pith. sign in

arxiv: 2606.00643 · v1 · pith:SRM4L7VVnew · submitted 2026-05-30 · 📊 stat.ML · cs.LG· cs.NA· math.NA· math.OC· math.ST· stat.TH

Taming the Loss Landscape of PINNs with Noisy Feynman-Kac Supervision: Operator Preconditioning and Non-Asymptotic Error Bounds

Nathanael Tepakbong , Hanyu Hu , Chengyu Liu , Xiang Zhou This is my paper

Pith reviewed 2026-06-28 18:16 UTC · model grok-4.3

classification 📊 stat.ML cs.LGcs.NAmath.NAmath.OCmath.STstat.TH
keywords PINNsFeynman-Kacpreconditioningloss landscapenon-asymptotic boundstanh networksMonte CarloPDE solving
0
0 comments X

The pith

A pointwise data-fidelity term preconditions the PINN loss operator, reducing its condition number and enabling non-asymptotic L2 error bounds for FK-PINNs with tanh networks.

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

Standard PINNs often fail to converge on difficult PDEs because their loss landscapes are severely ill-conditioned by the underlying differential operator. The paper demonstrates that augmenting the loss with a pointwise data-fidelity term at a few domain points acts as an operator-level preconditioner. For suitable weights, this yields a substantially smaller condition number than the standard PINN loss, no matter how the pointwise labels are generated. When the PDE admits a Feynman-Kac representation, the labels can be obtained via Monte Carlo sampling of the FK functional to form FK-PINNs. Non-asymptotic L2 error bounds are then derived for tanh-activated networks trained with a finite number of gradient-descent steps, along with new pseudo-dimension bounds on the derivatives of such networks.

Core claim

The central claim is that the added pointwise data-fidelity term serves as an operator-level preconditioner for the PINN loss. Comparison bounds show that for appropriate weights the condition number is substantially smaller than that of the standard residual-plus-boundary loss, and this holds independently of the source of the pointwise labels. For the class of PDEs that admit a Feynman-Kac representation, Monte Carlo estimates of the FK functional supply the labels, producing FK-PINNs. For these networks with tanh activation, non-asymptotic L²(Ω) error bounds are obtained after finitely many gradient-descent steps. Pseudo-dimension bounds for the first- and second-order derivatives of tanh

What carries the argument

The pointwise data-fidelity supervision term added to residual and boundary losses, acting as an operator-level preconditioner.

Load-bearing premise

The PDE must belong to the class that admits a Feynman-Kac representation so Monte Carlo labels can be generated, and suitable weights for the data-fidelity term must exist to achieve the condition-number reduction.

What would settle it

A numerical computation showing that the condition number of the augmented loss exceeds that of the standard PINN loss for all weights on a test PDE, or that the observed L2 error after finite GD steps exceeds the derived bound by a large factor.

Figures

Figures reproduced from arXiv: 2606.00643 by Chengyu Liu, Hanyu Hu, Nathanael Tepakbong, Xiang Zhou.

