Higher-order deep solver of non-linear PDEs implied by a non-linear discrete Clark-Ocone formula

Abstract

In the present paper, we introduce a variant of the numerical scheme called the deep solver of PDE. Our scheme is based on a non-linear version of the discrete-time Clark--Ocone formula, which describes the convergent expansion of the error terms. Our new scheme incorporates the higher-order error terms, which we conjecture to stabilize the stochastic gradient descent procedure, and also the irregularities in the driver and the terminal function of the associated forward-backward stochastic differential equation.