# Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems

arXiv: https://arxiv.org/abs/2605.04126 · HTML: https://mlsanity.com/paper/2605.04126/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 32 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 14 of 14)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [Diffusion Models are Minimax Optimal Distribution Estimators](https://mlsanity.com/paper/2303.01861/) (2023)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Understanding Deep Convolutional Networks](https://mlsanity.com/paper/1601.04920/) (2016)
- [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Theory of Deep Convolutional Neural Networks II: Spherical Analysis](https://mlsanity.com/paper/2007.14285/) (2020)
- [Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces](https://mlsanity.com/paper/1905.10913/) (2019)
- [On the rates of convergence for learning with convolutional neural networks](https://mlsanity.com/paper/2403.16459/) (2024)
- [Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks](https://mlsanity.com/paper/2308.09605/) (2023)
- [Weak Physics Informed Neural Networks for Geometry Compatible Hyperbolic Conservation Laws on Manifolds](https://mlsanity.com/paper/2505.19036/) (2025)
- [Expressive Power of Deep Networks on Manifolds: Simultaneous Approximation](https://mlsanity.com/paper/2509.09362/) (2025)
- [Trace Regularity PINNs: Enforcing H^frac12(partial Omega) for Boundary Data](https://mlsanity.com/paper/2510.16817/) (2025)

## Built on by (most cited first, 1 of 1)

- [Boundary-Adapted PINNs for Elliptic Dirichlet Problems: H2(Omega) A Priori Error Bounds with Application to Mean Escape Time Computation](https://mlsanity.com/paper/2607.19167/) (2026)

## Abstract (arXiv)

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 H2s-1/2(partialmathcal Md)) 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.