FinNet: Solving Time-Independent Differential Equations with Finite Difference Neural Network

TL;DR

FinNet introduces a mesh-based finite difference approach integrated with deep learning to efficiently solve time-independent PDEs, overcoming limitations of traditional PINNs especially with minimal boundary constraints.

Contribution

This work presents FinNet, a novel method combining finite difference techniques with neural networks for solving PDEs, particularly effective for time-independent equations with limited boundary data.

Findings

FinNet achieves low error rates in solving PDEs.

It outperforms traditional PINNs in certain scenarios.

The method is mesh-free during prediction, enhancing efficiency.

Abstract

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.