Deep Picard Iteration for High-Dimensional Nonlinear PDEs

TL;DR

Deep Picard Iteration (DPI) introduces a simplified neural network approach for solving high-dimensional nonlinear PDEs by reformulating the problem into regression tasks and effectively managing gradient variance, achieving superior performance.

Contribution

The paper proposes DPI, a novel deep learning method that simplifies PDE solving via Picard iteration and gradient control variates, enhancing scalability and robustness.

Findings

Outperforms existing methods in up to 100 dimensions

Demonstrates robustness to hyperparameters

Effective in long horizon, 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 variancein the estimators of gradients by incorporating a control variate, supported by our 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.