Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed-form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the solution of PDEs based on a novel technique that combines the advantages of two recently emerging machine learning based approaches. First, physics-informed neural networks (PINNs) learn continuous solutions of PDEs and can be trained with little to no ground truth data. However, PINNs do not generalize well to unseen domains. Second, convolutional neural networks provide fast inference and generalize but either require large amounts of training data or a physics-constrained loss based on finite differences that can lead to inaccuracies and discretization artifacts. We leverage the advantages of both of these approaches by using Hermite spline kernels in order to continuously interpolate a grid-based state representation that can be handled by a CNN. This allows for training without any precomputed training data using a physics-informed loss function only and provides fast, continuous solutions that generalize to unseen domains. We demonstrate the potential of our method at the examples of the incompressible Navier-Stokes equation and the damped wave equation. Our models are able to learn several intriguing phenomena such as Karman vortex streets, the Magnus effect, Doppler effect, interference patterns and wave reflections. Our quantitative assessment and an interactive real-time demo show that we are narrowing the gap in accuracy of unsupervised ML based methods to industrial CFD solvers while being orders of magnitude faster.

本文介绍了一种新的技术方法，将机器学习的两种方法进行融合，通过物理知识驱动的神经网络和卷积神经网络相结合，解决了部分微分方程（PDE）的求解问题，实现了快速且连续的解决方案。通过在不需要预先计算训练数据的情况下，只使用物理信息的损失函数进行训练，同时展示了该方法在不可压缩Navier-Stokes方程和阻尼波动方程中的应用。

利用快速的、物理启发的 Hermite-Spline CNNs 接近没有数据的 PDEs 的 Spline-PINN 算法