16:00 - 17:00
Organised by Sam Blackburn.
Hosted by University of Exeter.
Title - Neural Networks for PDEs: From Paradigms to Numerical Analysis
Speaker - Professor Kris van der Zee (University of Nottingham)
In recent years there has been tremendous interest in the merging of machine-learning algorithms with traditional methods in computational science and mathematics. In this talk, I will focus on the use of neural networks for partial differential equations (PDEs). I will start by highlighting the three general paradigms: Learning the PDE solution, learning the PDE model, and learning the discretization method. All three aim to minimize a certain PDE-related cost functional over sets of neural-network functions.
Next, I will discuss the numerical analysis of PDE approximations generated by neural-network (quasi-) minimisers. I will present a recent abstract convergence theory, which is essentially an extension of Céa's fundamental quasi-optimality theory for Galerkin approximations in the Lax-Milgram setting.
Finally, I will consider Weak Adversarial Networks (WANs), a class of approximation methods that use two neural networks to approximate saddle-point formulations of PDEs. While this approach promises to be one of the most versatile for general PDE problems, several key numerical challenges remain open.
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