论文标题
Finnet：用有限差神经网络解决时间无关的微分方程
FinNet: Solving Time-Independent Differential Equations with Finite Difference Neural Network
论文作者
论文摘要
近年来，由于它们的网格柔性和计算效率，近年来，部分微分方程（PDE）的深度学习方法受到了很多关注。但是，到目前为止，大多数作品都集中在时间依赖性的非线性微分方程上。在这项工作中，我们使用众所周知的物理知情神经网络分析潜在问题的微分方程，而边界上的限制很小（即，约束仅在几个点上）。这种分析促使我们引入了一种名为Finnet的新技术，用于通过将有限的差异纳入深度学习来解决微分方程。即使我们在训练过程中使用网格，预测阶段也不是网状的。我们通过实验解决各种方程式来说明我们方法的有效性，这表明Finnet可以求解较低的错误率，即使Pinns无法解决。
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.
