Pith. sign in

REVIEW 3 major objections 5 minor 67 references

Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper claims that weak physics-informed neural networks approximate entropy solutions of geometry-compatible hyperbolic conservation laws on a $d$-dimensional Riemannian manifold with an $L^1$ error of order $n^{-1/(d+2)}(\log…

desk verdict A genuinely new manifold wPINN framework and a likely-valid Sobolev approximation theorem, but the main convergence rate rests on an empty test-function family and does not hold as stated. read the letter →

arxiv 2505.19036 v2 pith:7FIDWGYD submitted 2025-05-25 math.NA cs.NAstat.ML

Hanfei Zhou , Lei Shi This is my paper · ORCID
classification math.NAcs.NAstat.ML MSC 35L6565M1268T07
keywords weakphysics-informedneuralnetworksentropysolutionshyperbolicconservationlawsRiemannianmanifoldsgeometry-compatiblefluxconvergenceanalysiscurseofdimensionalityReLU
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

This paper tries to establish that a weak physics-informed neural network can approximate entropy solutions of geometry-compatible hyperbolic conservation laws on a $d$-dimensional Riemannian manifold with an $L^1$ error of order $n^{-1/(d+2)}(\log n)^{4/(d+2)}$, with high probability, using networks of depth $\sim \log n$ and width $\sim n^{(d+1)/(d+2)}$. The rate matches the minimax rate for Sobolev approximation in $d$-dimensional Euclidean space, and the network size depends only on the intrinsic manifold dimension, not on the ambient dimension $D$ in which the manifold is embedded. This matters because it says mesh-free neural solvers can in principle avoid the curse of dimensionality on low-dimensional manifolds while handling shocks and other nonsmooth solutions, where classical PINN theory is weak. The argument works through an error decomposition into approximation and quadrature errors, anchored by an $L^1$ contraction estimate for geometry-compatible conservation laws on manifolds.

What carries the argument

The machinery has four linked parts. First, the Kruzkov entropy residual $R_{\mathrm{int}}(u,\xi,c)$ on the manifold, built from $|u-c|$ and the geometry-compatible flux, is the training objective. Second, the test-function family $\xi_{\delta_1,\delta_2}$ with $\partial_t\xi\le -\delta_1$ and $\|\nabla_g\xi\|_\infty\le \delta_2$ yields the strong convexity inequality $\|(u-c)^4\|_{L^1(\Omega)}\le C_5 R_{\mathrm{int}}(u,\xi,c)$, which connects the residual to the $L^1$ error. Third, the $L^1$ contraction estimate via measure-valued solutions converts residual control at the final time into control of $\|u_n(\cdot,T)-u^*(\cdot,T)\|_{L^1(M^d)}$. Fourth, approximation theory supplies ReLU networks for Sobolev functions on manifolds through local-coordinate charts, partition of unity, multiplication networks, and spline interpolation in time to handle temporal error accumulation.

What would settle it

Evaluate the integral identity $\int_0^T \partial_t \xi(x,t)\,dt = \xi(x,T)-\xi(x,0)$. Under the paper's boundary condition both endpoint values are zero, so the integral must be zero; under the admissibility condition $\partial_t\xi\le -\delta_1<0$ it must be at most $-\delta_1 T<0$. Exhibiting any one admissible test function would resolve the contradiction; failing that, the strong convexity inequality (D.1) has no domain and the contraction estimate does not connect to the training loss.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: for a geometry-compatible flux on a smooth compact $d$-dimensional Riemannian manifold, the unique entropy solution $u^*$ can be recovered at the final time by a wPINN estimator $u_n$ satisfying $\|u_n(\cdot,T)-u^*(\cdot,T)\|_{L^1(M^d)} \lesssim n^{-1/(d+2)}(\log n)^{4/(d+2)}$ with probability at least $1-\exp(-n^{1-a}(\log n)^{4a})$. The estimator is built by minimizing a Kruzkov entropy residual against admissible test functions, and the proof shows the residual controls the $L^1$ error through a strong convexity inequality and an $L^1$ contraction theorem. A corollary improves the rate to $n^{-s/(2s+d)}(\log n)^{4s/(2s+d)}$ when $u^*(\cdot,t)\in W^s_1(M^d)$, with constants independent of the exponential-in-$T$ factor. Numerical experiments on the sphere confirm that standing shocks, moving shocks, rarefaction waves, and shock formation from smooth data are captured with reported $L^1$ test errors between 0.26% and 5.1%.

Load-bearing premise

The entire error bound rests on there being test functions that vanish on the full boundary of the time cylinder (so $\xi(x,0)=\xi(x,T)=0$) yet have time derivative at most $-\delta_1<0$ everywhere, a family the paper never shows to be non-empty and which the fundamental theorem of calculus makes empty as stated.

