AbstractPartial differential equations (PDEs) are ubiquitous in natural science and engineering problems. Traditional discrete methods for solving PDEs are usually time-consuming and labor-intensive due to the need for tedious mesh generation and numerical iterations. Recently, deep neural networks have shown new promise in cost-effective surrogate modeling because of their universal function approximation abilities. In this paper, we borrow the idea from physics-informed neural networks (PINNs) and propose an improved data-free surrogate model, DFS-Net. Specifically, we devise an attention-based neural structure containing a weighting mechanism to alleviate the problem of unstable or inaccurate predictions by PINNs. The proposed DFS-Net takes expanded spatial and temporal coordinates as the input and directly outputs the observables (quantities of interest). It approximates the PDE solution by minimizing the weighted residuals of the governing equations and data-fit terms, where no simulation or measured data are needed. The experimental results demonstrate that DFS-Net offers a good trade-off between accuracy and efficiency. It outperforms the widely used surrogate models in terms of prediction performance on different numerical benchmarks, including the Helmholtz, Klein–Gordon, and Navier–Stokes equations.
Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks
