MAGIC119: Recipes for Scientific Machine Learning

This module constitutes a very practical approach to machine learning and AI, and it aims at complementing the MAGIC117: Mathematical Foundations of AI module. See the Syllabus for detail information on the module content. Note that module includes Python coding.

Final Assessement

The assessment will consist of two or three project options presented in the examination paper. Each project will focus primarily on a specific machine learning recipe (see the tentative course content below) covered during the course and will typically be accompanied by a relevant dataset. Students will choose one of the three projects, either because it aligns with their research interests or simply because it matches their preferred area of interest.

Students will complete their chosen project by producing a Jupyter Notebook, similar in format to the MAGCI099 assessment. The notebook should include the code, data analysis, and results, together with a series of short reflective, essay-style responses. These prompts will ask students to present and interpret their findings, explain the methodology they have adopted, and critically evaluate both the approach and their understanding of the underlying machine learning techniques.

No specific prerequisites are needed, aside of basic programming skills and undergraduate-level mathematics.

(This is a new module, below you find the tentative syllabus organised lecture by lecture; note that it might change during the delivery of the course.)

Lecture 1 – Recipe 1: Learning a Physical Law from Data

The course opens with a brief introduction to the emergence of Scientific Machine Learning as a new paradigm in computational science, before immediately addressing the question: Can a machine learn a physical law directly from experimental data? Using a simple regression problem, students are introduced to the basic concepts of supervised learning and single-layer neural networks, laying the foundations for the remainder of the course.

Lecture 2 – Recipe 2: Learning Complex Nonlinear Relationships

Many scientific phenomena cannot be described by simple linear models. Building on the previous lecture, students explore multi-layer neural networks and learn how hidden layers enable the approximation of increasingly complex nonlinear relationships, introducing the concepts of deep learning, feature extraction and network training.

Lecture 3 – Recipe 3: Recognising Physical States

Many scientific datasets require the automatic identification of different physical regimes, such as laminar versus turbulent flow or different phases of matter. This lecture introduces supervised classification using neural networks and discusses how classification differs from regression, together with the appropriate loss functions and performance metrics.

Lecture 4 – Recipe 4: Understanding Scientific Images

Scientific observations are frequently represented as images, ranging from satellite data and microscopy to fluid simulations. This lecture introduces Convolutional Neural Networks (CNNs) and demonstrates how convolutional filters exploit spatial locality to detect meaningful patterns in scientific image data.

Lecture 5 – Recipe 5: Learning on Graphs and Forecasting Dynamical Systems

Many scientific datasets are defined on irregular computational meshes or evolve as time-dependent dynamical systems. This lecture introduces Graph Neural Networks (GNNs) for graph-structured data and recurrent architectures, including Long Short-Term Memory (LSTM) and Echo State Networks, for forecasting nonlinear and chaotic dynamics.

Lecture 6 – Recipe 6: Recovering Unknown Physical Parameters

Many scientific problems involve inferring unknown model parameters from indirect or incomplete observations rather than predicting future behaviour. Using examples from remote sensing and parameter estimation, this lecture introduces inverse problems and illustrates how machine learning can be combined with physical models to recover unknown quantities.

Lecture 7 – Recipe 7: Solving Partial Differential Equations without a Mesh

Can a neural network solve a differential equation without constructing a computational mesh? Using the heat equation as a prototype example, this lecture introduces Physics-Informed Neural Networks (PINNs), showing how differential equations, boundary conditions and observational data can be incorporated directly into the training process.

Lecture 8 – Recipe 8: Solving PDEs through Energy Minimisation

Many physical systems are naturally described by variational principles rather than differential equations alone. This lecture introduces the Deep Ritz Method, demonstrating how neural networks can approximate solutions of boundary value problems by minimising the associated energy functional.

Lecture 9 – Recipe 9: Learning Entire PDE Solvers

Instead of approximating a single solution of a differential equation, modern Scientific Machine Learning seeks to learn the mapping between entire families of inputs and outputs. This lecture introduces Neural Operators, with particular emphasis on Fourier Neural Operators (FNOs), illustrating how neural networks can learn complete solution operators for parameterised PDEs.

Lecture 10 – Recipe 10: Learning the Governing Equations of Dynamical Systems

The course concludes by considering the inverse problem of discovering the governing equations of a dynamical system directly from observations. Neural Ordinary Differential Equations (Neural ODEs) are introduced as a unifying framework connecting deep learning, continuous-time dynamical systems and optimal control, highlighting one of the most active research directions in Scientific Machine Learning.