Editorial extensions

If this is right

  • If Theorem 1 holds, wPINNs provide a mesh-free method for nonsmooth entropy solutions on manifolds with convergence rate $n^{-1/(d+2)}(\log n)^{4/(d+2)}$, independent of the ambient dimension $D$.
  • For entropy solutions with $W^s_1(M^d)$ regularity, the rate improves to $n^{-s/(2s+d)}(\log n)^{4s/(2s+d)}$ and the error constant no longer carries the exponential $e^{C_1T}$ factor.
  • The same proof scheme yields $L^p$ estimates by replacing the Kruzkov entropy with $|v-u|^p$, so the square-entropy loss used in the experiments falls within the theory.
  • Long-time prediction inherits a constant growing like $e^{C_1T}$, predicting error accumulation over long horizons and motivating time-discretized training.
  • On the sphere, the method reproduces standing and moving shocks, rarefactions, and shock formation from smooth data, with reported test errors between 0.26% and 5.1%.
  • The paper's own remarks point toward time-slicing and total-variation regularization as practical remedies for the exponential-in-time error growth.
  • The boundary-value experiments on spherical segments exercise Dirichlet boundary conditions that lie outside the closed-manifold theory developed in the main theorems.
  • The complexity bounds are stated with bounded or unbounded weight parameters, with the bounded case handled through a localization argument that avoids unbounded parameter growth.

Reading between the lines

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

  • Editorial inference: a robust repair of the test-function class is needed, because if $\xi$ is required to vanish at both $t=0$ and $t=T$, the condition $\partial_t\xi\le -\delta_1<0$ is contradictory; the theorem should either construct such functions or replace the full-boundary zero trace with a lateral-only boundary condition.
  • Editorial inference: the exponential-in-$T$ constant suggests a time-slicing strategy (training on short intervals and composing) as a natural way to reach long time horizons; the paper mentions this direction but does not analyze it.
  • Editorial inference: because the experiments impose Dirichlet boundary conditions on spherical segments, they exercise boundary handling outside the closed-manifold theory; a boundary-aware entropy formulation would be needed to close that gap.
  • Editorial inference: the claimed alleviation of the curse of dimensionality is relative to Euclidean minimax rates; genuine manifold-specific minimax rates are not yet known, so the optimality statement is a comparison with the Euclidean benchmark.
Share X Bluesky LinkedIn Reddit HN

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 a weak PINN (wPINN) formulation for entropy solutions of geometry-compatible hyperbolic conservation laws on compact Riemannian manifolds. Its main result, Theorem 1, claims that a suitably constructed ReLU-network estimator achieves, with high probability, an L1 error of order n^{-1/(d+2)} (log n)^{4/(d+2)} at the final time, with network size depending only on the intrinsic dimension d and not on the ambient dimension D. The proof is organized through an excess-risk decomposition into approximation error, based on neural-network approximation of Sobolev functions on manifolds together with temporal spline interpolation, and quadrature error, based on local Rademacher complexity. Numerical experiments on the sphere S^2 illustrate the method for standing and moving shocks, rarefaction waves, and sine-wave initial data.

Significance. If the convergence result were established, this would be a meaningful contribution: it would provide the first wPINN convergence analysis for hyperbolic conservation laws on manifolds, with explicit network constructions, rates matching the Euclidean minimax rate, and dependence only on the intrinsic dimension. The paper contains substantial technical work in the appendices, gives explicit constant dependencies, formulates a novel manifold test-function framework, and provides a public code repository for the numerical experiments. However, the central proof rests on a test-function family that is empty under the paper's own zero-trace definition, so the main theorem is not supported as written.

