10/04/2026 - 14:30 - ISBA C115 -
(University of Cambridge)
Will give a presentation on :
Abstract :
We consider the problem of performing statistical inference on the states of a nonlinear dynamical system, whose dynamics are governed by a nonlinear PDE, and corrupted by noise. These states further depend on an unknown initial condition which we model as a Gaussian random field, resulting in a priori random trajectories. Given discrete measurements of the system corrupted by generic (non-necessarily Gaussian) noise, one then wishes to update the prior trajectory to the best posterior update on the states of the dynamical system. This can be regarded as a problem of Bayesian inference in an infinite-dimensional regression context where the regression function is a solution to the PDE governing the dynamics and where the posterior distribution arises from a Gaussian process prior on the initial condition. The focus will be on so-called `dissipative’ systems for which, from a PDE perspective, inference on the system is expected to be poorly achieved. We will explain, however, why this is in fact good news for the statistician and that posterior trajectories concentrate at root(N)-rate around the ground truth trajectory of the system in a strong (supremum) sense and are asymptotically Gaussian with optimal covariance given by the Fisher information of the model. We will then explain how this allows one to construct uniform-in-time-and-space confidence bands for the true trajectory, and will briefly discuss computational hardness of running an MCMC in this setting.
