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

arXiv: https://arxiv.org/abs/2511.18555 · HTML: https://mlsanity.com/paper/2511.18555/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 102 references that are in the map.
- First posted on arXiv 23 Nov 2025.

## Builds on (most cited first, 30 of 40)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [A unified sparse optimization framework to learn parsimonious physics-informed models from data](https://mlsanity.com/paper/1906.10612/) (2019)
- [Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data](https://mlsanity.com/paper/2009.08810/) (2020)
- [A Unified Framework for Sparse Relaxed Regularized Regression: SR3](https://mlsanity.com/paper/1807.05411/) (2018)
- [Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization](https://mlsanity.com/paper/2206.00176/) (2022)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants](https://mlsanity.com/paper/2211.10575/) (2022)
- [Learning dynamical systems from data: a simple cross-validation perspective](https://mlsanity.com/paper/2007.05074/) (2020)
- [Numerical differentiation of noisy data: A unifying multi-objective optimization framework](https://mlsanity.com/paper/2009.01911/) (2020)
- [Physics-informed Spline Learning for Nonlinear Dynamics Discovery](https://mlsanity.com/paper/2105.02368/) (2021)
- [On the smoothness of nonlinear system identification](https://mlsanity.com/paper/1905.00820/) (2019)
- [Bayesian differential programming for robust systems identification under uncertainty](https://mlsanity.com/paper/2004.06843/) (2020)
- [Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discovery](https://mlsanity.com/paper/2301.12649/) (2023)
- [Rapid Bayesian identification of sparse nonlinear dynamics from scarce and noisy data](https://mlsanity.com/paper/2402.15357/) (2024)
- [Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty](https://mlsanity.com/paper/2210.08095/) (2022)
- [Generalized Kalman Smoothing: Modeling and Algorithms](https://mlsanity.com/paper/1609.06369/) (2016)
- [Simultaneous Identification and Denoising of Dynamical Systems](https://mlsanity.com/paper/2203.13837/) (2022)
- [A penalty method for PDE-constrained optimization in inverse problems](https://mlsanity.com/paper/1504.02249/) (2015)
- [Toward Efficient Kernel-Based Solvers for Nonlinear PDEs](https://mlsanity.com/paper/2410.11165/) (2024)

## Abstract (arXiv)

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.