Data based statistical closure of partial differential equations

Speaker: Chris Vales (Dartmouth)

Date: 4/13/26

Abstract: A challenging problem in the simulation of multiscale, multiphysics systems is how to incorporate dynamical information for degrees of freedom that are too expensive to resolve fully. Closure or parametrization methods aim to build surrogate models that can be used to approximate the contribution of such unresolved degrees of freedom to the dynamics of the coarse grain processes we can resolve with tractable computational cost.

In this talk, I will focus on a data-based, statistical closure scheme for problems governed by partial differential equations. Employing the mathematical framework of quantum mechanics, the scheme uses quantum density operators to encode statistical information about the unresolved degrees of freedom, and quantum observables to model their contribution to the resolved dynamics. I will develop a data-based implementation of the scheme, which relies on a compressed representation of the original dynamics that is invariant under dynamical symmetries. Finally, I will conclude with numerical results for a closure problem of the shallow water equations, demonstrating the accurate prediction of the main features of the dynamics for out of sample initial conditions.