# A Survey on Solving and Discovering Differential Equations Using Deep Neural Networks

arXiv: https://arxiv.org/abs/2304.13807 · HTML: https://mlsanity.com/paper/2304.13807/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 Apr 2023.

## Built on by (most cited first, 3 of 3)

- [From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyond](https://mlsanity.com/paper/2310.10121/) (2023)
- [An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning](https://mlsanity.com/paper/2408.13222/) (2024)
- [Grammar-based Ordinary Differential Equation Discovery](https://mlsanity.com/paper/2504.02630/) (2025)

## Abstract (arXiv)

Ordinary and partial differential equations (DE) are used extensively in scientific and mathematical domains to model physical systems. Current literature has focused primarily on deep neural network (DNN) based methods for solving a specific DE or a family of DEs. Research communities with a history of using DE models may view DNN-based differential equation solvers (DNN-DEs) as a faster and transferable alternative to current numerical methods. However, there is a lack of systematic surveys detailing the use of DNN-DE methods across physical application domains and a generalized taxonomy to guide future research. This paper surveys and classifies previous works and provides an educational tutorial for senior practitioners, professionals, and graduate students in engineering and computer science. First, we propose a taxonomy to navigate domains of DE systems studied under the umbrella of DNN-DE. Second, we examine the theory and performance of the Physics Informed Neural Network (PINN) to demonstrate how the influential DNN-DE architecture mathematically solves a system of equations. Third, to reinforce the key ideas of solving and discovery of DEs using DNN, we provide a tutorial using DeepXDE, a Python package for developing PINNs, to develop DNN-DEs for solving and discovering a classic DE, the linear transport equation.