发表机构
Pohang University of Science and Technology (POSTECH)（浦项科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对PINN在长时间PDE求解中因忽略时间因果性而失败的问题，提出因果强化的演化网络（CEENs），通过分区间顺序训练和积分形式损失函数，显著提升精度并降低计算成本，且支持并行加速。
AI 中文摘要
尽管物理信息神经网络（PINNs）日益流行，其在偏微分方程（PDEs）长时间积分中的适用性仍然受限。我们认为这一问题源于原始PINN公式中缺乏对时间因果性的考虑，导致在学习初始条件之前偏向于满足后期时间的控制方程，从而产生错误的解。为此，我们提出了一种将时间因果性无缝整合到训练过程中的新方法。受经典数值方法中体现时间因果性的启发，我们将时间域划分为不重叠的子区间，为每个子区间分配一个独特的神经网络，并基于这些子区间内PDEs的积分形式构建损失函数。所提出的网络按顺序训练，从初始时间步开始。我们的方法在多种原始PINN方法失败的PDE问题的长时间模拟中显著提高了精度，同时相比PINN方法需要更少的计算成本和内存。我们提供了一种并行化算法以进一步提高计算效率，在求解含时PDEs时显示出显著的加速效果。
英文摘要
Despite the growing popularity of physics-informed neural networks (PINNs), their applicability in the long-time integration of partial differential equations (PDEs) remains constrained. We argue that this problem stems from the lack of consideration of temporal causality in the original PINN formulation, resulting in a bias towards satisfying governing equations at later times before learning the initial condition and hence leading to erroneous solutions. To this end, we propose a novel method that seamlessly integrates temporal causality into the training process. Drawing inspiration from classical numerical methods where the temporal causality is reflected, we divide the time domain into nonoverlapping subintervals, assign a unique neural network to each subinterval, and construct a loss function founded on the integral form of PDEs within these subintervals. The proposed networks undergo sequential training, beginning with the initial time step. Our method demonstrates significant improvement in accuracy for long-time simulations of various PDE problems where the original PINN method fails while it requires less computational cost and memory compared to the PINN method. A parallelization algorithm is provided to further enhance the computational efficiency, showing a significant speedup for solving time-dependent PDEs.
Comments26 pages, 4 tables, 15 figures
Journal refComputer Methods in Applied Mechanics and Engineering 427 (2024) 117036
