# Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling

arXiv: https://arxiv.org/abs/2109.05364 · HTML: https://mlsanity.com/paper/2109.05364/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 58 references that are in the map.
- First posted on arXiv 11 Sep 2021.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Port-Hamiltonian Neural Networks for Learning Explicit Time-Dependent Dynamical Systems](https://mlsanity.com/paper/2107.08024/) (2021)
- [Structure-preserving neural networks](https://mlsanity.com/paper/2004.04653/) (2020)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data](https://mlsanity.com/paper/2012.02334/) (2020)
- [Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers](https://mlsanity.com/paper/2008.09768/) (2020)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Symplectic Neural Networks in Taylor Series Form for Hamiltonian Systems](https://mlsanity.com/paper/2005.04986/) (2020)
- [Time-Reversal Symmetric ODE Network](https://mlsanity.com/paper/2007.11362/) (2020)
- [Bayesian differential programming for robust systems identification under uncertainty](https://mlsanity.com/paper/2004.06843/) (2020)
- [Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy data](https://mlsanity.com/paper/2103.03385/) (2021)

## Built on by (most cited first, 27 of 27)

- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [Reversible and irreversible bracket-based dynamics for deep graph neural networks](https://mlsanity.com/paper/2305.15616/) (2023)
- [Interpretable Polynomial Neural Ordinary Differential Equations](https://mlsanity.com/paper/2208.05072/) (2022)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)
- [Pseudo-Hamiltonian Neural Networks with State-Dependent External Forces](https://mlsanity.com/paper/2206.02660/) (2022)
- [Think like a Scientist: Physics-guided LLM Agent for Equation Discovery](https://mlsanity.com/paper/2602.12259/) (2026)
- [Symmetry-Informed Governing Equation Discovery](https://mlsanity.com/paper/2405.16756/) (2024)
- [Learning Physics From Video: Unsupervised Physical Parameter Estimation for Continuous Dynamical Systems](https://mlsanity.com/paper/2410.01376/) (2024)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regression](https://mlsanity.com/paper/2312.14237/) (2023)
- [Discovering Symbolic Differential Equations with Symmetry Invariants](https://mlsanity.com/paper/2505.12083/) (2025)
- [Generalizing the SINDy approach with nested neural networks](https://mlsanity.com/paper/2404.15742/) (2024)
- [Equilibrium Conserving Neural Operators for Super-Resolution Learning](https://mlsanity.com/paper/2504.13422/) (2025)
- [Meta-learning Structure-Preserving Dynamics](https://mlsanity.com/paper/2508.11205/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Neural non-canonical Hamiltonian dynamics for long-time simulations](https://mlsanity.com/paper/2510.01788/) (2025)
- [Mining Causality from Continuous-time Dynamics Models: An Application to Tsunami Forecasting](https://mlsanity.com/paper/2210.04958/) (2022)
- [Pseudo-Hamiltonian system identification](https://mlsanity.com/paper/2305.06920/) (2023)
- [Interpretable Neural PDE Solvers using Symbolic Frameworks](https://mlsanity.com/paper/2310.20463/) (2023)
- [Function Spaces Without Kernels: Learning Compact Hilbert Space Representations](https://mlsanity.com/paper/2509.20605/) (2025)
- [Learning Physically Consistent Lagrangian Control Models Without Acceleration Measurements](https://mlsanity.com/paper/2512.03035/) (2025)
- [Parameter-varying neural ordinary differential equations with partition-of-unity networks](https://mlsanity.com/paper/2210.00368/) (2022)
- [Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamics](https://mlsanity.com/paper/2603.05201/) (2026)
- [From Data to Laws: Neural Discovery of Conservation Laws Without False Positives](https://mlsanity.com/paper/2603.20474/) (2026)
- [AC-SINDy: Compositional Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2604.18889/) (2026)
- [Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics](https://mlsanity.com/paper/2607.28939/) (2026)
- [Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysis](https://mlsanity.com/paper/2609.04329/) (2026)

## Abstract (arXiv)

Discovery of dynamical systems from data forms the foundation for data-driven modeling and recently, structure-preserving geometric perspectives have been shown to provide improved forecasting, stability, and physical realizability guarantees. We present here a unification of the Sparse Identification of Nonlinear Dynamics (SINDy) formalism with neural ordinary differential equations. The resulting framework allows learning of both "black-box" dynamics and learning of structure preserving bracket formalisms for both reversible and irreversible dynamics. We present a suite of benchmarks demonstrating effectiveness and structure preservation, including for chaotic systems.