Zeroth-Order Deep Learning Solves High-Dimensional PDEs

Solving high-dimensional partial differential equations (PDEs) with unknown coefficients is a significant challenge in scientific machine learning, particularly in areas like continuous-time reinforcement learning. Existing deep learning solvers often struggle with instability and error amplification due to repeated automatic differentiation, while probabilistic methods require explicit knowledge of data-generating dynamics, limiting their application in black-box scenarios. This research proposes a novel zeroth-order deep learning method that tackles these issues. It adopts a "representing-then-learning" approach, where solutions and their derivatives are learned even when the underlying PDE operators are only accessible through simulations and pointwise evaluations. The key innovation lies in using zeroth-order derivative (ZOD) estimators, derived from perturbed Monte Carlo trajectories, to generate targets for gradient and Hessian networks. The method is entirely model-free, relying solely on function evaluations. A statistical learning analysis provides a non-asymptotic error bound, breaking down the total error into discretization, approximation, statistical, and ZOD bias components. Numerical experiments confirm the method's competitive performance in both moderate and high-dimensional settings, offering a robust solution for complex PDE problems.