通过高阶导数约束提升物理信息神经网络的求解精度
Over-PINNs: Enhancing Physics-Informed Neural Networks via Higher-Order Partial Derivative Overdetermination of PDEs
- 利用自动微分生成高阶导数作为额外约束
- 在不显著增加计算成本下大幅提高PDE求解准确率
- 适合需要高精度物理模拟的研究者使用
偏微分方程（PDEs）是数学物理的核心。近年来，物理信息神经网络（PINNs）通过将物理定律嵌入神经网络训练，显著降低了对大规模数据集的依赖。然而，在处理复杂问题时，PINNs的精度仍有提升空间。为此，我们提出Over-PINNs框架，利用自动微分（AD）生成高阶辅助方程，引入额外的物理约束作为训练中的额外损失项，通过‘过定’方式增强模型捕捉物理信息的能力。数值结果表明，该方法在求解各类PDE时具有强泛化能力，显著提升了求解精度，且未带来显著的额外计算开销。
原文摘要 · Abstract (English)
Partial differential equations (PDEs) serve as the cornerstone of mathematical physics. In recent years, Physics-Informed Neural Networks (PINNs) have significantly reduced the dependence on large datasets by embedding physical laws directly into the training of neural networks. However, when dealing with complex problems, the accuracy of PINNs still has room for improvement. To address this issue, we introduce the Over-PINNs framework, which leverages automatic differentiation (AD) to generate higher-order auxiliary equations that impose additional physical constraints. These equations are incorporated as extra loss terms in the training process, effectively enhancing the model's ability to capture physical information through an "overdetermined" approach. Numerical results illustrate that this method exhibits strong versatility in solving various types of PDEs. It achieves a significant improvement in solution accuracy without incurring substantial additional computational costs.
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