Machine Learning for Scientific Computing and Numerical Analysis

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. Forward, inverse, and hybrid problems in scientific ML; supervised and unsupervised learning foundations.

2. Function Approximation :** Classical and Neural Approaches: Approximation theory for scientific computing; neural network approximation (universal approximation, depth–width trade-offs, spectral bias). Connections to classical basis.**

3. Optimization in Scientific Machine Learning: Gradient-based optimization (stochastic vs. deterministic) and generalization. Regularization, constrained optimization for PDEs, and implicit differentiation for inverse problems.

4. Physics-Informed Neural Networks (PINNs): Embedding PDE constraints and boundary conditions in loss functions. Failure modes (spectral bias) and mitigation strategies (adaptive weighting, domain decomposition).

5. Reduced Order Modelling: Projection-based methods (POD, reduced basis) and Galerkin approximation.

6 Operator Learning Frameworks: Learning mappings between function spaces (DeepONets, FNO, neural operators).

7. Randomised Linear Algebra for Scientific ML: Dimensionality reduction (Johnson–Lindenstrauss, sketching), randomized SVD, and random feature methods. Applications to NN training, kernel approximation. 8. Inverse Problems in Scientific ML: Ill-posedness and regularization; variational and Bayesian formulations. Learning inverse maps and hybrid physics–ML strategies for parameter identification and model discovery

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.

Project-based assessment, including a final presentation. The precise components and weighting still need confirmation. But last year we had 40% project work and 60% intermediate assessment (2 homeworks). For the final project no need to produce reports, only hand-in presentation slides and codes. For the intermediate homeworks, just hand in Jupyter notebooks.

Basics of numerical analysis and scientific computing, programming skills mainly Python.

• Videos will be provided
• Course notes will be provided

Watching lecture videos, flipped classroom discussions, practical implementation sessions, and project work.