Finite Difference Nets: A Deep Recurrent Framework for Solving Evolution PDEs

2021/04/16 by Cheng Chang, Liu Liu, Chang, Cheng +3
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations

paper · pdf · doi:10.48550/arxiv.2104.09625

openalex publication_date 2021/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data sets. We provide a new perspective, that is, a different type of architecture through exploring the possible connections between traditional numerical methods (such as finite difference schemes) and deep neural networks, particularly convolutional and fully-connected neural networks. Our proposed approach will show its effectiveness and efficiency in solving PDE models with an integral form, in particular, we test on one-way wave equations and system of conservation laws.

Citations

• Compressible Flow and Euler's Equations
• Deep Learning in Neural Networks: An Overview
• A guide to convolution arithmetic for deep learning
• Conditional Image Synthesis With Auxiliary Classifier GANs
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
• NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
• B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data
• Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks
• Discovery of Governing Equations with Recursive Deep Neural Networks

Related

• FiniteNet: A Fully Convolutional LSTM Network Architecture for Time-Dependent Partial Differential Equations
• DeepPropNet -- A Recursive Deep Propagator Neural Network for Learning Evolution PDE Operators
• PDE-Net: Learning PDEs from Data
• Neural-PDE: A RNN based neural network for solving time dependent PDEs
• Partial Differential Equations Meet Deep Neural Networks: A Survey