Publications (37)
DICE: Discrete inverse continuity equation for learning population dynamics
Tobias Blickhan, Jules Berman, Andrew Stuart +1
We introduce the Discrete Inverse Continuity Equation (DICE) method, a generative modeling approach that learns the evolution of a stochastic process from given sample populations…
Filtering Dynamical Systems Using Observations of Statistics
Eviatar Bach, Tim Colonius, Isabel Scherl +1
We consider the problem of filtering dynamical systems, possibly stochastic, using observations of statistics. Thus, the computational task is to estimate a time-evolving density $…
Multipole Graph Neural Operator for Parametric Partial Differential Equations
Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli +4
One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data…
Large Language Models: A Mathematical Formulation
Ricardo Baptista, Andrew Stuart, Son Tran
Large language models (LLMs) process and predict sequences containing text to answer questions, and address tasks including document summarization, providing recommendations, writi…
Kalman filtering and smoothing for linear wave equations with model error
Wonjung Lee, Damon McDougall, Andrew Stuart
Filtering is a widely used methodology for the incorporation of observed data into time-evolving systems. It provides an online approach to state estimation inverse problems when d…
Sequential Monte Carlo Methods for Bayesian Elliptic Inverse Problems
Alex Beskos, Ajay Jasra, Ege Muzaffer +1
In this article we consider a Bayesian inverse problem associated to elliptic partial differential equations (PDEs) in two and three dimensions. This class of inverse problems is i…