Representation of solutions of second-order linear equations in Barron space via Green's functions

Namkyeong Cho, Hyung Ju Hwang

Keywords: partial differential equations, neural networks, Barron norms, high dimension, approximation, regularity theory

Abstract: AI-based methods for solving high-dimensional partial differential equations (PDEs) have garnered significant attention as a promising approach to overcoming the curse of dimensionality faced by traditional techniques. This work establishes complexity estimates for the Barron norm of solutions of $d$-dimensional linear second-order PDEs, explicitly capturing the dependence on dimension. By leveraging well-developed theory for elliptic and parabolic equations, we represent the solutions of linear second-order equations using Green's functions. From these representations, we derive complexity bounds for the Barron norm of the solutions. Our results extend the prior work of Chen et al. (2021) in two key aspects. First, we consider more general elliptic and parabolic equations; specifically, we address both time-independent and time-dependent equations. Second, we provide sufficient conditions on the coefficients of the PDEs under which the solutions belong to Barron space rather than approximating the solutions via Barron functions in the $H^1$ norm. As a result, our approach yields theoretically improved results, providing a more intuitive understanding when approximating the solutions of PDEs via two-layer neural networks.

Primary Area: learning theory

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Submission Number: 3080