major comments (3)
  1. [§2.4, Lemma 14 (Appendix D), Appendix A] Lemma 14's strong-convexity inequality (D.1) is vacuous because the admissible test-function family ξ_{δ1,δ2} is empty inside W^{1,∞}_0(M_d × [0,T]). Appendix A defines W^{1,∞}_0 by zero trace on the full boundary, so any ξ in this space satisfies ξ(·,0)=ξ(·,T)=0. The defining condition ∂_t ξ ≤ −δ_1 < 0 then gives ξ(·,T) − ξ(·,0) = ∫_0^T ∂_t ξ dt ≤ −δ_1 T < 0, contradicting the zero trace at both endpoints. Consequently, Lemma 14, Lemma 16, Theorem 4, and the final step of Theorem 1, all of which invoke (D.1), do not provide the claimed control of the residual in terms of the L1 distance.
  2. [§2.3, Theorem 5] The paper does not reconcile the conflicting requirements on test functions: the contraction estimate in Theorem 5 removes boundary terms by requiring ξ ∈ W^{1,∞}_0(M_d × [0,T]), while Lemma 14 requires ∂_t ξ ≤ −δ_1 < 0, which is impossible on that space. If one drops the zero-trace condition to make ξ_{δ1,δ2} nonempty, then the integration by parts in Theorem 5 acquires nonzero boundary terms, and those terms are never estimated or incorporated into the loss. No alternative boundary treatment is provided, so the route from the strong-convexity inequality to the final L1 error bound is broken.
  3. [§3, Proof of Theorem 3] The reduction in the proof of Theorem 3 from a BV entropy solution to a function with finite W^{1,1} norm is not justified as stated. The text invokes [50] to replace u*(·,t) by a smooth approximant with controlled W^{1,1} norm, but then says 'we can assume u*(·,t) has finite W^{1,1} norm' and applies the W^1_1 approximation theorem directly to the entropy solution. Since the entropy solution is not itself in W^{1,1}, the argument needs an explicit two-step approximation chain with an L1 error estimate for u* − smooth approximant; the present text only gestures at this step, and the final network-size bound in (3.1) depends on this missing estimate.
minor comments (5)
  1. [Theorem 2] In the statement of Theorem 2, the width bound is written as W ≤ C ε^{-d/n}; this appears to be a typo for ε^{-d/s}. In addition, the same symbol s is used for the Sobolev smoothness exponent and for the network sparsity bound, which makes the statement confusing.
  2. [Theorem 1] The statement contains the phrase 'with probability at least at least'; one occurrence should be deleted.
  3. [Lemma 12 proof] The phrase 'almost everywrhere' in the proof of Lemma 12 is a typo for 'almost everywhere'.
  4. [Theorem 1] The family ξ_{δ1,δ2} is introduced without explicitly stating whether its elements are required to satisfy ξ ∈ W^{1,∞}_0(M_d × [0,T]). Since the loss R(u,ξ,c) in Definition 3 is only defined for such test functions, the statement should either impose this membership or explain how the loss is extended to test functions with nonzero boundary traces.
  5. [Remark 6] The proposed relaxation in Remark 6 replaces the pointwise condition on ∂_t ξ by an inequality involving (f(u)−f(c))/(u−c); because this expression depends on u, the resulting condition does not define a fixed test-function class and cannot be used in the max over ξ in the wPINN loss without further explanation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the convergence proof is constructive, uses the entropy solution only through external well-posedness facts, and does not fit any parameter to the target error.

full rationale

The paper's derivation chain is not circular. Theorem 1 is obtained from an explicit error decomposition into approximation error (Theorem 3), quadrature error (Theorem 4), and an L1 contraction estimate (Theorem 5). The approximation theorem is proved constructively via spline interpolation and ReLU-network approximation of Sobolev functions on manifolds, using only the temporal Lipschitz and total-variation bounds of entropy solutions imported from the external well-posedness theory [10]. The quadrature analysis replicates local Rademacher complexity arguments and proves the needed strong-convexity inequality (Lemma 14) rather than importing it as an unverified premise. The self-citation to [41] is motivational and the referenced lemma is re-proved in the present paper, so it is not load-bearing. The final rate n^{-1/(d+2)} emerges from balancing the explicit approximation error with the explicit quadrature error; no parameter is fitted to the claimed convergence rate. One genuine concern is flagged for correctness, not circularity: the test-function family ξ_{δ1,δ2} defined by ∂t ξ ≤ −δ1 < 0 appears to be empty when intersected with W^{1,∞}_0(M_d × [0,T]) because zero trace at t=0 and t=T is incompatible with a strictly negative time derivative. If so, Lemma 14's strong-convexity inequality is vacuous and Theorem 1 would not be established. That is an internal consistency problem, not a circular derivation: the inequality does not assume the theorem's conclusion. The paper is therefore not circular, but the empty-family issue is a serious correctness risk.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on established well-posedness theory, standard approximation-theoretic tools, and one ad hoc assumption about test functions that is not established and appears inconsistent with the paper's boundary definition. There are no invented physical entities. The free parameters delta1, delta2, delta3 are introduced to obtain strong convexity, not fitted to data.

