-kind self-similar free-boundary structure is introduced. 3.4 Stress Test: gCLM To probe the expressivity limits of ASYS, we consider a final stress test where the underlying equation features strong nonlocal coupling. The ge… view at source ↗
Figure 9
Figure 9. Figure 9: gCLM stress test: the 𝐿 2 -best candidate misses the late-time concentration. 0 1 2 3 4 5 6 7 8 9 10 outer-loop iteration 10 −4 10 −3 physics loss (ρgeo) gCLM 10 0 2 × 10 −1 3 × 10 −1 4 × 10 −1 6 × 10 −1 best-so-far validation L 2 SS-PINN physics loss (ρgeo) best-so-far validation L 2 L 2 -best candidate (b) Dual-axis trajectory: the physics loss decreases, but best-so-far validation 𝐿 2 remains high; the … view at source ↗
read the original abstract

Mathematicians understand a PDE solution through mathematical structures rather than tables of computed values. Historically, this has been the product of mathematical analysis, carried out by hand for each problem individually. Neither numerical simulation nor neural networks produce those structures directly. We propose Agentic Symbolic Search (ASYS), a prior-guided framework in which an agent translates PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs. The mathematical forms are refined under evolutionary search, while their continuous parameters are fit by gradient-based optimization. This makes the search an automated form of inductive-bias injection rather than blind symbolic regression. For problems with known analytical forms, ASYS recovers these forms naturally; for other problems, ASYS constructs analytical approximations which can guide mathematicians toward further analysis. In our experiments, across five problems spanning bounded dynamics, finite-time blow-up, and free-boundary focusing, ASYS produces interpretable representations, including a geometric interface formula for Allen-Cahn 2D dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up, in settings where no closed-form description was previously available. ASYS shows the possibility of a new paradigm for characterizing PDE solutions, beyond handcrafted analytical solutions, mesh-based numerical solutions, and neural network approximations.

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

2 major / 0 minor

Summary. The paper introduces Agentic Symbolic Search (ASYS), a framework in which an LLM-based agent translates PDE theory, constraints, and search experience into differentiable symbolic programs. These programs are refined via evolutionary search over their structure while continuous parameters are optimized by gradient descent. The central claim is that ASYS recovers known closed-form solutions on problems where they exist and, on five problems without prior closed forms (bounded dynamics, finite-time blow-up, free-boundary focusing), yields new interpretable representations such as a geometric interface formula for 2D Allen-Cahn dynamics and a nine-parameter contraction law for Keller-Segel chemotactic blow-up.

Significance. If the generated symbolic forms can be shown to encode dominant dynamics rather than incidental fits, the method would constitute a genuine advance in automated discovery of mathematical structure for PDEs, complementing both hand analysis and numerical/neural approaches. The absence of quantitative error metrics, baseline comparisons, ablation studies, or explicit residual definitions in the provided description, however, prevents assessment of whether the reported representations satisfy this standard.

major comments (2)
  1. Abstract: the claim that ASYS 'produces interpretable representations ... in settings where no closed-form description was previously available' is not accompanied by any quantitative error metrics, baseline comparisons, ablation studies, or explicit definition of how success (e.g., PDE residual, boundary-condition satisfaction, blow-up scaling) was measured; without these the data-to-claim link cannot be verified.
  2. Abstract (method description): the framework assumes an agent reliably converts PDE theory into 'testable differentiable symbolic programs' whose evolutionary refinement captures essential structure. No concrete operator sets, initial program templates, or residual definitions are supplied, leaving open the possibility that reported forms are over-parameterized approximations rather than mathematically useful characterizations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and will make revisions to improve clarity and verifiability of the claims.

read point-by-point responses

  1. Referee: Abstract: the claim that ASYS 'produces interpretable representations ... in settings where no closed-form description was previously available' is not accompanied by any quantitative error metrics, baseline comparisons, ablation studies, or explicit definition of how success (e.g., PDE residual, boundary-condition satisfaction, blow-up scaling) was measured; without these the data-to-claim link cannot be verified.