Figure 1
Figure 1. Figure 1: illustrates the potential V , forcing term f and corresponding wavefunction ψ. We report respectively in Figures 2 and 3 the wavefunctions learned by a standard PINN, and an FK-PINN, with both identical architecture training parameters and schedule, after 30000 + 15000 steps of Adam + L-BFGS. As the results clearly illustrate, while the standard PINN completely fails at learning the solution structure, due… view at source ↗
Figure 2
Figure 2. Figure 2: Numerical results for the standard PINN on the Schrodinger-type equation ¨ (7.1). Left side: PINN prediction Right side: absolute errors |ψ − uθ| [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical results for the FK-PINN on the Schrodinger-type equation ¨ (7.1). Left side: FK-PINN prediction Right side: absolute errors |ψ − uθ| 7.2. Summary table We now showcase the performance of FK-PINNs when compared to standard PINNs on a number of canonical PDEs. As we can see, the ability of this supervised approach to overcome the failure modes of PINNs is clear. We refer the reader to Appendix E fo… view at source ↗
Figure 4
Figure 4. Figure 4: The ground truth solution (Col.1), predicted 2D,3D results by PINNs (Col.2), 2D,3D absolute error by PINNs (Col.3), predicted 2D,3D results by FK-PINNs (Col.4), 2D,3D absolute error by FK-PINNs (Col.5) on Poisson equations [PITH_FULL_IMAGE:figures/full_fig_p054_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of PDE loss, BC loss of PINNs versus PDE loss, BC loss, Data loss of FK-PINNs for Poisson equation E.2.3. MEAN ESCAPE TIME We set the domain Ω as the regular hexagon centered at the origin with circumradius R = 2. We set V as a double-well potential function: V (x1, x2) = 1 4 (x 2 1 − 1)2 + α 2 x 2 2 , (E.13) where α = 1. The corresponding Mean Escape Time PDE is then given by: −∇V · ∇τ + β −1∆τ… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of Mean Escape Time PDE solutions learned by PINNs (left) and FK-PINNs(right) [PITH_FULL_IMAGE:figures/full_fig_p054_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Comparing the evolution of loss components for solving the Mean Escape Time problem between PINNs and FK-PINNs The training loss evolutions in [PITH_FULL_IMAGE:figures/full_fig_p055_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of Committor Function under Muller-Brown potential learned by PINNs (left) and FK-PINNs (right) ¨ [PITH_FULL_IMAGE:figures/full_fig_p056_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the convergence behavior of individual loss components in PINNs and FK-PINNs trained with Adam optimizer only Model Metric Poisson (E.2.2) Schrodinger-type ¨ (7) Mean Escape Time (E.2.3) Committor (E.2.4) PINNs L 2 Abs Err 1.333 ± 0.616 0.475 ± 0.148 16.56 ± 0.055 1.028 ± 0.283 L 2 Rel Err 0.322 ± 0.149 0.624 ± 0.195 1.007 ± 0.003 0.839 ± 0.661 H1 Abs Err 12.42 ± 4.345 2.893 ± 0.415 43.55 ± 0… view at source ↗
Figure 10
Figure 10. Figure 10: Condition numbers near a local minimum for PINNs and FK-PINNs for Poisson equation (left) and the Mean Escape Time Problem (right). The evolution of the Hessian condition number as the number of collocation points increases plotted in [PITH_FULL_IMAGE:figures/full_fig_p057_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Loss landscape of Standard PINN (left) and FK-PINN (right) trained on the Mean Escape Time PDE, next to a minimizer. 57 [PITH_FULL_IMAGE:figures/full_fig_p057_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparisons of different PINNs solving the Mean Escape Time PDE trained with (from left to right) Standard PINN (Adam + L-BFGS), Adam + L-BFGS + NNCG, ENGD, FK-PINN method MODEL L2 ERROR H1 ERROR PINNS (ADAM+L-BFGS) 1.002 ± 0.002 1.001 ± 0.001 PINNS (ADAM+L-BFGS+NNCG) 0.986 ± 0.011 1.006 ± 0.001 PINNS (ENGD) 1.016 ± 0.001 1.006 ± 0.014 FK-PINNS (ADAM+L-BFGS) 0.174 ± 0.004 0.699 ± 0.008 [PITH_FULL_IMAGE:f… view at source ↗
Figure 13
Figure 13. Figure 13: Sensitivity Analysis of Key Parameters in Feynman-Kac Monte Carlo Supervision. Figure (a): Fixed NMC=500, influence of ∆t on solution accuracy. Figure (b): Fixed ∆t = 1e − 3, influence of NMC on solution accuracy P DATA L2 ERROR H1 ERROR TIME (S) P DATA=0 1.003 ± 0.004 1.001 ± 0.001 319.7 ± 5.5 P DATA=0.001 0.189 ± 0.037 0.594 ± 0.069 456.3 ± 93.1 P DATA=0.01 0.120 ± 0.025 0.431 ± 0.143 665.7 ± 14.1 P DAT… view at source ↗
read the original abstract

Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator. We study PINNs augmented with a pointwise data-fidelity term, added at a few points in the domain to the standard residual and boundary losses. We show that this supervision term acts as an operator-level preconditioner: for suitable weights, our comparison bounds guarantee a substantially smaller condition number than under the standard PINN loss, independently of how the pointwise labels are obtained. For a broad class of PDEs admitting a Feynman-Kac (FK) representation, we generate such labels by Monte Carlo averages of the FK functional, resulting in what we call ``FK-PINNs", and using the excess risk decomposition approach, we derive non-asymptotic $L^2(\Omega)$-error bounds for FK-PINNs with $\tanh$ activation trained by finitely many steps of gradient descent. Along the way, we establish pseudo-dimension bounds for first- and second-order derivatives of $\tanh$ neural networks, which are of independent interest and, to the best of our knowledge, new. Numerical experiments on Poisson, Schr\"odinger, mean exit time, and committor problems corroborate the theory, and show that FK-PINNs can successfully solve PDEs for which standard PINNs exhibit severe failure modes.