free parameters (1)
  • delta1, delta2, delta3
    User-chosen constants satisfying delta1 - 3D Lip_f delta2 >= delta3 to force the pointwise lower bound in Lemma 14. They are not estimated from data, but their role is to make the strong convexity derivation work, and no admissible test function is shown to exist.
assumptions (5)
  • standard math Well-posedness and L1 contraction for geometry-compatible conservation laws on manifolds (Proposition 1, from [10])
    Used throughout as the external foundation for existence, uniqueness, TV bounds, and contraction of entropy solutions; this is a published theorem, not derived in the paper.
  • ad hoc to paper Admissible test functions xi with partial_t xi <= -delta1 < 0 exist in W^{1,infty}_0(Md x [0,T]), or in an appropriate variant, with ||nabla_g xi||_infty <= delta2
    Invoked in Theorem 1 and Lemma 14 to get the strong convexity inequality (D.1). This is not proved and appears false under the paper's own definition of W^{1,infty}_0, because zero trace at t=0 and t=T prevents a strictly negative derivative everywhere.
  • domain assumption The manifold admits smooth local coordinates satisfying Definition 5, including tubular neighborhood extensions
    Needed for Theorem 2 to approximate coordinate maps and Sobolev functions via charts; satisfied by compact smooth manifolds but imposes a specific atlas regularity.
  • domain assumption A BV function on the manifold can be replaced by a W^{1,1} function with controlled norm for approximation purposes
    Used in the proof of Theorem 3 via [50]; the paper states 'we can assume u* has finite W^1_1 norm' without carrying the smoothing error explicitly, so the assumption is only partially justified.
  • standard math Nash embedding theorem: compact manifold embeds isometrically into R^D
    Used to set up ambient-space neural networks and to justify D-dependence in constants.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds." pith.science (2026). https://pith.science/paper/7FIDWGYD

@misc{pith2026250519036,
  author       = {Pith},
  title        = {Pith review of: Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FIDWGYD}},
  note         = {Machine review of arXiv:2505.19036}
}
read the original abstract

Physics-informed neural networks (PINNs), owing to their mesh-free nature, offer a powerful approach for solving high-dimensional partial differential equations (PDEs) in complex geometries, including irregular domains. This capability effectively circumvents the challenges of mesh generation that traditional numerical methods face in high-dimensional or geometrically intricate settings. While recent studies have extended PINNs to manifolds, the theoretical foundations remain scarce. Existing theoretical analyses of PINNs in Euclidean space often rely on smoothness assumptions for the solutions. However, recent empirical evidence indicates that PINNs may struggle to approximate solutions with low regularity, such as those arising from nonlinear hyperbolic equations. In this paper, we develop a framework for PINNs tailored to the efficient approximation of weak solutions, particularly nonlinear hyperbolic equations defined on manifolds. We introduce a novel weak PINN (wPINN) formulation on manifolds that leverages the well-posedness theory to approximate entropy solutions of geometry-compatible hyperbolic conservation laws on manifolds. Employing tools from approximation theory, we establish a convergence analysis of the algorithm, including an analysis of approximation errors for time-dependent entropy solutions. This analysis provides insight into the accumulation of approximation errors over long time horizons. Notably, the network complexity depends only on the intrinsic dimension, independent of the ambient space dimension. Our results match the minimax rate in the d-dimensional Euclidean space, demonstrating that PINNs can alleviate the curse of dimensionality in the context of low-dimensional manifolds. Finally, we validate the performance of the proposed wPINN framework through numerical experiments, confirming its ability to efficiently approximate entropy solutions on manifolds.

Figures

Figures reproduced from arXiv: 2505.19036 by the authors.

Figure 1
Figure 1. Standing wave 16384, Nini = 16384, Nmin = 1, Nmax = 6, Nep = 5000, τmax = 0.015, and τmin = 0.01. The solution network consists 20 neurons per layer, and the test function network has 10 neurons per layer. A hyperparameter grid search is conducted for activation functions (ReLU, tanh, sin), solution network depths (Lθ = 4, 6), test function network depths (Lη = 2, 4), reset frequencies (r = 100, 200, 500, 1000), and… view at source ↗
Figure 2
Figure 2. Moving wave entropy are of the same order of magnitude, as shown in [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Standing shock solution predicted on the sphere. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Moving shock solution predicted on the sphere. [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]
Figure 5
Figure 5. Figure 5: Rarefaction wave and sine wave It is worth noting that the weak solution corresponding to this initial condition is not unique; the standing shock is also a weak solution. However, the rarefaction wave represents the unique entropy solution [PITH_FULL_IMAGE:figures/fu…
Figure 6
Figure 6. Figure 6: Rarefaction wave solution predicted on the sphere. [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Sine wave solution predicted on the sphere. [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Network construction: An illustration of the neural network architecture for ap [PITH_FULL_IMAGE:figures/full_fig_p033_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

67 extracted references · 59 canonical work pages

  1. [50]

