# Finite Difference Neural Networks: Fast Prediction of Partial Differential Equations

arXiv: https://arxiv.org/abs/2006.01892 · HTML: https://mlsanity.com/paper/2006.01892/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2020.

## Built on by (most cited first, 4 of 4)

- [Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations](https://mlsanity.com/paper/2205.09332/) (2022)
- [Learning time-dependent PDE solver using Message Passing Graph Neural Networks](https://mlsanity.com/paper/2204.07651/) (2022)
- [Characteristics-Informed Neural Networks for Forward and Inverse Hyperbolic Problems](https://mlsanity.com/paper/2212.14012/) (2022)
- [Multi-level datasets training method in Physics-Informed Neural Networks](https://mlsanity.com/paper/2504.21328/) (2025)

## Abstract (arXiv)

Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.