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Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks

Guanhang Lei, Zhen Lei, Lei Shi, Chenyu Zeng, Ding-Xuan Zhou

TL;DR

The paper develops a theoretical framework for solving PDEs on spheres with physics-informed convolutional neural networks (PICNNs). By combining spherical-harmonic–based Sobolev approximation, localization analysis of Rademacher and VC-dimension complexities, and a hybrid ReLU/ReLU$^k$ CNN architecture, it proves fast convergence rates: excess risk $\mathcal{R}(u_n) - \mathcal{R}(u^*) \lesssim n^{-a}(\log n)^{2a}$ and Sobolev error $\|u_n-u^*\|_{H^s(\mathbb{S}^{d-1})} \lesssim n^{-a/2}(\log n)^a$ with $a$ depending on problem parameters. The results are complemented by experiments on the 2D sphere that confirm the predicted polynomial rates and demonstrate strategies to circumvent the curse of dimensionality when the ground truth is highly smooth or exhibits favorable structure. The work provides the first rigorous convergence analysis for PINNs with CNN architectures solving PDEs on manifolds and suggests pathways to extend the approach to general manifolds. Overall, it advances the theoretical understanding and practical reliability of PINN-based solvers for high-dimensional PDEs on curved geometries.

Abstract

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on surfaces, including spheres. However, theoretical understanding of the numerical performance of PINNs, especially PINNs on surfaces or manifolds, is still lacking. In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere. By using and improving the latest approximation results of deep convolutional neural networks and spherical harmonic analysis, we prove an upper bound for the approximation error with respect to the Sobolev norm. Subsequently, we integrate this with innovative localization complexity analysis to establish fast convergence rates for PICNN. Our theoretical results are also confirmed and supplemented by our experiments. In light of these findings, we explore potential strategies for circumventing the curse of dimensionality that arises when solving high-dimensional PDEs.

Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks

TL;DR

The paper develops a theoretical framework for solving PDEs on spheres with physics-informed convolutional neural networks (PICNNs). By combining spherical-harmonic–based Sobolev approximation, localization analysis of Rademacher and VC-dimension complexities, and a hybrid ReLU/ReLU CNN architecture, it proves fast convergence rates: excess risk and Sobolev error with depending on problem parameters. The results are complemented by experiments on the 2D sphere that confirm the predicted polynomial rates and demonstrate strategies to circumvent the curse of dimensionality when the ground truth is highly smooth or exhibits favorable structure. The work provides the first rigorous convergence analysis for PINNs with CNN architectures solving PDEs on manifolds and suggests pathways to extend the approach to general manifolds. Overall, it advances the theoretical understanding and practical reliability of PINN-based solvers for high-dimensional PDEs on curved geometries.

Abstract

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on surfaces, including spheres. However, theoretical understanding of the numerical performance of PINNs, especially PINNs on surfaces or manifolds, is still lacking. In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere. By using and improving the latest approximation results of deep convolutional neural networks and spherical harmonic analysis, we prove an upper bound for the approximation error with respect to the Sobolev norm. Subsequently, we integrate this with innovative localization complexity analysis to establish fast convergence rates for PICNN. Our theoretical results are also confirmed and supplemented by our experiments. In light of these findings, we explore potential strategies for circumventing the curse of dimensionality that arises when solving high-dimensional PDEs.
Paper Structure (27 sections, 24 theorems, 269 equations, 6 figures)

This paper contains 27 sections, 24 theorems, 269 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Main Result
  3. PINN for General PDEs
  4. PINN for PDEs on Spheres
  5. The CNN Architectures
  6. Main Result
  7. Error Decomposition
  8. Approximation Error Analysis
  9. Statistical Error Analysis
  10. Convergence Analysis: Proof of \ref{['main theorem']}
  11. Experiments
  12. Smooth Solution on 2-D Sphere
  13. Influence of Smoothness
  14. Overcoming the Curse of Dimensionality
  15. The Laplace-Beltrami operator and Sobolev spaces on spheres
  16. ...and 12 more sections

Key Result

Lemma 1

(Theorem 12.8 in Agmon1959Estimates) Let $s \in \mathbb{N}$ and $\mathcal{L}$ denote a uniformly elliptic operator of order $2s$ possessing bounded coefficients, with its leading coefficients being continuous. Furthermore, suppose $\mathcal{L}$ is weakly positive semi-definite. For a sufficiently la holds for every $u \in H^{2s}(\Omega) \cap H^{s}_0(\Omega)$.

Figures (6)

  • Figure 1: The relative loss history of PICNN model for $u^* = x_1x_2x_3$ on 2-D sphere. Each epoch contains $128$ training iterations.
  • Figure 2: Scatter plots on the test points for $u^* = x_1x_2x_3$ on 2-D sphere. From left to right, we show the predicted solution given by the PICNN, the true solution $u^*$ and the absolute error.
  • Figure 3: The log-log plot with an estimated convergence slope for $u^* = x_1x_2x_3$ on 2-D sphere. Each point shows the average over $10$ replicates.
  • Figure 4: The log-log plot for solutions with different smoothness order on the 2-D sphere. Each point shows the average over $5$ replicates.
  • Figure 6: The log-log plot for smooth solutions on $d-1$ dimension sphere. Each point shows the average over $5$ replicates.
  • ...and 1 more figures

Theorems & Definitions (41)

  • Lemma 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Theorem 1
  • Lemma 3
  • Lemma 4
  • Remark 3
  • Lemma 5
  • Lemma 6
  • ...and 31 more