Study on a Fast Solver for Poisson’s Equation Based on Deep Learning Technique

TLDR

Fast and efficient computational electromagnetic simulation is a long‑standing challenge. In this article, we propose a data‑driven model to solve Poisson’s equation that leverages the learning capacity of deep learning techniques. A deep convolutional neural network is trained on data generated from finite‑difference solvers to predict electric potential for varied excitations and permittivity distributions in 2‑D and 3‑D models, with a carefully designed cost function that yields reliable simulation, significant speedup, and good accuracy. Numerical experiments demonstrate that the ConvNet achieves less than 3 % average relative error in both 2‑D and 3‑D simulations while significantly reducing computation time, confirming deep neural networks’ strong learning capacity for numerical simulations and enabling fast solvers for computational electromagnetic problems.

Abstract

Fast and efficient computational electromagnetic simulation is a long-standing challenge. In this article, we propose a data-driven model to solve Poisson's equation that leverages the learning capacity of deep learning techniques. A deep convolutional neural network (ConvNet) is trained to predict the electric potential with different excitations and permittivity distribution in 2-D and 3-D models. With a careful design of cost function and proper training data generated from finite-difference solvers, the proposed network enables a reliable simulation with significant speedup and fairly good accuracy. Numerical experiments show that the same ConvNet architecture is effective for both 2-D and 3-D models, and the average relative prediction error of the proposed ConvNet model is less than 3% in both 2-D and 3-D simulations with a significant reduction in computation time compared to the finite-difference solver. This article shows that deep neural networks have a good learning capacity for numerical simulations. This could help us to build some fast solvers for some computational electromagnetic problems.