Physics-Informed Neural Networks (PINNs) and Deep Operator Network (DeepONet) for beginners

Introduction

This mini-course is designed to provide graduate students and researchers with an introduction to state-of-the-art, data-driven algorithms for the numerical approximation of a wide array of control and optimization problems governed by both linear and nonlinear partial differential equations (PDEs). Coverage includes a comprehensive exploration of the theory and convergence analysis of Physics-Informed Neural Networks (PINNs) and Deep Operator Network (DeepONet). PINNs are applied to problems where initial and boundary conditions remain fixed, while DeepONet is used to approximate operators that map these conditions to optimal controls or designs. The numerical implementation of these algorithms using the deepXDE libray in Python is considered as well.

The course summarizes the lectures given by Francisco Periago at the XXI Jacques-Louis Lions Hispano-French School on Numerical Simulation in Physics and Engineering held at Universidad de Castilla-La Mancha (Spain) on July 7-11, 2025. FP acknowledges the invitation by SEMA (Spanish Society of Applied Mathematics) and SMAI (French Society for Applied and Industrial Mathematics). He is also imdebted to the organizers for their kindness and hospitality.

This repository contains all the material for the course, namely, notes, slides, videos, and Python scripts.

Acknowledgements: Work supported by grant PID2022-141957OA-C22 funded by MCIN/AEI/10.13039/501100011033, by "ERDF A way of making Europe", and the Autonomous Community of the Regi'on of Murcia, Spain, through the programme for the development of scientific and technical research by competitive groups (21996/PI/22), included in the Regional Program for the Promotion of Scientific and Technical Research of Fundaci'on S'eneca -- Agencia de Ciencia y Tecnolog'ia de la Regi'on de Murcia.

Remark: This material has been created by using Centro de Producción de Contenidos Digitales from the Universidad Politécnica de Cartagena, within the projects Erasmus Plus, INDIe 2018-1-ES01-KA201-050924, and INDIe4All 2020-1-ES01-KA201-083177, supported by the European Commission.

License: Atribución-CompartirIgual 4.0 Internacional de Creative Commons. CC BY-SA 4.0.

Contents

1. Preliminaries: Python installation for the course.
2. Physics-Informed Neural Networks. Part I: Theory.
3. Physics-Informed Neural Networks. Part II: Python implementation.
4. Deep Operator Network. Part I: Theory
5. Deep Operator Network. Part II: Phyton implementation

Dataset

In the following link you will find the data files needed in some examples. ```

Dataset

Citation

If you have followed this course, please cite it as:

@book {gar26,
	AUTHOR = {Garc\'{\i}a-Cervera, C. J. and Kessler, M. and Pedregal, P. and Periago, F.},
	TITLE = {Physics-Informed Neural Networks and Deep Operator Network for beginners},
	SERIES = {Lecture Notes of the XXI Jacques-Louis Lions Spanish-French School Ciudad Real, 2025},
	VOLUME = {},
	PUBLISHER = {SEMA-SIMA Springer Series},
	YEAR = {2026},
	PAGES = {},
	ISBN = {},
	MRCLASS = {},
	MRNUMBER = {},
}