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而兴起，在多种PDE问题上展现出令人鼓舞的结果。然而，训练PINNs求解多维PDE并逼近高度复杂解函数存在根本性限制。这些具有挑战性的PDE所需的训练点（配置点）数量急剧增长，但由于高昂的计算成本和庞大的内存开销而受到严重制约。为解决这一问题，我们提出了一种面向PINNs的网络架构与训练算法。所提出的可分离PINN（SPINN）方法基于逐轴运算，显著减少了多维PDE中的网络传播次数，有别于传统PINN的点式处理方式。我们还提出使用前向模式自动微分来降低计算PDE残差的计算成本，从而在单个商用GPU上实现了大量配置点（>10^7）的处理。实验结果表明，在多维PDE中，计算成本大幅降低（在相同配置点数量下，实际运行时间减少62倍，FLOPs减少1,394倍），同时实现了更高的精度。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显著优于此前最佳方法（单GPU上9分钟 vs 10小时），且保持准确性。最后，我们展示了SPINN能够精确获得高度非线性多维PDE——（3+1）维纳维-斯托克斯方程的解。可视化结果与代码请参见https://jwcho5576.github.io/spinn.github.io/。