Learning the Brownian Motion in Machine Learning for Parametric Parabolic PDEs: A Backward Quadrature Approach

We propose a backward quadrature learning (BQL) method for solving high dimensional parametric semilinear parabolic partial differential equations (PDEs). To the best of our knowledge, this work is the first to extend the applicability of BSDE-based stochastic numerical methods to this class of PDEs. Specifically, inspired by the approach of learning the random variables (LRV)~\cite{BeckerJentzenMullerWurstemberger2024}, BQL incorporates learnable quadrature for Brownian increments at each time level into the backward recursion of BSDEs, enabling more efficient layerwise approximation of local conditional expectations. Furthermore, BQL introduces a continuation surrogate to facilitate the transfer of value-function information from subsequent to preceding time levels, avoiding nested quadrature while controlling the propagation of local approximation errors. Our main theoretical contribution is the derivation of a global error bound that explicitly characterizes the decomposition of the total error, together with a proof of first-order mean-square convergence under suitable approximation and integrability assumptions. Finally, numerical experiments demonstrate that BQL achieves competitive solution accuracy while substantially reducing the amortized computational cost per query. For the HJB benchmark across the tested dimensions, with $Q=7$ parameter queries, BQL achieves computational cost reductions of approximately 40-fold and 4-fold compared with P$^2$INN~[2] and FBSNN~[3], respectively.