    Miranda, Jr., D

    M. Miranda, Jr., D. Pallara, F. Paronetto, and M. Preunkert. Heat semigroup and functions of bounded variation on Riemannian manifolds. Journal f¨ ur die Reine und Angewandte Mathematik, 613:99–119, 2007

    work page 2007

  2. [1]

    Hyperbolic conservation laws on manifolds: Total variation estimates and the finite volume method

    Paulo Amorim, Matania Ben-Artzi, and Philippe G LeFloch. Hyperbolic conservation laws on manifolds: Total variation estimates and the finite volume method. Methods and Applications of Analysis , 12(1):291–324, 2005

    work page 2005

  3. [2]

    Neural operator: Graph kernel network for partial differential equations

    Anima Anandkumar, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Nikola Kovachki, Zongyi Li, Burigede Liu, and Andrew Stuart. Neural operator: Graph kernel network for partial differential equations. In ICLR 2020 Workshop on Integration of Deep Neural Models and Differential Equations , 2019. 51

    work page 2020

  4. [3]

    Bartlett

    Martin Anthony and Peter L. Bartlett. Neural Network Learning: Theoretical Founda- tions. Cambridge University Press, Cambridge, 1999

    work page 1999

  5. [4]

    Intrinsic finite element method for advection-diffusion-reaction equations on surfaces

    Elena Bachini, Matthew W Farthing, and Mario Putti. Intrinsic finite element method for advection-diffusion-reaction equations on surfaces. Journal of Computational Physics , 424:109827, 2021

    work page 2021

  6. [5]

    Local Rademacher complex- ities

    Peter L Bartlett, Olivier Bousquet, and Shahar Mendelson. Local Rademacher complex- ities. Annals of Statistics , 33(4):1497–1537, 2005

    work page 2005

  7. [6]

    Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks.Journal of Machine Learning Research, 20(1):2285–2301, 2019

    Peter L Bartlett, Nick Harvey, Christopher Liaw, and Abbas Mehrabian. Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks.Journal of Machine Learning Research, 20(1):2285–2301, 2019

    work page 2019

  8. [7]

    Physics-informed neural networks for shell structures

    Jan-Hendrik Bastek and Dennis M Kochmann. Physics-informed neural networks for shell structures. European Journal of Mechanics. A, Solids , 97:104849, 2023

    work page 2023

Show all 67 references
  1. [8]

    The quiet revolution of numerical weather prediction

    Peter Bauer, Alan Thorpe, and Gilbert Brunet. The quiet revolution of numerical weather prediction. Nature, 525(7567):47–55, 2015

    2015

  2. [9]

    Hyperbolic conservation laws on the sphere

    Matania Ben-Artzi, Joseph Falcovitz, and Philippe G LeFloch. Hyperbolic conservation laws on the sphere. a geometry-compatible finite volume scheme. Journal of Computa- tional Physics , 228(16):5650–5668, 2009

    2009

  3. [10]

    Matania Ben-Artzi and Philippe G. LeFloch. Well-posedness theory for geometry- compatible hyperbolic conservation laws on manifolds. Annales de l’Institut Henri Poincar´ e C. Analyse Non Lin´ eaire, 24(6):989–1008, 2007

    2007

  4. [11]

    Navier–Stokes, fluid dy- namics, and image and video inpainting

    Marcelo Bertalmio, Andrea L Bertozzi, and Guillermo Sapiro. Navier–Stokes, fluid dy- namics, and image and video inpainting. In Proceedings of the 2001 IEEE Computer So- ciety Conference on Computer Vision and Pattern Recognition. CVPR 2001 , volume 1, pages I–I. IEEE, 2001

    2001

  5. [12]

    Error estimates for deep learning methods in fluid dynamics

    Animikh Biswas, Jing Tian, and Suleyman Ulusoy. Error estimates for deep learning methods in fluid dynamics. Numerische Mathematik , 151(3):753–777, 2022

    2022

  6. [13]

    A divergence-conforming finite ele- ment method for the surface Stokes equation

    Andrea Bonito, Alan Demlow, and Martin Licht. A divergence-conforming finite ele- ment method for the surface Stokes equation. SIAM Journal on Numerical Analysis , 58(5):2764–2798, 2020

    2020

  7. [14]

    The Mathematical Theory of Finite Element Methods

    Susanne C Brenner. The Mathematical Theory of Finite Element Methods . Springer, 2008

    2008

  8. [15]

    Improving weak PINNs for hyperbolic conserva- tion laws: Dual norm computation, boundary conditions and systems

