Physics Informed Neural Network using Finite Difference Method

2026-02-25

AI Summary

This work proposes a novel paradigm for physics-informed neural networks (PINNs) by systematically integrating the finite difference method (FDM) to replace automatic differentiation in constructing partial differential equation (PDE) loss terms. Traditional PINNs rely on automatic differentiation, which entails complex implementation and substantial computational overhead. In contrast, the proposed FDM-PINN framework significantly simplifies model implementation and enhances training efficiency. Demonstrated on canonical PDEs such as the Laplace and Burgers equations, FDM-PINN not only outperforms purely data-driven deep learning models lacking physical constraints but also achieves accuracy comparable to conventional automatic differentiation-based PINNs while substantially reducing computational cost. This approach establishes a new, efficient, and lightweight pathway for physics-driven modeling.

Technology Category

Machine Learning: Learning with ManifoldsComputer Vision: Diffusion Models for VisionNatural Language Processing: Learning & Optimization for NLP

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsUser Modeling, Personalization and Recommendation: Federated recommendation systems and personalizationSocial Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

In recent engineering applications using deep learning, physics-informed neural network (PINN) is a new development as it can exploit the underlying physics of engineering systems. The novelty of PINN lies in the use of partial differential equations (PDE) for the loss function. Most PINNs are implemented using automatic differentiation (AD) for training the PDE loss functions. A lesser well-known study is the use of finite difference method (FDM) as an alternative. Unlike an AD based PINN, an immediate benefit of using a FDM based PINN is low implementation cost. In this paper, we propose the use of finite difference method for estimating the PDE loss functions in PINN. Our work is inspired by computational analysis in electromagnetic systems that traditionally solve Laplace's equation using successive over-relaxation. In the case of Laplace's equation, our PINN approach can be seen as taking the Laplacian filter response of the neural network output as the loss function. Thus, the implementation of PINN can be very simple. In our experiments, we tested PINN on Laplace's equation and Burger's equation. We showed that using FDM, PINN consistently outperforms non-PINN based deep learning. When comparing to AD based PINNs, we showed that our method is faster to compute as well as on par in terms of error reduction.

Problem

• Physics-Informed Neural Network
• Finite Difference Method
• Partial Differential Equations
• Loss Function
• Automatic Differentiation

Innovation

• Physics-Informed Neural Network
• Finite Difference Method
• Partial Differential Equations
• Laplacian Filter
• Computational Efficiency