Statistical Foundations for Scientific Machine Learning and PDEs

Seminar in Mathematics

Abstract: Many data collection processes in natural scientific settings are described by partial differential equations (PDEs) and stochastic differential equations (SDEs). In these settings, key statistical tasks such as the estimation of unknown high-dimensional parameters, prediction and uncertainty quantification have given rise to sophisticated frequentist and Bayesian statistical methodology, which in turn rely on high-dimensional computational algorithms such as Markov Chain Monte Carlo (MCMC). In this talk, we discuss recent mathematical results in this context. In particular, we discuss dimension-free statistical convergence results for non-linear "operator" regression with neural networks, and polynomial-time mixing guarantees for high-dimensional posterior sampling.