Deep Picard Iteration for High-Dimensional Nonlinear PDEs

Abstract. We present the Deep Picard Iteration (DPI) method, a new deep learning approach for solving high-dimensional partial differential equations (PDEs). The core innovation of DPI lies in its use of Picard iteration to reformulate the typically complex training objectives of neural network–based PDE solutions into much simpler, standard regression tasks based on function values and gradients. This design not only greatly simplifies the optimization process but also offers the potential for further scalability through parallel data generation. Crucially, to fully realize the benefits of regressing on both function values and gradients in the DPI method, we address the issue of infinite variance in the estimators of gradients by incorporating a control variate, supported by theoretical analysis. Our experiments on problems up to 100 dimensions demonstrate that DPI consistently outperforms existing state-of-the-art methods, with greater robustness to hyperparameters, particularly in challenging scenarios with long time horizons and strong nonlinearity. The code is available at https://github.com/DeepOptimalControl/DeepPicardIteration . Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and Data Available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/DeepOptimalControl/DeepPicardIteration and in the supplementary materials ( M169312_Supplementary_Materials_1.pdf [ 8.54MB], ( DeepPicardIteration-main.zip [ 102KB]), linked from the main article webpage. [Formula: see text]