    Aidan Chaumet and Jan Giesselmann. Improving weak PINNs for hyperbolic conserva- tion laws: Dual norm computation, boundary conditions and systems. SMAI Journal of Computational Mathematics , 10:373–401, 2024

    2024

  9. [16]

    A comparison study of deep Galerkin method and deep Ritz method for elliptic problems with different boundary conditions

    Jingrun Chen, Rui Du, and Keke Wu. A comparison study of deep Galerkin method and deep Ritz method for elliptic problems with different boundary conditions. Communica- tions in Mathematical Research , 36(3):354–376, 2020

    2020

  10. [17]

    Efficient approximation of deep ReLU networks for functions on low dimensional manifolds

    Minshuo Chen, Haoming Jiang, Wenjing Liao, and Tuo Zhao. Efficient approximation of deep ReLU networks for functions on low dimensional manifolds. Advances in Neural Information Processing Systems, 32, 2019. 52

    2019

  11. [18]

    Nonparametric regression on low-dimensional manifolds using deep ReLU networks: Function approximation and statistical recovery

    Minshuo Chen, Haoming Jiang, Wenjing Liao, and Tuo Zhao. Nonparametric regression on low-dimensional manifolds using deep ReLU networks: Function approximation and statistical recovery. IMA, Information and Inference , 11(4):1203–1253, 2022

    2022

  12. [19]

    Physics-informed machine learning for reduced-order modeling of nonlinear problems

    Wenqian Chen, Qian Wang, Jan S Hesthaven, and Chuhua Zhang. Physics-informed machine learning for reduced-order modeling of nonlinear problems. Journal of Compu- tational Physics , 446:110666, 2021

    2021

  13. [20]

    δ-PINNs: Physics- informed neural networks on complex geometries

    Francisco Sahli Costabal, Simone Pezzuto, and Paris Perdikaris. δ-PINNs: Physics- informed neural networks on complex geometries. Engineering Applications of Artificial Intelligence, 127:107324, 2024

    2024

  14. [21]

    Error estimates for physics- informed neural networks approximating the Navier–Stokes equations

    Tim De Ryck, Ameya D Jagtap, and Siddhartha Mishra. Error estimates for physics- informed neural networks approximating the Navier–Stokes equations. IMA Journal of Numerical Analysis, 44(1):83–119, 2024

    2024

  15. [22]

    On the approximation of functions by tanh neural networks

    Tim De Ryck, Samuel Lanthaler, and Siddhartha Mishra. On the approximation of functions by tanh neural networks. Neural Networks, 143:732–750, 2021

    2021

  16. [23]

    Error analysis for physics-informed neural networks (PINNs) approximating kolmogorov PDEs

    Tim De Ryck and Siddhartha Mishra. Error analysis for physics-informed neural networks (PINNs) approximating kolmogorov PDEs. Advances in Computational Mathematics , 48(6):79, 2022

    2022

  17. [24]

    Generic bounds on the approximation error for physics-informed (and) operator learning

    Tim De Ryck and Siddhartha Mishra. Generic bounds on the approximation error for physics-informed (and) operator learning. Advances in Neural Information Processing Systems, 35:10945–10958, 2022

    2022

  18. [25]

    Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning

    Tim De Ryck and Siddhartha Mishra. Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning. Acta Numerica , 33:633–713, 2024

    2024

  19. [26]

    wPINNs: Weak physics in- formed neural networks for approximating entropy solutions of hyperbolic conservation laws

    Tim De Ryck, Siddhartha Mishra, and Roberto Molinaro. wPINNs: Weak physics in- formed neural networks for approximating entropy solutions of hyperbolic conservation laws. SIAM Journal on Numerical Analysis , 62(2):811–841, 2024

    2024

  20. [27]

    Reproducing kernel Hilbert spaces on manifolds: Sobolev and diffusion spaces

    Ernesto De Vito, Nicole M¨ ucke, and Lorenzo Rosasco. Reproducing kernel Hilbert spaces on manifolds: Sobolev and diffusion spaces. Analysis and Applications , 19(03):363–396, 2021

    2021

  21. [28]

    The deep Ritz method: A deep learning-based numerical algo- rithm for solving variational problems

    Weinan E and Bing Yu. The deep Ritz method: A deep learning-based numerical algo- rithm for solving variational problems. Communications in Mathematics and Statistics , 6(1):1–12, 2018

    2018

  22. [29]

    The shallow water equations and their application to realistic cases

    Pilar Garc ´ ıa-Navarro, J Murillo, J Fern´ andez-Pato, Isabel Echeverribar, and Mario Morales-Hern´ andez. The shallow water equations and their application to realistic cases. Environmental Fluid Mechanics , 19:1235–1252, 2019

