Deep learning approaches for partial differential equations (PDEs) have received much attention in recent years due to their mesh-freeness and computational efficiency. However, most of the works so far have concentrated on time-dependent nonlinear differential equations. In this work, we analyze potential issues with the well-known Physic Informed Neural Network for differential equations with little constraints on the boundary (i.e., the constraints are only on a few points). This analysis motivates us to introduce a novel technique called FinNet, for solving differential equations by incorporating finite difference into deep learning. Even though we use a mesh during training, the prediction phase is mesh-free. We illustrate the effectiveness of our method through experiments on solving various equations, which shows that FinNet can solve PDEs with low error rates and may work even when PINNs cannot.
翻译:近年来,部分差异方程式(PDEs)的深层次学习方法因其网状自由度和计算效率而备受关注。然而,迄今为止,大部分工程都集中在时间依赖的非线性差异方程式上。在这项工作中,我们分析了众所周知的Physic-知情神经网络在差异方程式方面的潜在问题,这些差异方程式对边界几乎没有多少限制(即,制约因素只局限在几个点上 ) 。这一分析促使我们引入了一种叫FinNet的新技术,通过将有限差异纳入深层次学习来解决差异方程式。尽管我们在培训中使用网状,但预测阶段是无网状的。我们通过各种方程式的解决实验来说明我们的方法的有效性,这表明FinNet能够以低误差率解决PDEs,甚至在PINNs无法解决时,也可能奏效。