Regularity of Second-Order Elliptic PDEs in Spectral Barron Spaces
This provides a theoretical foundation for approximating PDE solutions with neural networks in high dimensions, though it is incremental as it builds on existing Barron space theory.
The paper tackles the regularity of solutions to second-order elliptic PDEs in spectral Barron spaces, showing that under mild conditions, solutions gain two additional orders of Barron regularity, enabling approximation by two-layer neural networks with cosine activations where width is independent of spatial dimension.
We establish a regularity theorem for second-order elliptic PDEs on $\mathbb{R}^{d}$ in spectral Barron spaces. Under mild ellipticity and smallness assumptions, the solution gains two additional orders of Barron regularity. As a corollary, we identify a class of PDEs whose solutions can be approximated by two-layer neural networks with cosine activation functions, where the width of the neural network is independent of the spatial dimension.