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

0 major / 3 minor

Summary. The manuscript claims that augmenting the standard PINN loss with a pointwise data-fidelity term (generated via Monte Carlo sampling from the Feynman-Kac representation for PDEs admitting such a form) acts as an operator-level preconditioner. For suitable weights, comparison bounds show a substantially smaller condition number than the standard PINN loss, independently of label source. Using excess-risk decomposition, the authors derive non-asymptotic L²(Ω) error bounds for FK-PINNs with tanh activations trained by finitely many gradient-descent steps. New pseudo-dimension bounds for first- and second-order derivatives of tanh networks are established as auxiliary results. Numerical experiments on Poisson, Schrödinger, mean-exit-time, and committor problems are reported to corroborate the claims.

Significance. If the central claims hold, the work supplies a concrete mechanism for improving loss conditioning in PINNs together with non-asymptotic error guarantees that do not rely on asymptotic regimes. The independence of the preconditioning effect from the label source and the provision of new pseudo-dimension bounds for network derivatives are notable strengths. The approach is restricted to the class of PDEs admitting Feynman-Kac representations, but within that class the results appear to offer both theoretical and practical value for problems where standard PINNs fail.

minor comments (3)
  1. [Abstract] The abstract states that 'comparison bounds guarantee a substantially smaller condition number' but does not indicate the dependence of the weight choice on the operator or on the number of supervision points; a brief clarifying sentence would improve readability.
  2. [Introduction / Related work] The pseudo-dimension bounds are described as 'of independent interest and, to the best of our knowledge, new.' A short comparison with existing bounds for ReLU or other activations in the related-work section would help situate the contribution.
  3. [Numerical experiments] Numerical experiments are said to 'corroborate the theory,' yet the manuscript does not report the Monte-Carlo sample size used to generate FK labels or the precise schedule for the supervision weights; these details are needed for reproducibility.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed summary of our manuscript and the positive assessment of its significance. The recommendation for minor revision is appreciated. No specific major comments appear in the report, so we provide no point-by-point responses below. We will incorporate any minor editorial changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's central claims rest on comparison bounds for the preconditioning effect of the added supervision term (independent of label source) and on excess-risk plus pseudo-dimension arguments for the non-asymptotic L2 error bounds under finite GD steps. These steps are presented as derived from standard statistical learning tools and operator analysis rather than from fitted parameters renamed as predictions or from self-citation chains. The Feynman-Kac representation is an external assumption on the PDE class, not a self-referential definition. No load-bearing step reduces by construction to the paper's own inputs or prior self-citations.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of a Feynman-Kac representation for the PDE class and on the availability of suitable supervision weights; no new physical entities are postulated.

free parameters (1)
  • supervision weights
    Chosen to guarantee the condition-number reduction; their specific values are not derived from first principles.
axioms (2)
  • domain assumption Target PDE admits a Feynman-Kac representation
    Required to generate pointwise labels via Monte Carlo sampling of the FK functional.
  • domain assumption Networks use tanh activation
    Used to obtain the stated non-asymptotic error bounds and pseudo-dimension results.

pith-pipeline@v0.9.1-grok · 5825 in / 1346 out tokens · 33320 ms · 2026-06-28T18:16:15.012192+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

75 extracted references · 3 canonical work pages

  1. [1]

    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

  2. [2]

    Journal of Scientific Computing , volume=

    Scientific machine learning through physics--informed neural networks: Where we are and what’s next , author=. Journal of Scientific Computing , volume=. 2022 , publisher=

    2022

  3. [3]

    Advances in neural information processing systems , volume=

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

  4. [4]

    SIAM Journal on Scientific Computing , volume=

    Understanding and mitigating gradient flow pathologies in physics-informed neural networks , author=. SIAM Journal on Scientific Computing , volume=. 2021 , publisher=

