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 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.

翻译：本文提出深度Picard迭代（DPI）方法，这是一种用于求解高维偏微分方程（PDEs）的新型深度学习框架。DPI的核心创新在于利用Picard迭代，将基于神经网络的PDE求解中通常复杂的训练目标，重新表述为基于函数值及其梯度的、更为简单的标准回归任务。这一设计不仅极大简化了优化过程，还通过并行数据生成展现出进一步扩展的潜力。关键在于，为充分发挥DPI方法中同时对函数值和梯度进行回归的优势，我们通过引入控制变量解决了梯度估计量方差无限的问题，并提供了相应的理论分析支持。在维度高达100的各类问题上进行的实验表明，DPI方法在性能上持续优于现有最先进方法，且对超参数具有更强的鲁棒性，尤其是在具有长时间跨度和强非线性的挑战性场景中。