Linearly Constrained Gaussian Processes with Boundary Conditions
This work addresses the challenge of incorporating complex prior constraints in Bayesian machine learning for applications in physics and engineering, though it appears incremental as it builds on existing parametrization techniques.
The authors tackled the problem of encoding prior knowledge from linear partial differential equations and boundary conditions into Gaussian process priors, resulting in a fully algorithmic construction that ensures only valid solutions are represented without approximation.
One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear partial differential equations together with their boundary conditions. We construct multi-output Gaussian process priors with realizations in the solution set of such systems, in particular only such solutions can be represented by Gaussian process regression. The construction is fully algorithmic via Gröbner bases and it does not employ any approximation. It builds these priors combining two parametrizations via a pullback: the first parametrizes the solutions for the system of differential equations and the second parametrizes all functions adhering to the boundary conditions.