    2021

  5. [5]

    Computer Methods in Applied Mechanics and Engineering , volume=

    A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2023 , publisher=

    2023

  6. [6]

    The Twelfth International Conference on Learning Representations , year=

    An operator preconditioning perspective on training in physics-informed machine learning , author=. The Twelfth International Conference on Learning Representations , year=

  7. [7]

    International Conference on Machine Learning , pages=

    Challenges in Training PINNs: A Loss Landscape Perspective , author=. International Conference on Machine Learning , pages=. 2024 , organization=

    2024

  8. [8]

    Journal of Computational Physics , volume=

    When and why PINNs fail to train: A neural tangent kernel perspective , author=. Journal of Computational Physics , volume=. 2022 , publisher=

    2022

  9. [9]

    arXiv preprint arXiv:2410.06308 , year=

    Quantifying training difficulty and accelerating convergence in neural network-based PDE solvers , author=. arXiv preprint arXiv:2410.06308 , year=

    work page arXiv

  10. [10]

    The Thirty-ninth Annual Conference on Neural Information Processing Systems , year=

    Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective , author=. The Thirty-ninth Annual Conference on Neural Information Processing Systems , year=

  11. [11]

    International Conference on Machine Learning , pages=

    Achieving high accuracy with PINNs via energy natural gradient descent , author=. International Conference on Machine Learning , pages=. 2023 , organization=

    2023

  12. [12]

    Forty-second International Conference on Machine Learning , year=

    Learn Singularly Perturbed Solutions via Homotopy Dynamics , author=. Forty-second International Conference on Machine Learning , year=

  13. [13]

    2025 , url=

    Nilo Schwencke and Cyril Furtlehner , booktitle=. 2025 , url=

    2025

  14. [14]

    Stochastic differential equations: an introduction with applications , pages=

    Stochastic differential equations , author=. Stochastic differential equations: an introduction with applications , pages=. 2003 , publisher=

    2003

  15. [15]

    2014 , publisher=

    Brownian motion and stochastic calculus , author=. 2014 , publisher=

    2014

  16. [16]

    Sabelfeld , title =

    Karl K. Sabelfeld , title =. 1991 , series =

    1991

  17. [17]

    2022 , publisher=

    Partial differential equations , author=. 2022 , publisher=

    2022

  18. [18]

    2016 , publisher=

    Monte-Carlo methods and stochastic processes: from linear to non-linear , author=. 2016 , publisher=

    2016

  19. [19]

    2022 , publisher=

    Monte Carlo Methods for Partial Differential Equations With Applications to Electronic Design Automation , author=. 2022 , publisher=

    2022

  20. [20]

    Texts in applied mathematics , volume=

    Stochastic processes and applications , author=. Texts in applied mathematics , volume=. 2014 , publisher=

    2014

  21. [21]

    Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , month =

    Kendall, Alex and Gal, Yarin and Cipolla, Roberto , title =. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) , month =

  22. [22]

    Applied and Computational Harmonic Analysis , volume=

    Loss landscapes and optimization in over-parameterized non-linear systems and neural networks , author=. Applied and Computational Harmonic Analysis , volume=. 2022 , publisher=

    2022

  23. [23]

    arXiv preprint arXiv:2405.13738 , year=

    Interpolation with deep neural networks with non-polynomial activations: necessary and sufficient numbers of neurons , author=. arXiv preprint arXiv:2405.13738 , year=

    work page arXiv

  24. [24]

    On the Optimal Memorization Power of Re

    Gal Vardi and Gilad Yehudai and Ohad Shamir , booktitle=. On the Optimal Memorization Power of Re. 2022 , url=

    2022

  25. [25]

    Linear convergence of gradient and proximal-gradient methods under the polyak-

    Karimi, Hamed and Nutini, Julie and Schmidt, Mark , booktitle=. Linear convergence of gradient and proximal-gradient methods under the polyak-. 2016 , organization=

    2016

  26. [26]

    Zhurnal vychislitel'noi matematiki i matematicheskoi fiziki , volume=

    Gradient methods for minimizing functionals , author=. Zhurnal vychislitel'noi matematiki i matematicheskoi fiziki , volume=. 1963 , publisher=

    1963

  27. [27]

