物理信息神经网络的因果性引导自适应采样方法
Causality-guided adaptive sampling method for physics-informed neural networks
AI总结：
本文提出因果性引导自适应采样方法（Causal AS），利用加权PDE残差动态选择配置点，并结合时间对齐驱动更新参数方案，在多个含时方程上显著提升了物理信息神经网络的求解精度。
摘要：
与纯数据驱动方法相比，物理信息神经网络（PINNs）——一种已被证明用于求解偏微分方程（PDEs）的强大工具——的一个关键特征是将PDE约束嵌入到损失函数中。用于评估PDE残差的配置点的选择和分布对PINNs的性能至关重要。此外，因果训练目前是一种流行的训练模式。在这项工作中，我们提出了用于PINNs的因果性引导自适应采样（Causal AS）方法。鉴于因果训练的特点，我们使用加权PDE残差作为选择配置点的指标，以聚焦于正在训练区域内PDE残差较大的区域。对于所涉及的参数$p$，我们开发了时间对齐驱动更新（TADU）方案来对其进行动态更新，而不是简单地将其固定为常数。每次选择的配置点将在下一次自适应采样步骤之前被释放，以避免先前选择的配置点造成的累积效应并降低计算成本。为了说明Causal AS方法的有效性，我们将其应用于求解含时方程，包括Allen-Cahn方程、NLS方程、KdV方程和mKdV方程。在训练过程中，我们采用时间推进技术，并通过将输入坐标嵌入傅里叶展开来严格施加周期性边界条件，以缓解优化挑战。数值结果表明，预测解与真实解具有极好的一致性。与类似工作——R3采样的因果扩展（Causal R3）相比，我们提出的Causal AS方法在精度上表现出显著优势。
英文摘要：
Compared to purely data-driven methods, a key feature of physics-informed neural networks (PINNs) - a proven powerful tool for solving partial differential equations (PDEs) - is the embedding of PDE constraints into the loss function. The selection and distribution of collocation points for evaluating PDE residuals are critical to the performance of PINNs. Furthermore, the causal training is currently a popular training mode. In this work, we propose the causality-guided adaptive sampling (Causal AS) method for PINNs. Given the characteristics of causal training, we use the weighted PDE residuals as the indicator for the selection of collocation points to focus on areas with larger PDE residuals within the regions being trained. For the hyper-parameter $p$ involved, we develop the temporal alignment driven update (TADU) scheme for its dynamic update beyond simply fixing it as a constant. The collocation points selected at each time will be released before the next adaptive sampling step to avoid the cumulative effects caused by previously chosen collocation points and reduce computational costs. To illustrate the effectiveness of the Causal AS method, we apply it to solve time-dependent equations, including the Allen-Cahn equation, the NLS equation, the KdV equation and the mKdV equation. During the training process, we employe a time-marching technique and strictly impose the periodic boundary conditions by embedding the input coordinates into Fourier expansion to mitigate optimization challenges. Numerical results indicate that the predicted solution achieves an excellent agreement with the ground truth. Compared to a similar work, the causal extension of R3 sampling (Causal R3), our proposed Causal AS method demonstrates a significant advantage in accuracy.
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