Bayesian Numerical Methods for Nonlinear Partial Differential Equations
This work addresses the problem of efficient and probabilistic numerical solutions for nonlinear PDEs, which is incremental as it builds on earlier linear PDE methods.
The paper tackles the challenge of solving nonlinear partial differential equations (PDEs) as a Bayesian inference problem, extending prior linear methods to handle nonlinear cases with high computational costs. It demonstrates that meaningful probabilistic uncertainty quantification can be achieved while controlling the number of evaluations of PDE components, using a Matérn process prior identified through novel theoretical analysis.
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of the PDE is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.