Fenrir: Physics-Enhanced Regression for Initial Value Problems
This work addresses a specific challenge in computational physics and machine learning for researchers dealing with parameter estimation in differential equations, though it appears incremental as it builds on existing probabilistic numerics.
The authors tackled the problem of parameter estimation in ordinary differential equations by converting initial value problems into Gauss-Markov processes, reducing it to hyperparameter estimation, which is easier. The method handles partial observations and avoids local optima better than some classical approaches, with experimental results showing it is on par or moderately better than competing methods.
We show how probabilistic numerics can be used to convert an initial value problem into a Gauss--Markov process parametrised by the dynamics of the initial value problem. Consequently, the often difficult problem of parameter estimation in ordinary differential equations is reduced to hyperparameter estimation in Gauss--Markov regression, which tends to be considerably easier. The method's relation and benefits in comparison to classical numerical integration and gradient matching approaches is elucidated. In particular, the method can, in contrast to gradient matching, handle partial observations, and has certain routes for escaping local optima not available to classical numerical integration. Experimental results demonstrate that the method is on par or moderately better than competing approaches.