# Sparse identification of nonlinear dynamics with library optimization mechanism: Recursive long-term prediction perspective

arXiv: https://arxiv.org/abs/2507.18220 · HTML: https://mlsanity.com/paper/2507.18220/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 71 references that are in the map.
- First posted on arXiv 24 Jul 2025.

## Builds on (most cited first, 21 of 21)

- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [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)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [A unified sparse optimization framework to learn parsimonious physics-informed models from data](https://mlsanity.com/paper/1906.10612/) (2019)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [SINDy-RL: Interpretable and Efficient Model-Based Reinforcement Learning](https://mlsanity.com/paper/2403.09110/) (2024)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification](https://mlsanity.com/paper/2410.16528/) (2024)
- [Prediction with Approximated Gaussian Process Dynamical Models](https://mlsanity.com/paper/2006.14551/) (2020)
- [Learning Parametric Koopman Decompositions for Prediction and Control](https://mlsanity.com/paper/2310.01124/) (2023)
- [An Online Data-Driven Method to Locate Forced Oscillation Sources from Power Plants Based on Sparse Identification of Nonlinear Dynamics (SINDy)](https://mlsanity.com/paper/2207.05356/) (2022)
- [Learning-based Event-triggered MPC with Gaussian processes under terminal constraints](https://mlsanity.com/paper/2110.12214/) (2021)
- [Sparse Identification of Nonlinear Dynamics with Side Information (SINDy-SI)](https://mlsanity.com/paper/2310.04227/) (2023)
- [Polytopic Autoencoders with Smooth Clustering for Reduced-order Modelling of Flows](https://mlsanity.com/paper/2401.10620/) (2024)
- [Practical one-shot data-driven design of fractional-order PID controller: Fictitious reference signal approach](https://mlsanity.com/paper/2311.05869/) (2023)
- [Simple controller design to achieve iso-damping robustness: Non-iterative data-driven approach based on fractional-order reference model](https://mlsanity.com/paper/2409.20375/) (2024)
- [Data-driven discovery of mechanical models directly from MRI spectral data](https://mlsanity.com/paper/2411.06958/) (2024)
- [Sparse Identification of Nonlinear Dynamics Enhanced by Ensemble Learning, Multi-Step Prediction Evaluation, Elite Strategy, and Classification Techniques for Applications to Industrial Systems](https://mlsanity.com/paper/2503.05154/) (2025)
- [Fractional-order controller tuning via minimization of integral of time-weighted absolute error without multiple closed-loop tests](https://mlsanity.com/paper/2507.12987/) (2025)

## Built on by (most cited first, 2 of 2)

- [Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning](https://mlsanity.com/paper/2603.23854/) (2026)
- [AC-SINDy: Compositional Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2604.18889/) (2026)

## Abstract (arXiv)

The sparse identification of nonlinear dynamics (SINDy) approach can discover the governing equations of dynamical systems based on measurement data, where the dynamical model is identified as the sparse linear combination of the given basis functions. A major challenge in SINDy is the design of a library, which is a set of candidate basis functions, as the appropriate library is not trivial for many dynamical systems. To overcome this difficulty, this study proposes SINDy with library optimization mechanism (SINDy-LOM), which is a combination of the sparse regression technique and the novel learning strategy of the library. In the proposed approach, the basis functions are parametrized. The SINDy-LOM approach involves a two-layer optimization architecture: the inner-layer, in which the data-driven model is extracted as the sparse linear combination of the candidate basis functions, and the outer-layer, in which the basis functions are optimized from the viewpoint of the recursive long-term (RLT) prediction accuracy; thus, the library design is reformulated as the optimization of the parametrized basis functions. The dynamical model obtained by SINDy-LOM has good interpretability and usability, as this approach yields a parsimonious closed-form model. The library optimization mechanism significantly reduces user burden. The RLT perspective improves the reliability of the resulting model compared with the traditional SINDy approach that can only ensure the one-step-ahead prediction accuracy. The effectiveness of the proposed approach is verified through numerical experiments.