Abstract
Partial 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.
Highlights
Partial differential equations (PDEs) are ubiquitous in natural science and engineering problems
Tartakovsky et al.13 presented a physics-informed machine learning approach, PICKLE, for elliptic diffusion equations. This approach uses conditional Karhunen–Loeve expansion to minimize the partial differential equations (PDEs) residuals and approximate the observed parameters and states. Ahalpara14 developed a surrogate model for solving the Korteweg–de Vries (KdV) equation using a genetic algorithm
Based on the observation that existing surrogate models tend to obtain unstable prediction results in different subdomains, we propose a weighting mechanism to calibrate the weight of input coordinates in the loss function
Summary
Partial differential equations (PDEs) are ubiquitous in natural science and engineering problems. 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. To fulfill this role, Raissi et al. employed machine learning techniques (Gaussian process and Bayesian regression) to devise functional representations for linear/nonlinear operators in physical and mathematical problems. Due to the limited approximating capacity of machine learning techniques, these models may not guarantee the desired prediction result and tend to yield an inaccurate solution for complex nonlinear PDE systems
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