Machine Learning for Scientific Computing and Numerical Analysis

Students should regsiter via EduXchange link

This MSc course introduces and develops advanced methods at the intersection of machine learning and scientific computing, with a special emphasis on solving and analyzing forward and inverse problems governed by partial differential equations (PDEs). Students will learn how to combine classical numerical methods with modern neural-network architectures to approximate functions, operators, and solution maps, while critically assessing stability, generalization, and interpretability.

Syllabus

1. Introduction to Machine Learning for Scientific Computing Motivation and scope: data-driven vs. physics-based modelling. Problem settings in scientific ML: forward problems, inverse problems, and hybrid approaches. Supervised and unsupervised learning fundamentals.

2. Function Approximation: Classical and Neural Approaches Approximation theory foundations, Neural network approximation theory: universal approximation theorems, depth–width trade-offs, spectral bias.

3. Optimization in Scientific Machine Learning Gradient-based methods: stochastic vs. deterministic optimization. Regularization strategies: L², early stopping, dropout, weight decay. Constrained optimization for PDE constraints and implicit differentiation for inverse problems.

4. Physics-Informed Neural Networks (PINNs) Encoding PDEs and boundary/initial conditions in loss functions. Analysis of failure modes including spectral bias and ill-conditioning. Mitigation strategies: adaptive loss weighting, domain decomposition.

5. Operator Learning Frameworks Learning mappings between infinite-dimensional function spaces. Architectures such as DeepONets and Fourier Neural Operators (FNO). Comparison with reduced-order modelling and classical numerical solvers.

6. Inverse Problems in Scientific ML Ill-posedness and the need for regularization. Variational and Bayesian formulations for inverse problems. Learning inverse maps using autoencoders, normalizing flows, and hybrid physics–ML models. Case studies in parameter identification and model discovery.

7. Advanced and Emerging Topics Explainability and interpretability in physics-based ML. Perspectives on open research challenges and future directions in scientific ML.

By the end of this course the students will be able to: (i) understand the role of ML in numerical analysis and PDE based modelling (ii) analyse and implement function approximation schemes using both classical and neural-network methods; (iii) choose appropriate strategies to implement and assess accuracy, generalisation and stability.

• Course notes will be provided
• Videos will be provided