Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems

Hanfei Zhou; Lei Shi

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arXiv · 2605.04126

Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems

Hanfei Zhou·Lei Shi

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Abstract

This paper develops convolutional neural network (CNN) methods for simultaneous Sobolev approximation and elliptic boundary value problems on compact Riemannian manifolds. We prove approximation estimates for single- and multichannel CNNs, with rates governed by the intrinsic dimension and the smoothness gap. Motivated by elliptic stability, we propose a physics-informed CNN framework with a spectral boundary loss. The boundary residual is expanded in boundary Laplace--Beltrami eigenmodes and penalized by Sobolev trace weights, matching the natural $\mathcal H^{2s-1/2}(\partial\mathcal M^d)$ trace norm for $2s$-order elliptic problems. This avoids smooth auxiliary constructions for exact boundary enforcement and singular Sobolev--Slobodeckij double integrals, while allowing FFT-based or precomputed spectral implementations. We also derive an error decomposition separating approximation, generalization, and spectral truncation errors, showing that the proposed loss is aligned with localized fast-rate generalization analysis. Numerical experiments on the upper hemisphere and upper half-torus demonstrate improved accuracy, convergence, and stability over standard PINNs, with one to two orders of magnitude gains for high-frequency boundary data.

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BibTeX RIS

Hanfei Zhou, Lei Shi. 2026-05-05. Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems. https://arxiv.org/abs/2605.04126

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