A joint optimization approach to identifying sparse dynamics using least squares kernel collocation
Alexander W. Hsu, Ike W. Griss Salas, Jacob M. Stevens-Haas, J. Nathan Kutz, Aleksandr Aravkin, Bamdad Hosseini
TL;DR
The paper tackles learning autonomous ODEs from scarce, partial, and noisy data by jointly estimating the state trajectory and the governing dynamics within a reproducing kernel Hilbert space framework. It introduces JSINDy, an all-at-once collocation method that enforces self-consistency between the estimated state and a sparse, library-based dynamics, and leverages a representer theorem to reduce to a finite-dimensional optimization solved via alternating Levenberg–Marquardt steps and sparsifying iterations. The approach demonstrates robustness across scenarios including very low sampling rates, partial observability, higher-order ODEs, and model misspecification, achieving accurate state trajectories and sparse coefficient recovery across Lorenz, Lotka–Volterra, Van der Pol, and Duffing-type systems. It provides a principled relaxation of the ODE constraint that stabilizes optimization and improves robustness to noise, while offering flexible sparsification strategies and the potential for Bayesian model selection to further enhance variable recovery. Limitations include computational cost from dense kernel matrices, suggesting future work on state-space GP representations and scalable algorithms to extend applicability to longer time horizons.
Abstract
We develop an all-at-once modeling framework for learning systems of ordinary differential equations (ODE) from scarce, partial, and noisy observations of the states. The proposed methodology amounts to a combination of sparse recovery strategies for the ODE over a function library combined with techniques from reproducing kernel Hilbert space (RKHS) theory for estimating the state and discretizing the ODE. Our numerical experiments reveal that the proposed strategy leads to significant gains in terms of accuracy, sample efficiency, and robustness to noise, both in terms of learning the equation and estimating the unknown states. This work demonstrates capabilities well beyond existing and widely used algorithms while extending the modeling flexibility of other recent developments in equation discovery.