    Neural Networks , volume=

    On the approximation of functions by tanh neural networks , author=. Neural Networks , volume=. 2021 , publisher=

    2021

  28. [28]

    Communications in Computational Physics , year=

    A rate of convergence of physics informed neural networks for the linear second order elliptic pdes , author=. Communications in Computational Physics , year=. doi:10.4208/cicp.OA-2021-0186 , number=

    work page doi:10.4208/cicp.oa-2021-0186 2021

  29. [29]

    Conference on learning theory , pages=

    A priori generalization analysis of the deep Ritz method for solving high dimensional elliptic partial differential equations , author=. Conference on learning theory , pages=. 2021 , organization=

    2021

  30. [30]

    2012 , publisher=

    Matrix analysis , author=. 2012 , publisher=

    2012

  31. [31]

    Proceedings of the 36th International Conference on Machine Learning , pages =

    Gradient Descent Finds Global Minima of Deep Neural Networks , author =. Proceedings of the 36th International Conference on Machine Learning , pages =. 2019 , volume =

    2019

  32. [32]

    Journal of Machine Learning Research , volume=

    Piratenets: Physics-informed deep learning with residual adaptive networks , author=. Journal of Machine Learning Research , volume=

  33. [33]

    1991 , publisher=

    Functional Analysis , author=. 1991 , publisher=

    1991

  34. [34]

    Constructive approximation , volume=

    Learning theory estimates via integral operators and their approximations , author=. Constructive approximation , volume=. 2007 , publisher=

    2007

  35. [35]

    Bernoulli , volume=

    On the convergence of PINNs , author=. Bernoulli , volume=. 2025 , publisher=

    2025

  36. [36]

    1995 , publisher=

    Positive harmonic functions and diffusion , author=. 1995 , publisher=

    1995

  37. [37]

    First time to exit of a continuous It

    Bouchard, Bruno and Geiss, Stefan and Gobet, Emmanuel , journal=. First time to exit of a continuous It

  38. [38]

    2018 , publisher=

    High-dimensional probability: An introduction with applications in data science , author=. 2018 , publisher=

    2018

  39. [39]

    Stochastic processes and their applications , volume=

    Weak approximation of killed diffusion using Euler schemes , author=. Stochastic processes and their applications , volume=. 2000 , publisher=

    2000

  40. [40]

    Stochastic Processes and Their Applications , volume=

    Stopped diffusion processes: boundary corrections and overshoot , author=. Stochastic Processes and Their Applications , volume=. 2010 , publisher=

    2010

  41. [41]

    Neurocomputing , volume=

    Improved physics-informed neural network in mitigating gradient-related failures , author=. Neurocomputing , volume=. 2025 , publisher=

    2025

  42. [42]

    Nonlinearity , volume=

    Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning , author=. Nonlinearity , volume=. 2021 , publisher=

    2021

  43. [43]

    Communications in Mathematics and Statistics , volume=

    Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations , author=. Communications in Mathematics and Statistics , volume=. 2017 , publisher=

    2017

  44. [44]

    International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing , year=

    Stochastic methods for solving high-dimensional partial differential equations , author=. International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing , year=

  45. [45]

    , author=

    Towards a Theory of Transition Paths. , author=. Journal of Statistical Physics , volume=

  46. [46]

    Markov Processes: Volume 1 , pages=

    Markov processes , author=. Markov Processes: Volume 1 , pages=. 1965 , publisher=

    1965

  47. [47]

    2009 , publisher=

    Markov processes: characterization and convergence , author=. 2009 , publisher=

    2009

  48. [48]

    Applebaum, David , year=. L

  49. [49]

    2006 , publisher=

    Controlled Markov processes and viscosity solutions , author=. 2006 , publisher=

    2006

  50. [50]

    Kloeden and Eckhard Platen , title =

    Peter E. Kloeden and Eckhard Platen , title =. 1992 , doi =

    1992

  51. [51]

    2004 , publisher=

    Monte Carlo methods in financial engineering , author=. 2004 , publisher=

    2004

  52. [52]

    Systems & control letters , volume=

    Adapted solution of a backward stochastic differential equation , author=. Systems & control letters , volume=. 1990 , publisher=

    1990

  53. [53]

    Mathematical finance , volume=

    Backward stochastic differential equations in finance , author=. Mathematical finance , volume=. 1997 , publisher=

