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

TL;DR

This paper reviews physics-informed neural networks (PINNs) for solving differential equations, demonstrating their effectiveness with minimal data on weakly nonlinear problems and highlighting limitations on strongly nonlinear cases.

Contribution

It provides a comprehensive tutorial and benchmark analysis of PINNs, emphasizing their advantages and challenges in solving differential equations with limited data.

Findings

PINNs work well with weak nonlinearity and minimal data

Strong nonlinearity requires additional training data

Benchmark tests compare PINNs to traditional methods

Abstract

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.