    2019

  23. [30]

    A combined PDE and texture synthesis approach to inpainting

    Harald Grossauer. A combined PDE and texture synthesis approach to inpainting. In Computer Vision-ECCV 2004: 8th European Conference on Computer Vision, Prague, Czech Republic, May 11-14, 2004. Proceedings, Part II 8 , pages 214–224. Springer, 2004. 53

    2004

  24. [31]

    Error bounds for approximations with deep ReLU neural networks in W s,p norms

    Ingo G¨ uhring, Gitta Kutyniok, and Philipp Petersen. Error bounds for approximations with deep ReLU neural networks in W s,p norms. Analysis and Applications , 18(5):803– 859, 2020

    2020

  25. [32]

    Pre-training strategy for solving evolution equations based on physics-informed neural networks

    Jiawei Guo, Yanzhong Yao, Han Wang, and Tongxiang Gu. Pre-training strategy for solving evolution equations based on physics-informed neural networks. Journal of Com- putational Physics , 489:112258, 2023

    2023

  26. [33]

    Numerical Prediction and Dynamic Meteorology

    George Joseph Haltiner and Roger Terry Williams. Numerical Prediction and Dynamic Meteorology. Wiley, 1980

    1980

  27. [34]

    Error analysis of higher order trace finite element methods for the surface Stokes equation

    Thomas Jankuhn, Maxim A Olshanskii, Arnold Reusken, and Alexander Zhiliakov. Error analysis of higher order trace finite element methods for the surface Stokes equation. Journal of Numerical Mathematics , 29(3):245–267, 2021

    2021

  28. [35]

    Neural operator: Learning maps between function spaces with applications to PDEs

    Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhat- tacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to PDEs. Journal of Machine Learning Re- search, 24(89):1–97, 2023

    2023

  29. [36]

    First order quasilinear equations in several independent variables

    Stanislav N Kruˇ zkov. First order quasilinear equations in several independent variables. Mathematics of the USSR-Sbornik , 10(2):217, 1970

    1970

  30. [37]

    Low dimensional approximation and generalization of mul- tivariate functions on smooth manifolds using deep ReLU neural networks

    Demetrio Labate and Ji Shi. Low dimensional approximation and generalization of mul- tivariate functions on smooth manifolds using deep ReLU neural networks. Neural Net- works, 174:106223, 2024

    2024

  31. [38]

    Deep learning

    Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521(7553):436–444, May 2015

    2015

  32. [39]

    Probability in Banach Spaces: Isoperimetry and Processes

    Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: Isoperimetry and Processes. Springer Science & Business Media, 2013

    2013

  33. [40]

    John M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Math- ematics. Springer, New York, second edition, 2013

    2013

  34. [41]

    Solving PDEs on spheres with physics-informed convolutional neural networks.Applied and Computational Harmonic Analysis, 74:101714, 2025

    Guanhang Lei, Zhen Lei, Lei Shi, Chenyu Zeng, and Ding-Xuan Zhou. Solving PDEs on spheres with physics-informed convolutional neural networks.Applied and Computational Harmonic Analysis, 74:101714, 2025

    2025

  35. [42]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Borislavov Kovachki, Kamyar Azizzadenesheli, Burigede liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. In International Conference on Learning Rep- resentations, 2021

    2021

  36. [43]

    Physics-informed neural operator for learning partial differential equations

    Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. Physics-informed neural operator for learning partial differential equations. ACM/JMS Journal of Data Science , 1(3):1–27, 2024

    2024

  37. [44]

    Higher- order quasi-Monte Carlo training of deep neural networks

    Marcello Longo, Siddhartha Mishra, T Konstantin Rusch, and Christoph Schwab. Higher- order quasi-Monte Carlo training of deep neural networks. SIAM Journal on Scientific Computing, 43(6):A3938–A3966, 2021. 54

    2021

  38. [45]

    Deep network approxima- tion for smooth functions

    Jianfeng Lu, Zuowei Shen, Haizhao Yang, and Shijun Zhang. Deep network approxima- tion for smooth functions. SIAM Journal on Mathematical Analysis , 53(5):5465–5506, 2021

    2021

  39. [46]

    Learning nonlinear operators via DeepONet based on the universal approximation theo- rem of operators

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation theo- rem of operators. Nature Machine Intelligence , 3:218 – 229, 2019

    2019

  40. [47]

    Machine learn- ing for elliptic PDEs: Fast rate generalization bound, neural scaling law and minimax optimality

    Yiping Lu, Haoxuan Chen, Jianfeng Lu, Lexing Ying, and Jose Blanchet. Machine learn- ing for elliptic PDEs: Fast rate generalization bound, neural scaling law and minimax optimality. In International Conference on Learning Representations , 2022

