Convergence Rates for a Deep Learning Algorithm for Semilinear PDEs

Abstract

We derive convergence rates for a deep solver for semilinear partial differential equations which is based on a Feynman-Kac representation in terms of a forward-backward stochastic differential equation and a discretization in time. We show that the error of the deep solver is bounded in terms of its loss functional, hence yielding a direct measure to judge the quality in numerical applications, and that the loss functional converges sufficiently fast to zero to guarantee that the approximation error vanishes in the limit. As a consequence of these results, we show that the deep solver has a strong convergence rate of order 1/2.

Keywords: semilinear PDE, forward-backward SDE, deep solver, strong convergence rate