# DeepXDE: A deep learning library for solving differential equations

## TL;DR

DeepXDE is a Python library that facilitates solving various types of differential equations using physics-informed neural networks, offering an accessible, flexible, and efficient tool for research and education in scientific machine learning.

## Contribution

The paper introduces DeepXDE, a comprehensive Python library for PINNs that supports complex geometries, inverse problems, and various PDE types, enhancing usability and educational value.

## Key findings

- DeepXDE effectively solves forward and inverse PDE problems.
- PINNs with RAR improve training efficiency.
- DeepXDE demonstrates versatility across multiple PDE examples.

## Abstract

Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs), which embed a PDE into the loss of the neural network using automatic differentiation. The PINN algorithm is simple, and it can be applied to different types of PDEs, including integro-differential equations, fractional PDEs, and stochastic PDEs. Moreover, from the implementation point of view, PINNs solve inverse problems as easily as forward problems. We propose a new residual-based adaptive refinement (RAR) method to improve the training efficiency of PINNs. For pedagogical reasons, we compare the PINN algorithm to a standard finite element method. We also present a Python library for PINNs, DeepXDE, which is designed to serve both as an education tool to be used in the classroom as well as a research tool for solving problems in computational science and engineering. Specifically, DeepXDE can solve forward problems given initial and boundary conditions, as well as inverse problems given some extra measurements. DeepXDE supports complex-geometry domains based on the technique of constructive solid geometry, and enables the user code to be compact, resembling closely the mathematical formulation. We introduce the usage of DeepXDE and its customizability, and we also demonstrate the capability of PINNs and the user-friendliness of DeepXDE for five different examples. More broadly, DeepXDE contributes to the more rapid development of the emerging Scientific Machine Learning field.