用物理约束训练的DeepONet高效求解非线性偏微分方程
DeepONet for Solving Nonlinear Partial Differential Equations with Physics-Informed Training
- 通过分支-主干结构学习算子映射，统一处理多类非线性PDE
- 复杂分支网络提升性能，简单主干网络更利于泛化
- 首次给出物理信息模型的泛化误差理论边界，适合研究者参考
本文研究算子学习方法DeepONet在求解非线性偏微分方程（PDE）中的应用。与传统需为每类PDE单独训练神经网络的方法不同，算子学习可在不重训练的情况下实现跨PDE泛化。本研究聚焦物理信息训练下的DeepONet性能，重点关注两个方面：(1) 深度分支网络与主干网络的逼近能力；(2) Sobolev范数下的泛化误差。结果表明，复杂分支网络带来显著性能提升，而主干网络保持较简结构时效果最佳。此外，通过分析导数的Rademacher复杂度与伪维数，推导出DeepONet求解非线性PDE的泛化误差上界。该工作填补了关键理论空白，为广泛物理信息机器学习模型提供了严格的误差估计。
原文摘要 · Abstract (English)
In this paper, we investigate the applications of operator learning, specifically DeepONet, for solving nonlinear partial differential equations (PDEs). Unlike conventional function learning methods that require training separate neural networks for each PDE, operator learning enables generalization across different PDEs without retraining. This study examines the performance of DeepONet in physics-informed training, focusing on two key aspects: (1) the approximation capabilities of deep branch and trunk networks, and (2) the generalization error in Sobolev norms. Our results show that complex branch networks provide substantial performance gains, while trunk networks are most effective when kept relatively simple. Furthermore, we derive a bound on the generalization error of DeepONet for solving nonlinear PDEs by analyzing the Rademacher complexity of its derivatives in terms of pseudo-dimension. This work bridges a critical theoretical gap by delivering rigorous error estimates. This paper fills a theoretical gap by providing error estimates for a wide range of physics-informed machine learning models and applications.
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