    Authors: We agree that the abstract would be strengthened by including quantitative support. The full manuscript reports PDE residual norms, comparisons to numerical solutions, and explicit success criteria (residual thresholds, boundary-condition satisfaction, and scaling exponents) in Sections 4–5. We will revise the abstract to incorporate key error metrics and a concise definition of the residual measure used. revision: yes

  2. Referee: Abstract (method description): the framework assumes an agent reliably converts PDE theory into 'testable differentiable symbolic programs' whose evolutionary refinement captures essential structure. No concrete operator sets, initial program templates, or residual definitions are supplied, leaving open the possibility that reported forms are over-parameterized approximations rather than mathematically useful characterizations.

    Authors: The manuscript specifies the operator sets (arithmetic, differentiation, integration, and PDE-specific operators) in Section 3.1, initial program templates derived from theory in Section 3.2, and residual definitions (including boundary terms and blow-up indicators) in Section 4.2. To make these elements immediately visible, we will add a brief summary of the operator library and residual computation to the abstract. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper describes ASYS as an agent-driven search process that translates external PDE theory, constraints, and search experience into symbolic programs, which are then refined by evolutionary search and gradient-based parameter fitting. The central outputs (e.g., recovered known forms or new approximations such as the geometric interface or nine-parameter contraction law) are generated results of this search rather than quantities defined by construction from the inputs or from fitted parameters renamed as predictions. No self-definitional equations, load-bearing self-citations, uniqueness theorems imported from the authors, or ansatzes smuggled via citation appear in the abstract or described framework. The derivation chain remains self-contained as a method for inductive-bias injection, with experiments serving as external validation on benchmark problems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on the assumption that an agent can produce usable differentiable symbolic programs from PDE theory; no explicit free parameters, additional axioms, or invented physical entities are described.

axioms (1)
  • domain assumption An agent can translate PDE theory, public problem constraints, and accumulated search experience into testable differentiable symbolic programs
    This premise is required for the framework to generate candidate expressions that can then be refined.

pith-pipeline@v0.9.1-grok · 5766 in / 1264 out tokens · 35274 ms · 2026-06-26T17:53:22.754861+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

71 extracted references · 1 canonical work pages

  1. [1]

    Updating quasi-

    Nocedal, Jorge , journal=. Updating quasi-. 1980 , doi=

    1980

  2. [2]

    arXiv preprint arXiv:2605.09018 , year=

    Evolutionary Ensemble of Agents , author=. arXiv preprint arXiv:2605.09018 , year=

    Pith/arXiv arXiv

  3. [3]

    arXiv preprint arXiv:2604.23472 , year=

    Escher-Loop: Mutual Evolution by Closed-Loop Self-Referential Optimization , author=. arXiv preprint arXiv:2604.23472 , year=

    Pith/arXiv arXiv

  4. [4]

    1992 , publisher=

    Genetic Programming: On the Programming of Computers by Means of Natural Selection , author=. 1992 , publisher=

    1992

  5. [5]

    Nature , volume=

    Mathematical discoveries from program search with large language models , author=. Nature , volume=. 2024 , publisher=

    2024

  6. [6]

    2025 , doi=

    Novikov, Alexander and V. 2025 , doi=

    2025

  7. [7]

    , booktitle=

    Shojaee, Parshin and Meidani, Kazem and Gupta, Shashank and Farimani, Amir Barati and Reddy, Chandan K. , booktitle=. 2025 , note=. doi:10.48550/arXiv.2404.18400 , pages=

    work page doi:10.48550/arxiv.2404.18400 2025

  8. [8]

    2025 , doi=

    Xia, Shijie and Sun, Yuhan and Liu, Pengfei , journal=. 2025 , doi=

    2025

  9. [9]

    2026 , note=

    Li, Shanda and Marwah, Tanya and Shen, Junhong and Sun, Weiwei and Risteski, Andrej and Yang, Yiming and Talwalkar, Ameet , journal=. 2026 , note=

    2026

  10. [10]

