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型偏微分方程求解器出现，在多种PDE求解中展现出令人鼓舞的结果。然而，在训练PINNs求解多维PDE及逼近高度复杂解函数时存在根本性局限。这类挑战性PDE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与沉重的内存开销，实际可用点数严重受限。为克服这一难题，我们提出用于PINN的网络架构与训练算法。所提出的可分离PINN（SPINN）采用基于坐标轴的处理方式，与传统PINN逐点处理不同，能显著减少多维PDE中的网络传播次数。我们还提出利用前向模式自动微分降低PDE残差计算的计算成本，使得在单块商用GPU上可实现超过10^7个配置点的训练。实验结果表明，在多维PDE中，该方法在相同配置点数下大幅降低计算成本（实际耗时降低62倍，FLOPs降低1394倍），同时取得更优精度。此外，我们证明SPINN能以显著快于现有最优方法的速度求解混沌(2+1)维纳维-斯托克斯方程（单GPU运行9分钟对比10小时），并保持精度。最后，我们展示SPINN能精确获得高度非线性的多维PDE——(3+1)维纳维-斯托克斯方程的解。可视化结果与代码请见https://jwcho5576.github.io/spinn.github.io/。