Upcoming Event: Oden Institute Seminar

Finite-difference least square method for solving Hamilton-Jacobi equations using neural networks

Carlos Esteve Yagüe, University of Alicante

Abstract

In recent years, advancements in deep learning and new optimisation algorithms have motivated the use of artificial neural networks to solve non-linear problems in high-dimensional setups. One of the crucial steps during the implementation of any deep learning method is the choice of the loss functional, which is used to train the neural network parameters, typically through a gradient-based method. In this talk, I will consider the approximation of the viscosity solution for Hamilton-Jacobi equations by means of an artificial neural network. I will discuss the choice of the loss functional, which should be such that any critical point approximates the viscosity solution. I will present some recent results concerning loss functionals involving a consistent and monotone numerical Hamiltonian of Lax-Friedrichs type. Using the numerical diffusion built in the numerical Hamiltonian, we are able to prove that any critical point solves the associated finite-difference problem and, therefore, approximates the viscosity solution. I will also present a method in which the numerical diffusion of the numerical scheme is decreased during the training, allowing for approximations with less numerical diffusion.

Biography

Carlos Esteve Yagüe earned his Mathematics degree from the University of Alicante. In 2014, he pursued a master program in mathematics at Université Sorbonne Paris Nord, supported by a scholarship from the Paris Mathematics Foundation. He completed his Ph.D. in 2019 at the same university under the guidance of Professor Philippe Souplet. Following his doctorate, Esteve Yagüe joined Enrique Zuazua's research team at the Autonomous University of Madrid and the University of Deusto in Bilbao. In January 2022, he began a postdoctoral position at the University of Cambridge, working with Carola-Bibiane Schönlieb's group. His research focuses at the interface between partial differential equations, machine learning and inverse problems in image analysis, as well as exploring optimal control problems and game theory. In 2023, Esteve Yagüe was awarded the Ramón y Cajal grant, from the Spanish government, facilitating his incorporation into the University of Alicante's Department of Mathematics.