    1997

  54. [54]

    Probability theory and related fields , volume=

    A probabilistic approach to one class of nonlinear differential equations , author=. Probability theory and related fields , volume=. 1991 , publisher=

    1991

  55. [55]

    Annales de l’Institut Henri Poincar

    Branching diffusion representation of semilinear PDEs and Monte Carlo approximation , author=. Annales de l’Institut Henri Poincar

  56. [56]

    Stochastic Processes and their Applications , volume=

    Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method , author=. Stochastic Processes and their Applications , volume=. 2020 , publisher=

    2020

  57. [57]

    Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences , volume=

    Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs , author=. Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences , volume=. 2007 , publisher=

    2007

  58. [58]

    Stochastic Processes and their applications , volume=

    Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations , author=. Stochastic Processes and their applications , volume=. 2004 , publisher=

    2004

  59. [59]

    I , author=

    Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I , author=. Communications on pure and applied mathematics , volume=. 1959 , publisher=

    1959

  60. [60]

    Bartlett and Nick Harvey and Christopher Liaw and Abbas Mehrabian , title =

    Peter L. Bartlett and Nick Harvey and Christopher Liaw and Abbas Mehrabian , title =. Journal of Machine Learning Research , year =

  61. [61]

    Journal of Computer and System Sciences , volume=

    Polynomial bounds for VC dimension of sigmoidal and general Pfaffian neural networks , author=. Journal of Computer and System Sciences , volume=. 1997 , publisher=

    1997

  62. [62]

    2009 , publisher=

    Neural network learning: Theoretical foundations , author=. 2009 , publisher=

    2009

  63. [63]

    2019 , publisher=

    High-dimensional statistics: A non-asymptotic viewpoint , author=. 2019 , publisher=

    2019

  64. [64]

    2013 , publisher=

    Probability in Banach Spaces: isoperimetry and processes , author=. 2013 , publisher=

    2013

  65. [65]

    Applied and Computational Harmonic Analysis , volume=

    Solving PDEs on spheres with physics-informed convolutional neural networks , author=. Applied and Computational Harmonic Analysis , volume=. 2025 , publisher=

    2025

  66. [66]

    Machine Learning For Elliptic

    Yiping Lu and Haoxuan Chen and Jianfeng Lu and Lexing Ying and Jose Blanchet , booktitle=. Machine Learning For Elliptic. 2022 , url=

    2022

  67. [67]

    , author=

    Transition-path theory and path-finding algorithms for the study of rare events. , author=. Annual review of physical chemistry , volume=

  68. [68]

    Aditya Prakash , booktitle=

    Zhiyuan Zhao and Xueying Ding and B. Aditya Prakash , booktitle=. 2024 , url=

    2024

  69. [69]

    Advances in neural information processing systems , volume=

    Visualizing the loss landscape of neural nets , author=. Advances in neural information processing systems , volume=

  70. [70]

    SIAM Journal on Numerical Analysis , volume=

    Value-gradient based formulation of optimal control problem and machine learning algorithm , author=. SIAM Journal on Numerical Analysis , volume=. 2023 , publisher=

    2023

  71. [71]

    Journal of Computational Physics , volume=

    PINN training using biobjective optimization: The trade-off between data loss and residual loss , author=. Journal of Computational Physics , volume=. 2023 , publisher=

    2023

  72. [72]

    SIAM Journal on Scientific Computing , volume=

    Deep splitting method for parabolic PDEs , author=. SIAM Journal on Scientific Computing , volume=. 2021 , publisher=

    2021

  73. [73]

    Journal of Computational Physics , volume=

    A derivative-free method for solving elliptic partial differential equations with deep neural networks , author=. Journal of Computational Physics , volume=. 2020 , publisher=

    2020

  74. [74]

    SIAM Journal on Scientific Computing , volume=

    Deep Picard iteration for high-dimensional nonlinear PDEs , author=. SIAM Journal on Scientific Computing , volume=. 2026 , publisher=

    2026

  75. [75]

    IEEE Transactions on Neural Networks and Learning Systems , volume=

    Rigorous a posteriori error bounds for PDE-defined PINNs , author=. IEEE Transactions on Neural Networks and Learning Systems , volume=. 2023 , publisher=

    2023