Barron Space Representations for Elliptic PDEs with Homogeneous Boundary Conditions
This addresses the computational challenge of solving high-dimensional PDEs in fields like physics and finance, though it is incremental as it builds on existing Barron space frameworks.
The paper tackles the approximation of high-dimensional elliptic PDEs with homogeneous boundary conditions by proving that solutions can be efficiently approximated by two-layer neural networks under Barron space assumptions, avoiding the curse of dimensionality.
We study the approximation complexity of high-dimensional second-order elliptic PDEs with homogeneous boundary conditions on the unit hypercube, within the framework of Barron spaces. Under the assumption that the coefficients belong to suitably defined Barron spaces, we prove that the solution can be efficiently approximated by two-layer neural networks, circumventing the curse of dimensionality. Our results demonstrate the expressive power of shallow networks in capturing high-dimensional PDE solutions under appropriate structural assumptions.