Deep Learning for PDEs
Course Description
Artificial intelligence (AI) is currently changing public life and science in an unprecedented way. One of the central methods used are artificial neural networks, often summarized under the term 'deep learning', that are modeled after the human brain. There is currently intense research from various sides to develop a mathematical foundation of AI, for example to analyze phenomena such as missing robustness, understand the behavior of training algorithms, or explain the decisions of an AI algorithm. Additionally, AI-based methods are increasingly being used in mathematical fields such as (partial) differential equations.
Differential equations are mathematical equations that describe the relationship between a function and its derivatives with respect to one or more independent variables. In other words, a differential equation is an equation that involves the derivatives of a function. Differential equations arise naturally in many areas of science and engineering, and they are used to model a wide range of physical phenomena, including motion, heat transfer, fluid flow, chemical reactions, and many more. They are also applied in many other fields, such as economics, finance, and biology. Based on their properties, differential equations can be classified into several categories, such as order, linearity, and type of coefficients.
Solving differential equations is an important topic in mathematics, and there are many methods for finding solutions to different types of differential equations. These methods include analytical techniques, such as separation of variables and the method of integrating factors, as well as numerical methods, such as Euler's method and the Runge-Kutta method. The runtime of the numerical methods differ from method to method and from equation to equation and is usually expressed in terms of the discretization parameters. The complexity of these numerical schemes increases with the difficulty of the equations they seek to solve, making them computationally expensive. Although methods like the finite element method are quite successful for many classes of partial differntial equations, they require fine-tuning to handle different types of conditions and often suffer from problems like the curse of dimensionality.
In this course, we will present selected topics on deep learning methods for partial differential equations based on selected papers. These methods range from purely data-driven models such as Fourier Neural Operator to purely model-based approaches such as Physics Informed Neural Networks. We will begin by reviewing some terms from the theory of partial differential equations and the theory of deep learning. We will then cover selected topics such as operator learning, physics informed neural networks, neural networks for parametric PDEs, the deep Ritz method, etc.
Prerequisites
The course is targeted at Master students from mathematics. Basic knowledge of PDE theory (as e.g. covered in the course 'Einführung in partielle Differentialgleichungen (WP23)', in particular, theory of classical and weak solutions, i.e. Sobolev spaces, Bochner Spaces (or Sobolev-Lebesgue spaces), evolution equations of first and second order, etc.) and deep learning is highly recommended.
Schedule and Venue
Lecture: Mo, 14.00-16.00 (Room B 047) and Thu, 16.00-18.00 (Room B 047)
Exercise Class: Wed, 12.00-14.00 (Room B 045)
Tutorial: TBA
Office Hours: Wed, 14.00-15.00 (Bacho, Room 514 in Akademiestr. 7) and Thu 12.30-13.30 (Fono, Room 514 in Akademiestr. 7); please announce via a short e-mail that you intend to come.
Exam: TBA
Creditable Modules
The lecture is designed as a 9 ECTS course consisting of lectures and exercise classes. The following options are available.
Master in Financial and Insurance Mathematics:
- (PStO 2021) WP12 Advanced Topics in Mathematics A
- (PStO 2019) WP13 Advanced Topics in Mathematics A
Master in Mathematics:
- (PStO 2021) WP35 Fortgeschrittene Themen aus der künstlichen Intelligenz und Data Science
Registration
Please register for our course (https://uni2work.ifi.lmu.de/course/S23/MI/DLPDE) on uni2work: access key - DLPDE2023.