MOCK: an Algorithm for Learning Nonparametric Differential Equations via Multivariate Occupation Kernel Functions

Victor William Rielly, Kamel Lahouel, Ethan Lew, Nicholas Fisher, Vicky Geneva Haney, Michael Lee Wells, Bruno Michel Jedynak

Abstract: Learning a nonparametric system of ordinary differential equations from trajectories in a $d$-dimensional state space requires learning $d$ functions of $d$ variables. Explicit formulations often scale quadratically in $d$ unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach, the multivariate occupation kernel method (MOCK), using the implicit formulation provided by vector-valued reproducing kernel Hilbert spaces. The solution for the vector field relies on multivariate occupation kernel functions associated with the trajectories and scales linearly with the dimension of the state space. We validate through experiments on a variety of simulated and real datasets ranging from 2 to 1024 dimensions, and provide an example with a divergence-free vector field. MOCK outperforms all other comparators on 3 of the 9 datasets on full trajectory prediction and 4 out of the 9 datasets on next-point prediction.

Submission Length: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=zduPsND4Sy&noteId=Ni43hzewI5

Changes Since Last Submission:

See https://openreview.net/forum?id=fjVIp2Z9RS&noteId=oKGpw1Z5d7

We confirm that all authors have approved the manuscript for submission. We also confirm that the content of the manuscript has not been published or submitted for publication elsewhere.

Video: https://drive.google.com/file/d/173U-a-awTd6YCn4gTsZ4Av7Mouypcfjs/view?usp=sharing

Code: https://drive.google.com/file/d/1p36HH00dHGLuBeHEfhJyMlfxjv1At5La/view

Assigned Action Editor: ~Tongliang_Liu1

Submission Number: 3885