On the Representation of Solutions to Elliptic PDEs in Barron Spaces
This work addresses the challenge of solving high-dimensional PDEs numerically, which is important for applications in physics and engineering, but it is incremental as it builds on existing Barron space theory.
This paper tackles the problem of approximating solutions to high-dimensional elliptic PDEs by proving that under certain conditions, the solution is close to a Barron function, with dimension-explicit polynomial bounds on the Barron norm. As a result, the solution can be approximated by a two-layer neural network with a dimension-explicit convergence rate in the H^1 norm.
Numerical solutions to high-dimensional partial differential equations (PDEs) based on neural networks have seen exciting developments. This paper derives complexity estimates of the solutions of $d$-dimensional second-order elliptic PDEs in the Barron space, that is a set of functions admitting the integral of certain parametric ridge function against a probability measure on the parameters. We prove under some appropriate assumptions that if the coefficients and the source term of the elliptic PDE lie in Barron spaces, then the solution of the PDE is $ε$-close with respect to the $H^1$ norm to a Barron function. Moreover, we prove dimension-explicit bounds for the Barron norm of this approximate solution, depending at most polynomially on the dimension $d$ of the PDE. As a direct consequence of the complexity estimates, the solution of the PDE can be approximated on any bounded domain by a two-layer neural network with respect to the $H^1$ norm with a dimension-explicit convergence rate.