LAUSR

In this article, we develop a hybrid physics-informed neural network (hybrid PINN) for partial differential equations (PDEs). We borrow the idea from the convolutional neural network (CNN) and finite volume methods. Unlike the physics-informed neural network (PINN) and its variations, the method proposed in this article uses an approximation of the differential operator to solve the PDEs instead of automatic differentiation (AD). The approximation is given by a local fitting method, which is the main contribution of this article. As a result, our method has been proved to have a convergent rate. This will also avoid the issue that the neural network gives a bad prediction, which sometimes happened in PINN. To the author's best knowledge, this is the first work that the machine learning PDE's solver has a convergent rate, such as in numerical methods. The numerical experiments verify the correctness and efficiency of our algorithm. We also show that our method can be applied in inverse problems and surface PDEs, although without proof.