# Solving differential equations using physics informed deep learning: a hand-on tutorial with benchmark tests

arXiv: https://arxiv.org/abs/2302.12260 · HTML: https://mlsanity.com/paper/2302.12260/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 7 references that are in the map.
- First posted on arXiv 23 Feb 2023.

## Builds on (most cited first, 2 of 2)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)

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

- [Unraveling particle dark matter with Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.17597/) (2025)
- [A comprehensive analysis of PINNs: Variants, Applications, and Challenges](https://mlsanity.com/paper/2505.22761/) (2025)
- [Solving higher-order Lane-Emden-Fowler type equations using physics-informed neural networks: benchmark tests comparing soft and hard constraints](https://mlsanity.com/paper/2307.07302/) (2023)
- [Solving Nonlinear Energy Supply and Demand System Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2412.17001/) (2024)

## Abstract (arXiv)

We revisit the original approach of using deep learning and neural networks to solve differential equations by incorporating the knowledge of the equation. This is done by adding a dedicated term to the loss function during the optimization procedure in the training process. The so-called physics-informed neural networks (PINNs) are tested on a variety of academic ordinary differential equations in order to highlight the benefits and drawbacks of this approach with respect to standard integration methods. We focus on the possibility to use the least possible amount of data into the training process. The principles of PINNs for solving differential equations by enforcing physical laws via penalizing terms are reviewed. A tutorial on a simple equation model illustrates how to put into practice the method for ordinary differential equations. Benchmark tests show that a very small amount of training data is sufficient to predict the solution when the non linearity of the problem is weak. However, this is not the case in strongly non linear problems where a priori knowledge of training data over some partial or the whole time integration interval is necessary.