    2026 , doi=

    Du, Jianda and Sun, Youran and Yang, Haizhao , journal=. 2026 , doi=

    2026

  11. [11]

    and Gholami, Amir , journal=

    Gaonkar, Saarth and Zheng, Xiang and Xi, Haocheng and Tiwari, Rishabh and Keutzer, Kurt and Morozov, Dmitriy and Mahoney, Michael W. and Gholami, Amir , journal=. 2025 , doi=

    2025

  12. [12]

    and Karniadakis, George Em , journal=

    Toscano, Juan Diego and Chen, Daniel T. and Karniadakis, George Em , journal=. 2025 , doi=

    2025

  13. [13]

    2026 , doi=

    Toscano, Juan Diego and Chai, Zhaojie and Karniadakis, George Em , journal=. 2026 , doi=

    2026

  14. [14]

    2025 , doi=

    Liu, Jianming and Zhu, Ren and Xu, Jian and Ding, Kun and Zhang, Xu-Yao and Meng, Gaofeng and Liu, Cheng-Lin , journal=. 2025 , doi=

    2025

  15. [15]

    2025 , note=

    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 , booktitle=. 2025 , note=

    2025

  16. [16]

    Journal of Computational Physics , volume=

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , author=. Journal of Computational Physics , volume=. 2019 , publisher=

    2019

  17. [17]

    International Conference on Learning Representations , year=

    Fourier Neural Operator for Parametric Partial Differential Equations , author=. International Conference on Learning Representations , year=

  18. [18]

    Learning nonlinear operators via

    Lu, Lu and Jin, Pengzhan and Pang, Guofei and Zhang, Zhongqiang and Karniadakis, George Em , journal=. Learning nonlinear operators via. 2021 , doi=

    2021

  19. [19]

    Proceedings of the National Academy of Sciences , volume=

    In-context Operator Learning with Data Prompts for Differential Equation Problems , author=. Proceedings of the National Academy of Sciences , volume=. 2023 , doi=

    2023

  20. [20]

    , journal=

    Yang, Liu and Osher, Stanley J. , journal=. 2024 , doi=

    2024

  21. [21]

    arXiv preprint arXiv:2404.19756 , year=

    Liu, Ziming and Wang, Yixuan and Vaidya, Sachin and Ruehle, Fabian and Halverson, James and Solja. arXiv preprint arXiv:2404.19756 , year=

    Pith/arXiv arXiv

  22. [22]

    arXiv preprint arXiv:2001.04385 , year=

    Universal differential equations for scientific machine learning , author=. arXiv preprint arXiv:2001.04385 , year=

    Pith/arXiv arXiv 2001

  23. [23]

    Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric

    Wang, Yongji and Lai, Ching-Yao and G. Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric. Physical Review Letters , volume=. 2023 , doi=

    2023

  24. [24]

    arXiv preprint arXiv:2509.14185 , year=

    Discovery of Unstable Singularities , author=. arXiv preprint arXiv:2509.14185 , year=

    arXiv

  25. [25]

    Journal of Computational Physics , volume=

    Multi-stage neural networks: Function approximator of machine precision , author=. Journal of Computational Physics , volume=. 2024 , doi=

    2024

  26. [26]

    , journal=

    Wang, Yixuan and Liu, Ziming and Li, Zongyi and Anandkumar, Anima and Hou, Thomas Y. , journal=. High Precision. 2025 , doi=

    2025

  27. [27]

    Investigating the Ability of

    Kumar, Dibyakanti and Mukherjee, Anirbit , journal=. Investigating the Ability of. 2024 , doi=

    2024

  28. [28]

    Proceedings of the National Academy of Sciences , volume=

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems , author=. Proceedings of the National Academy of Sciences , volume=. 2016 , doi=

    2016

  29. [29]

    Interpretable Machine Learning for Science with

    Cranmer, Miles , journal=. Interpretable Machine Learning for Science with. 2023 , doi=

    2023

  30. [30]

