I study theoretical and computational aspects of fractional-order partial differential equations (FPDEs).
FPDEs govern the mascroscopic properties of systems undergoing
anomalous diffusion. They also model nonlocality, memory effects (nonlocality in time), and can also be used to continuously interpolate
between standard models/PDEs.
My research interests include:
Machine Learning of FPDEs from Data. In the past decades, a vast number of systems in hydrology, telecommunications, finance, viscoelastics, biology, and plasma physics have been discovered to exhibit heavy-tailed statistics and anomalous diffusion, suggesting they are governed by FPDEs. Machine Learning can allow us to discover new FPDE models from data, bypassing the analytical difficulties of Fractional Calculus.
Path Integral Monte Carlo Methods for Fractional Quantum Mechanics. We have developed a fractional PIMC method, allowing us to study for the first time realistic systems governed by the fractional Schrodinger equation.
Boundary Conditions for fractional Laplacians on Bounded Domains, especially the Stochastic (Levy) Processes that underlie such Boundary Value Problems. This is both a way to understand boundary conditions, and to solve fractional elliptic and parabolic BVPs using Monte Carlo methods based on Feynman-Kac formulas.
Approximation theory of neural networks.
I work in the
groups of my Ph.D. advisor,