The course aims to study the mathematical aspects of techniques, aimed at solving partial differential equations, based on neural networks, data-driven learning, and nonlinear approximation. In particular, the course explores topics in approximation theory, model order reduction, and optimization. The methods under consideration will be analysed theoretically and implemented numerically.
Learning Objectives: To understand and know how to manipulate neural networks from a mathematical point of view; to know the classical results in the approximation theory of neural networks with particular focus on quantitative results; to be familiar with different architectures of neural operators for model reduction for partial differential equations; to be able to demonstrate quantitative approximation results in this field; to know and be able to implement optimization methods for highly non-convex problems.
Course Prerequisites
Basic knowledge of numerical analysis, mathematical analysis, partial differential equations. A basic knowledge of python or similar languages is advisable
Teaching Methods
Classroom lectures, tutorials in the computer lab, study of research papers, seminars. The topics presented may vary according to the students' preferences
Assessment Methods
Oral exam and report. Every student will be able to implement the numerical methods presented during the course, focusing on some extensions, applications, or studying in details some theoretical aspects, also using the most recent scientific literature suggested by the lecturers.
Texts
Notes prepared by the lecturer, available on the course web page. Scientific papers provided by the lecturer.
Contents
We will present, from the points of view of theory and implementation, several techniques, aimed at solving partial differential equations, based on neural networks, data-driven learning, and nonlinear approximation. In particular, the course explores topics in approximation theory, model order reduction, and optimization.
Example of topics that can be covered in the course:
Theory - introduction to neural networks - general approximation theorems (universal approximation. etc.) - ReLU networks, approximation results with ReLU networks - approximation of functions in high-dimensional domains - optimization methods, backpropagation - PINNs for inverse problems - approximation of parametric problems with ReLU networks - neural operators for model reduction applied to PDEs (e.g., deepONets, Fourier Neural Operators, Graph Neural Networks, UNets, etc.) - other nonlinear methods of solving EDPs (e.g., kernel methods, Gaussian process regression)
Applications - implementation and optimization of neural networks - implementation of neural operators
Course Language
Italian
More information
Office hours, in presence or online, can be arranged by appointment. The lecturer's notes are available on the course page.
