Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions

Abstract

We study the complexity of approximating high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube using Barron spaces. Under suitable Barron assumptions on the coefficients and forcing term, we prove that the solutions can be approximated to any prescribed accuracy \(\varepsilon>0\) by two-layer neural networks whose widths and relevant parameters are bounded by \(\mathcal{O}\bigl(d^{C|\log\varepsilon|}\bigr)\). Consequently, we identify a class of elliptic PDEs that can be approximated by shallow neural networks without suffering from the curse of dimensionality.