    2020 , doi=

    Udrescu, Silviu-Marian and Tegmark, Max , journal=. 2020 , doi=

    2020

  31. [31]

    International Conference on Learning Representations , year=

    Deep Symbolic Regression: Recovering Mathematical Expressions from Data via Risk-Seeking Policy Gradients , author=. International Conference on Learning Representations , year=

  32. [32]

    Journal of Machine Learning Research , volume=

    Finite Expression Method for Solving High-Dimensional Partial Differential Equations , author=. Journal of Machine Learning Research , volume=. 2025 , note=

    2025

  33. [33]

    Neuro-Symbolic

    Oikonomou, Orestis and Lingsch, Levi and Grund, Dana and Mishra, Siddhartha and Kissas, Georgios , journal=. Neuro-Symbolic. 2025 , doi=

    2025

  34. [34]

    Proceedings of the 42nd International Conference on Machine Learning , series=

    Closed-form Solutions: A New Perspective on Solving Differential Equations , author=. Proceedings of the 42nd International Conference on Machine Learning , series=. 2025 , note=

    2025

  35. [35]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    An Interpretable Approach to the Solutions of High-Dimensional Partial Differential Equations , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=. 2024 , doi=

    2024

  36. [36]

    1996 , doi=

    Scaling, Self-Similarity, and Intermediate Asymptotics , author=. 1996 , doi=

    1996

  37. [37]

    Physical Review Letters , volume=

    Universality and scaling in gravitational collapse of a massless scalar field , author=. Physical Review Letters , volume=. 1993 , doi=

    1993

  38. [38]

    Numerical simulation of singular solutions to the two-dimensional cubic

    Sulem, P-L and Sulem, C and Patera, A , journal=. Numerical simulation of singular solutions to the two-dimensional cubic. 1984 , doi=

    1984

  39. [39]

    LeMesurier, B. J. and Papanicolaou, G. C. and Sulem, Catherine and Sulem, Pierre-Louis , journal=. Focusing and Multi-Focusing Solutions of the Nonlinear. 1988 , doi=

    1988

  40. [40]

    On universality of blow-up profile for

    Merle, Frank and Rapha. On universality of blow-up profile for. Inventiones Mathematicae , volume=. 2004 , doi=

    2004

  41. [41]

    The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear

    Merle, Frank and Rapha. The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear. Annals of Mathematics , volume=. 2005 , doi=

    2005

  42. [42]

    Universal pinching of 3

    Eggers, Jens , journal=. Universal pinching of 3. 1993 , doi=

    1993

  43. [43]

    2015 , publisher=

    Singularities: Formation, Structure, and Propagation , author=. 2015 , publisher=

    2015

  44. [44]

    Nonlinearity , volume=

    The Role of Self-Similarity in Singularities of Partial Differential Equations , author=. Nonlinearity , volume=. 2009 , doi=

    2009

  45. [45]

    Potentially Singular Solutions of the 3

    Luo, Guo and Hou, Thomas Y , journal=. Potentially Singular Solutions of the 3. 2014 , doi=

    2014

  46. [46]

    , journal=

    Chen, Jiajie and Hou, Thomas Y. , journal=. Stable Nearly Self-Similar Blowup of the 2. 2022 , doi=

    2022

  47. [47]

    , journal=

    Chen, Jiajie and Hou, Thomas Y. , journal=. Stable Nearly Self-Similar Blowup of the 2. 2025 , doi=

    2025

  48. [48]

    , journal=

    Chen, Jiajie and Hou, Thomas Y. , journal=. Singularity Formation in 3. 2025 , doi=

    2025

  49. [49]

    Acta Metallurgica , volume=

    A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening , author=. Acta Metallurgica , volume=. 1979 , doi=

    1979

  50. [50]

    On the blowing up of solutions of the

    Fujita, Hiroshi , journal=. On the blowing up of solutions of the

  51. [51]

    Journal of Theoretical Biology , volume=

    Initiation of slime mold aggregation viewed as an instability , author=. Journal of Theoretical Biology , volume=. 1970 , doi=

