Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems

arXiv cs.LG / 5/7/2026

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Key Points

The paper introduces CNN-based methods to simultaneously approximate manifold functions and solve elliptic boundary value problems on compact Riemannian manifolds.
It proves Sobolev approximation rates for both single- and multichannel CNNs, where convergence depends on intrinsic manifold dimension and smoothness rather than the ambient dimension, reducing the curse of dimensionality.
Building on the theory, the authors propose a physics-informed CNN (PICNN) tailored to boundary value problems by targeting a “boundary-norm mismatch” common in standard PINNs.
They use a spectral boundary loss derived from the boundary Laplace–Beltrami operator to control Sobolev trace errors via weighted frequency energies tied to boundary eigenvalue decay.
Experiments show that the proposed PICNN approach improves accuracy, convergence, and stability compared with standard PINN formulations.

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.