Adam Hayes

Reliable AI for Solving Partial Differential Equations

Project Summary

The aim of this project is to develop new algorithms and theoretical foundations towards reliable and transparent AI based solutions for nonlinear differential equations describing complex phenomena in nature and engineering. The proposed approach will also incorporate well understood and widely studied techniques from classical methods which will result in hybrid AI methods. Together, these aspects will provide theoretical guarantees that assure reliability of computational simulations making them suitable for scientific, engineering and medical applications.

Unlike most state-of-the-art machine learning approaches for computational simulation, the methods developed in this project will not be fully black-box methods. By incorporating physical constraints and classical numerical techniques, these hybrid computational methods will not only be fast, but also provide a degree of transparency that is lacking in current approaches. Additionally, incorporating physical constraints will also allow the algorithm to converge to the correct solution faster, even when only few training data is available.

The algorithm development will be complemented by the mathematical analysis of the methods which will establish theoretical guarantees on the convergence of the algorithm and the conservation of desirable properties of the solution.

Research Interests

Machine Learning, Deep Learning, Bayesian Machine Learning, Numerical Methods for Solving Partial Differential Equations, AI for Solving Partial Differential Equations

Background

Graduated from University of Birmingham with a Class I MSci in Mathematics.

Undergraduate Masters dissertation focused on using Non-Linear Feed-Forward Neural Networks to solve Partial Differential Equations.

Supervisors

Dr Lisa Maria Kreusser

Dr Xi Chen

Adam Hayes