A joint optimization approach to identifying sparse dynamics using least squares kernel collocation

The identification of ordinary differential equations (ODEs) and dynamical systems is a fundamental problem in control [32, 59, 60], data assimilation [42, 84], and more recently in scientific machine learning (ML) [11, 72, 74]. While algorithms such as Sparse Identification of Nonlinear Dynamics (SINDy) and its variants [46] are widely used by practitioners, they often fail in scenarios where observations of the state of the system are scarce, indirect, and noisy. In such scenarios modifications to SINDy-type methods are required to enforce additional constraints on the recovered equations to make them consistent with the observational data. Put simply, traditional SINDy-type methods work in two steps: (1) the data is used to filter the state of the system and estimate the derivatives, and (2) the filtered state is used to learn the underlying dynamics. In the regime of scarce, noisy and incomplete data, step 1 is inaccurate, which can propagate to poor results in the subsequent step 2. In this paper, we propose an all-at-once approach to filtering and equation learning based on collocation in a reproducing kernel Hilbert space (RKHS) which we term Joint SINDy (JSINDy), and shows that the issues above can be mitigated by performing both steps together. This joins a broader class of dynamics-informed methods that integrate the governing equations directly into the learning objective, either as hard constraints or as least-squares relaxations, which couples the problems of state estimation and model discovery. Representative examples include physics-informed and sparse-regression frameworks based on neural networks, splines, kernels, finite differences, and adjoint methods [21, 27, 39, 41, 72, 73, 88].