Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems
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 ex...
| Authors | Hanfei Zhou, Lei Shi |
|---|---|
| Primary category | cs.LG |
| Published | 5 May 2026 |
| Last updated at source | 21 June 2026 |
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