    2022

  41. [48]

    Deep learning observables in computa- tional fluid dynamics

    Kjetil O Lye, Siddhartha Mishra, and Deep Ray. Deep learning observables in computa- tional fluid dynamics. Journal of Computational Physics , 410:109339, 2020

    2020

  42. [49]

    A few notes on statistical learning theory

    Shahar Mendelson. A few notes on statistical learning theory. In Advanced Lectures on Machine Learning: Machine Learning Summer School 2002 Canberra, Australia, Febru- ary 11–22, 2002 Revised Lectures , pages 1–40. Springer, 2003

    2002

  43. [51]

    Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs

    Siddhartha Mishra and Roberto Molinaro. Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs. IMA Journal of Numerical Analysis , 42(2):981–1022, 2022

    2022

  44. [52]

    Enhancing accuracy of deep learning algo- rithms by training with low-discrepancy sequences.SIAM Journal on Numerical Analysis, 59(3):1811–1834, 2021

    Siddhartha Mishra and T Konstantin Rusch. Enhancing accuracy of deep learning algo- rithms by training with low-discrepancy sequences.SIAM Journal on Numerical Analysis, 59(3):1811–1834, 2021

    2021

  45. [53]

    The imbedding problem for Riemannian manifolds

    John Nash. The imbedding problem for Riemannian manifolds. Annals of Mathematics , 63:20–63, 1956

    1956

  46. [54]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics , 378:686–707, 2019

    2019

  47. [55]

    Deep ReLU network approximation of functions on a manifold

    Johannes Schmidt-Hieber. Deep ReLU network approximation of functions on a manifold. ArXiv, abs/1908.00695, 2019

    1908 arXiv

  48. [56]

    Nonparametric regression using deep neural networks with ReLU activation function

    Johannes Schmidt-Hieber. Nonparametric regression using deep neural networks with ReLU activation function. Annals of Statistics , 48(4):1875, 2020

    2020

  49. [57]

    Deep network approximation characterized by number of neurons

    Zuowei Shen. Deep network approximation characterized by number of neurons. Com- munications in Computational Physics , 28(5), 2020

    2020

  50. [58]

    On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs

    Yeonjong Shin, J´ erˆ ome Darbon, and George Em Karniadakis. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs. Communications in Computational Physics , 28(5), 2020. 55

    2020

  51. [59]

    Topics in Variational PDE Image Segmentation, Inpainting and Denoising

    Bing Song. Topics in Variational PDE Image Segmentation, Inpainting and Denoising . University of California, Los Angeles, 2003

    2003

  52. [60]

    Singular Integrals and Differentiability Properties of Functions

    Elias M Stein. Singular Integrals and Differentiability Properties of Functions. Princeton University Press, 1970

    1970

  53. [61]

    Generating textures on arbitrary surfaces using reaction-diffusion

    Greg Turk. Generating textures on arbitrary surfaces using reaction-diffusion. ACM SIGGRAPH Computer Graphics , 25(4):289–298, 1991

    1991

  54. [62]

    A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

    Chenxi Wu, Min Zhu, Qinyang Tan, Yadhu Kartha, and Lu Lu. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering , 403:115671, 2023

    2023

  55. [63]

    Nearly optimal VC-dimension and pseudo- dimension bounds for deep neural network derivatives

    Yahong Yang, Haizhao Yang, and Yang Xiang. Nearly optimal VC-dimension and pseudo- dimension bounds for deep neural network derivatives. Advances in Neural Information Processing Systems, 36:21721–21756, 2023

    2023

  56. [64]

    Error bounds for approximations with deep ReLU networks

    Dmitry Yarotsky. Error bounds for approximations with deep ReLU networks. Neural Networks, 94:103–114, 2017

    2017

  57. [65]

    Weak adversarial networks for high-dimensional partial differential equations

    Yaohua Zang, Gang Bao, Xiaojing Ye, and Haomin Zhou. Weak adversarial networks for high-dimensional partial differential equations. Journal of Computational Physics , 411:109409, 2020

    2020

  58. [66]

    Classification with deep neural networks and logistic loss

    Zihan Zhang, Lei Shi, and Ding-Xuan Zhou. Classification with deep neural networks and logistic loss. Journal of Machine Learning Research , 25(125):1–117, 2024

    2024

  59. [67]

    Deep distributed convolutional neural networks: Universality

    Ding-Xuan Zhou. Deep distributed convolutional neural networks: Universality. Analysis and Applications, 16(06):895–919, 2018. 56

    2018

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.