Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems

TL;DR

This paper introduces CNN methods for approximating functions and solving boundary value problems on manifolds, overcoming dimensionality issues and improving accuracy with spectral boundary losses.

Contribution

It develops a theoretical framework for manifold CNN approximation and proposes a spectral boundary loss for physics-informed CNNs addressing boundary value problems.

Findings

Manifold CNNs achieve approximation rates governed by intrinsic dimension.

Spectral boundary loss improves stability and accuracy of PINNs on manifolds.

Numerical experiments show enhanced convergence over standard PINNs.

Abstract

This paper develops convolutional neural network (CNN) methods for simultaneous approximation and elliptic boundary value problems on compact Riemannian manifolds. We establish simultaneous Sobolev approximation results for single- and multichannel CNNs, showing that manifold functions and their derivatives can be approximated with rates governed by the intrinsic dimension and the smoothness gap, rather than by the ambient dimension, thereby mitigating the curse of dimensionality. Building on this approximation theory, we propose a physics-informed CNN (PICNN) framework specially designed for boundary value problems. The main numerical issue is a boundary-norm mismatch: standard PINNs usually impose boundary data through low-order, often L2-type, penalties, whereas elliptic stability requires Sobolev trace control. We address this by introducing a spectral boundary loss based on the boundary Laplace-Beltrami operator, which represents trace errors as weighted frequency energies and relates truncation error to boundary eigenvalue decay. This avoids smooth auxiliary constructions required by exact boundary enforcement and singular double integrals arising in Sobolev-Slobodeckij penalties, while enabling implementations based on Fast Fourier Transforms (FFTs) or precomputed spectral bases on structured boundaries. Numerical experiments demonstrate improved accuracy, convergence, and stability over standard PINNs.