Title: Causality-guided adaptive sampling method for physics-informed neural networks Show Chinese title

Title: 基于因果关系的物理信息神经网络自适应采样方法

Abstract: 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. Show Chinese abstract

Abstract: 与纯数据驱动的方法相比，物理信息神经网络（PINNs）——一种用于求解偏微分方程（PDEs）的经过验证的强大工具——的一个关键特性是将PDE约束嵌入到损失函数中。 用于评估PDE残差的配点的选择和分布对于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。 在本工作中，我们为PINNs提出了因果引导的自适应采样（Causal AS）方法。 鉴于因果训练的特点，我们使用加权PDE残差作为配点选择的指标，以关注正在训练区域中具有较大PDE残差的区域。 对于涉及的超参数$p$，我们开发了时间对齐驱动更新（TADU）方案，用于其动态更新，而不仅仅是将其固定为常数。 每次时间选择的配点将在下一次自适应采样步骤之前释放，以避免由先前选择的配点引起的累积效应并减少计算成本。 为了说明Causal AS方法的有效性，我们将它应用于求解时间依赖方程，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，预测解与真实值有很好的一致性。 与类似的工作相比，因果扩展的R3采样（Causal R3），我们提出的Causal AS方法在准确性方面表现出显著的优势。