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.

翻译：物理信息神经网络（PINNs）近期作为有前景的数据驱动型偏微分方程求解器涌现，在各类偏微分方程上展现出令人鼓舞的成果。然而，训练PINNs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）数量显著增长，但由于高昂的计算成本和沉重的内存开销，其数量受到严重限制。为解决这一问题，我们提出了一种适用于PINNs的网络架构与训练算法。所提出的方法——可分离PINN（SPINN）——基于逐轴方式运作，与常规PINN的逐点处理不同，能大幅降低多维偏微分方程中的网络传播次数。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成本，从而在单个商用GPU上实现大量配置点（>10^7）。实验结果表明，在多维偏微分方程中，计算成本显著降低（在相同配置点数量下，实际运行时间降低62倍，FLOPs降低1,394倍），同时实现了更优的精度。此外，我们证明SPINN求解混沌(2+1)维纳维-斯托克斯方程的速度远快于先前最佳方法（单GPU上9分钟对比10小时），且保持了精度。最后，我们展示了SPINN能够准确获取高度非线性多维偏微分方程——(3+1)维纳维-斯托克斯方程——的解。