Gaussian Process Approximation for Differential Equations

Sammanfattning

This thesis develops a Bayesian framework based on Gaussian processes for inferring parameters of differential operators from data and reconstructing the corresponding solution together with its uncertainty. Such inverse problems are often ill-posed and sensitive to noise, which makes classical deterministic inverse approaches unstable and limits their ability to quantify uncertainty. The setup considers an unknown function u(x) and a forcing term f(x) related through a differential operator acting on u(x). The proposed approach avoids the explicit numerical solution of differential equations by constructing a Gaussian process approximation of the unknown functions. Gaussian process priors are used to encode smoothness assumptions over the input domain. This enables estimation of the operator parameters by minimizing the negative marginal likelihood of the Gaussian process model. Using the prior together with the data likelihood, a posterior distribution over the functions is obtained. The posterior distribution provides solutions to the differential equation through point estimates and associated measures of uncertainty. The methodology is applied to several differential equations, including the heat equation, an integro-differential equation and the Allen–Cahn reaction-diffusion equation. The parameters are inferred from data. The numerical experiments show that, for the linear test cases, the operator parameters can be recovered with high accuracy and the corresponding solutions can be reconstructed reliably, even when the data are scarce and noisy. The results improve as the amount of training data increases and deteriorate as the noise level increases. For the nonlinear Allen–Cahn reaction-diffusion equation, the method remains applicable, but the parameter estimates and reconstructions are less accurate and require substantially more training data.