    1970

  52. [52]

    European Journal of Applied Mathematics , volume=

    A Self-Similar Solution to the Focusing Problem for the Porous Medium Equation , author=. European Journal of Applied Mathematics , volume=. 1993 , publisher=

    1993

  53. [53]

    Singularity Formation and Global Well-Posedness for the Generalized

    Chen, Jiajie , journal=. Singularity Formation and Global Well-Posedness for the Generalized. 2020 , doi=

    2020

  54. [54]

    Mathematics of Control, Signals and Systems , volume=

    Approximation by superpositions of a sigmoidal function , author=. Mathematics of Control, Signals and Systems , volume=. 1989 , doi=

    1989

  55. [55]

    Neural Networks , volume=

    Approximation capabilities of multilayer feedforward networks , author=. Neural Networks , volume=. 1991 , doi=

    1991

  56. [56]

    Approximation theory of the

    Pinkus, Allan , journal=. Approximation theory of the. 1999 , doi=

    1999

  57. [57]

    Machine Learning , volume=

    Approximation and estimation bounds for artificial neural networks , author=. Machine Learning , volume=. 1994 , doi=

    1994

  58. [58]

    Conference on Learning Theory (COLT) , pages=

    Benefits of depth in neural networks , author=. Conference on Learning Theory (COLT) , pages=. 2016 , note=

    2016

  59. [59]

    Advances in Neural Information Processing Systems (NeurIPS) , pages=

    The expressive power of neural networks: A view from the width , author=. Advances in Neural Information Processing Systems (NeurIPS) , pages=. 2017 , note=

    2017

  60. [60]

    Acta Numerica , volume=

    Nonlinear approximation , author=. Acta Numerica , volume=. 1998 , doi=

    1998

  61. [61]

    1822 , publisher=

    Th\'eorie analytique de la chaleur , author=. 1822 , publisher=

  62. [62]

    Mathematische Annalen , volume=

    \"Uber die partiellen Differenzengleichungen der mathematischen Physik , author=. Mathematische Annalen , volume=. 1928 , doi=

    1928

  63. [63]

    Tellus , volume=

    Numerical integration of the barotropic vorticity equation , author=. Tellus , volume=. 1950 , doi=

    1950

  64. [64]

    Physical Review Letters , volume=

    Interaction of ``solitons'' in a collisionless plasma and the recurrence of initial states , author=. Physical Review Letters , volume=. 1965 , doi=

    1965

  65. [65]

    Journal of the Atmospheric Sciences , volume=

    Deterministic nonperiodic flow , author=. Journal of the Atmospheric Sciences , volume=. 1963 , doi=

    1963

  66. [66]

    Advances in Neural Information Processing Systems (NeurIPS) , year=

    Characterizing possible failure modes in physics-informed neural networks , author=. Advances in Neural Information Processing Systems (NeurIPS) , year=

  67. [67]

    International Conference on Machine Learning (ICML) , pages=

    On the spectral bias of neural networks , author=. International Conference on Machine Learning (ICML) , pages=. 2019 , note=

    2019

  68. [68]

    Progress of Theoretical Physics Supplement , volume=

    Initial Value Problems of One-Dimensional Self-Modulation of Nonlinear Waves in Dispersive Media , author=. Progress of Theoretical Physics Supplement , volume=. 1974 , publisher=

    1974

  69. [69]

    Mathematische Annalen , volume=

    Singularity Patterns in a Chemotaxis Model , author=. Mathematische Annalen , volume=. 1996 , publisher=

    1996

  70. [70]

    arXiv preprint arXiv:2606.08405 , year=

    Self-Evolving Scientific Agent Discovers Generalizable Physically-Reasoned Fluid Control , author=. arXiv preprint arXiv:2606.08405 , year=

    Pith/arXiv arXiv

  71. [71]

    International Conference on Learning Representations (ICLR) , year=

    Adam: A Method for Stochastic Optimization , author=. International Conference on Learning Representations (ICLR) , year=