论文标题
使用无网状离散化的物理信息神经网络（PINN）加速培训
Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations
论文作者
论文摘要
我们提出了一种用于加速物理信息神经网络（PINNS）的新技术：离散训练的PINN（DT-PINNS）。已知通过训练期间自动分化的PINN损失函数中部分导数项的重复计算在计算上是昂贵的，尤其是对于高阶导数而言。通过使用无网状径向基函数 - 限制差异（RBF-FD）计算出的高阶精确数值离散化来替换这些精确的空间衍生物来训练DT-PINN，并通过稀疏Matrix矢量乘法应用。 RBF-FD的使用允许在放置在不规则域几何上的点云样本上训练DT细菌。此外，尽管传统的PINN（香草杆）通常在GPU上以32位的浮点（FP32）进行存储和培训，但我们表明，对于DT-Pinns，使用GPU上的FP64导致训练时间比FP32 vanilla-pinns具有可比性的精度相当。我们通过一系列实验证明了DT细菌的效率和准确性。首先，我们探讨了网络深度对具有随机权重的神经网络的数值和自动分化的影响，并表明三阶准确度及以上的RBF-FD近似值更有效，同时非常准确。然后，我们将DT-Pinns与线性和非线性泊松方程式上的香草细菌进行比较，并表明DT-Pinns在消费者GPU上更快的训练时间更快地损失了相似的损失。最后，我们还证明，通过使用RBF-FD离散空间衍生物并使用自动分化为时间衍生物，可以通过离散空间衍生物来获得PINN解决方案（时空问题）的相似结果。我们的结果表明，FP64 DT-Pinns为FP32香草细菌提供了优越的成本准确性。
We present a new technique for the accelerated training of physics-informed neural networks (PINNs): discretely-trained PINNs (DT-PINNs). The repeated computation of partial derivative terms in the PINN loss functions via automatic differentiation during training is known to be computationally expensive, especially for higher-order derivatives. DT-PINNs are trained by replacing these exact spatial derivatives with high-order accurate numerical discretizations computed using meshless radial basis function-finite differences (RBF-FD) and applied via sparse-matrix vector multiplication. The use of RBF-FD allows for DT-PINNs to be trained even on point cloud samples placed on irregular domain geometries. Additionally, though traditional PINNs (vanilla-PINNs) are typically stored and trained in 32-bit floating-point (fp32) on the GPU, we show that for DT-PINNs, using fp64 on the GPU leads to significantly faster training times than fp32 vanilla-PINNs with comparable accuracy. We demonstrate the efficiency and accuracy of DT-PINNs via a series of experiments. First, we explore the effect of network depth on both numerical and automatic differentiation of a neural network with random weights and show that RBF-FD approximations of third-order accuracy and above are more efficient while being sufficiently accurate. We then compare the DT-PINNs to vanilla-PINNs on both linear and nonlinear Poisson equations and show that DT-PINNs achieve similar losses with 2-4x faster training times on a consumer GPU. Finally, we also demonstrate that similar results can be obtained for the PINN solution to the heat equation (a space-time problem) by discretizing the spatial derivatives using RBF-FD and using automatic differentiation for the temporal derivative. Our results show that fp64 DT-PINNs offer a superior cost-accuracy profile to fp32 vanilla-PINNs.
