Physics-informed neural networks (PINNs) have recently emerged as promising data-driven PDE solvers showing encouraging results on various PDEs. However, there is a fundamental limitation of training PINNs to solve multi-dimensional PDEs and approximate highly complex solution functions. The number of training points (collocation points) required on these challenging PDEs grows substantially, but it is severely limited due to the expensive computational costs and heavy memory overhead. To overcome this issue, we propose a network architecture and training algorithm for PINNs. The proposed method, separable PINN (SPINN), operates on a per-axis basis to significantly reduce the number of network propagations in multi-dimensional PDEs unlike point-wise processing in conventional PINNs. We also propose using forward-mode automatic differentiation to reduce the computational cost of computing PDE residuals, enabling a large number of collocation points (>10^7) on a single commodity GPU. The experimental results show drastically reduced computational costs (62x in wall-clock time, 1,394x in FLOPs given the same number of collocation points) in multi-dimensional PDEs while achieving better accuracy. Furthermore, we present that SPINN can solve a chaotic (2+1)-d Navier-Stokes equation significantly faster than the best-performing prior method (9 minutes vs 10 hours in a single GPU), maintaining accuracy. Finally, we showcase that SPINN can accurately obtain the solution of a highly nonlinear and multi-dimensional PDE, a (3+1)-d Navier-Stokes equation. For visualized results and code, please see https://jwcho5576.github.io/spinn.github.io/.

翻译：物理信息神经网络（PINNs）近年来作为有前景的数据驱动偏微分方程求解器出现，在各类偏微分方程上展现了令人鼓舞的结果。然而，训练PINNs以求解多维偏微分方程并逼近高度复杂的解函数存在根本性局限。在这些具有挑战性的偏微分方程上所需的训练点（配置点）数量大幅增长，但由于高昂的计算成本和巨大的内存开销而受到严重制约。为解决这一问题，我们提出了一种用于PINNs的网络架构和训练算法。所提出的方法——可分离PINN（SPINN）——基于逐轴方式操作，显著减少多维偏微分方程中的网络传播次数，这与传统PINNs中逐点处理的方式不同。我们还提出使用前向模式自动微分来降低计算偏微分方程残差的计算成本，从而在单块商用GPU上支持大量配置点（>10^7）。实验结果表明，在多维偏微分方程中，计算成本大幅降低（在相同配置点数量下，挂钟时间减少62倍，FLOPs减少1,394倍），同时实现了更优的精度。此外，我们展示SPINN能够以显著快于先前最佳方法的速度求解混沌（2+1）维纳维-斯托克斯方程（单GPU上9分钟对比10小时），并保持精度。最后，我们演示SPINN能够精确获得高度非线性和多维偏微分方程——（3+1）维纳维-斯托克斯方程——的解。可视化结果和代码请参见 https://jwcho5576.github.io/spinn.github.io/。