Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems

ArXi:2605.04126v1 Announce Type: new This paper develops convolutional neural network (CNN) methods for simultaneous approximation and elliptic boundary value problems on compact Riemannian manifolds. We establish simultaneous Sobole 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.