Scientific Machine Learning Course
Course taught at Toulouse-INP (ENSEEIHT), spring 2026. The course introduces the mathematical and algorithmic foundations of scientific machine learning (SciML): approximation of functions with neural networks and the associated optimization problems, physics-informed neural networks (PINNs) for forward and inverse problems, data-driven equation discovery, operator learning (DeepONet, Fourier neural operators), and hybrid methods that couple learned components with classical numerical solvers.
Companion book chapter
A. Kopaničáková and E. Riccietti, Introduction to optimization methods for training SciML models, 2026. [PDF] [HAL]
Slides
The lectures were delivered using the board. The slides below provide complementary material but do not replace the lecture notes. Full lecture notes are available upon request.
- Module 1 — Function approximation with neural networks[slides]
- Module 2 — Physics-informed neural networks [slides]
- Module 3 — Operator learning [slides]
- Module 4 — Hybrid methods: coupling numerical solvers with learned components [slides]
Practical sessions (TPs)
Hands-on Jupyter notebooks (PyTorch). Some TPs were created by Dr. Valentin Mercier. These are student versions, i.e., the exercises are intended to be completed by the students.
The Python environment can be set up with setup_env.sh
- TP1 — Neural-network surrogates in PyTorch [notebook]
- TP2 — Physics-informed neural networks [notebook]
- TP3 — Inverse problems with PINNs [notebook]
- TP4 — Data-driven equation discovery (SINDy, Burgers' equation) [notebook]
- TP5 — Fourier neural operators [notebook]
- TP6 — Hybrid iterative solvers (HINTS: DeepONet + relaxation methods) [notebook]
How to cite
If you use the lecture notes, slides, or practical sessions, please cite the companion book chapter:
@unpublished{kopanicakova2026optimsciml,
title = {Introduction to optimization methods for training SciML models},
author = {Kopani{\v c}{\'a}kov{\'a}, Alena and Riccietti, Elisa},
year = {2026},
note = {Preprint hal-05459235},
url = {https://ens-lyon.hal.science/hal-05459235}
}
To refer to the course material itself:
@misc{kopanicakova_sciml_course,
title = {Scientific Machine Learning: lecture notes and practical sessions},
author = {Kopani{\v c}{\'a}kov{\'a}, Alena},
year = {2026},
howpublished = {\url{https://kopanicakova.github.io/teaching/sciml/}},
note = {Course taught at Toulouse-INP (ENSEEIHT), 2024--2026}
}