Master Degree Project Presentation: Parameter inference of PDEs with Gaussian processes in a probabilistic framework

Abstract: Partial differential equations (PDEs) are widely used to model real-world phenomena. Such models often depend on parameters with physical or biological interpretations that are difficult, or even impossible, to measure directly. This thesis presents a Bayesian framework for learning parameters in PDEs from data.

The foundations of Bayesian inference and posterior sampling using Hamiltonian Monte Carlo is presented, including a convergence result of the sampling algorithm to the posterior distribution.

It is then explained how prior knowledge encoded by PDEs can be incorporated into probabilistic models through Gaussian processes and their mean square derivatives, enabling inference in parameters. PDE-informed Gaussian processes avoid explicit numerical solutions of the governing PDEs and, while they have primarily been validated on simulated data, this thesis applies the approach to a real-world dataset assumed to follow a Fisher-KPP diffusion equation.

These PDE-informed Gaussian process models are implemented in Stan, a probabilistic programming framework designed for efficient posterior sampling through NUTS, a variant of HMC sampling. The implemented code produces uncertainty quantification for not only PDE parameters, but also for the PDE solution and data noise. The implementation is made readily available for reuse and extension. The results demonstrate both the potential of the method and the practical limitations encountered in real-world applications.