A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

Solving high-dimensional, fully nonlinear partial differential equations (PDEs) is a fundamental challenge in scientific machine learning. Traditional methods often require explicit knowledge of the PDE's internal structure (model-based) or rely on automatic differentiation to compute derivatives, which can become unstable and computationally expensive in high dimensions. This paper addresses the problem of solving "black-box" PDEs where the underlying operators and coefficients are unknown, and only simulation-based data is available.

The authors propose a "representing-then-learning" framework that treats the PDE as a black box. Instead of inferring the PDE coefficients, they use a simulator to generate stochastic trajectories. The core innovation is the use of zeroth-order derivative (ZOD) estimators—which rely solely on function evaluations at perturbed points—to generate targets for gradient and Hessian neural networks. This approach avoids the instability of automatic differentiation. The method integrates this with approximate value iteration, reducing the nonlinear PDE into a sequence of linear problems that are solved iteratively.

The study provides a comprehensive statistical learning analysis, establishing non-asymptotic error bounds in (weighted) Sobolev space. The authors decompose the total error into discretization, approximation, statistical, and ZOD bias components. They demonstrate that their method achieves competitive performance in both moderate and high-dimensional settings. Crucially, they derive the sample complexity for learning solutions and their derivatives, providing the first theoretical convergence guarantees for this class of model-free, fully nonlinear PDE solvers.