# Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization

arXiv: https://arxiv.org/abs/2206.00176 · HTML: https://mlsanity.com/paper/2206.00176/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 72 references that are in the map.
- First posted on arXiv 1 Jun 2022.

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

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [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)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017)
- [A unified sparse optimization framework to learn parsimonious physics-informed models from data](https://mlsanity.com/paper/1906.10612/) (2019)
- [Sparse Regression at Scale: Branch-and-Bound rooted in First-Order Optimization](https://mlsanity.com/paper/2004.06152/) (2020)
- [Sparse High-Dimensional Regression: Exact Scalable Algorithms and Phase Transitions](https://mlsanity.com/paper/1709.10029/) (2017)
- [A Unified Framework for Sparse Relaxed Regularized Regression: SR3](https://mlsanity.com/paper/1807.05411/) (2018)
- [Sparse Regression: Scalable algorithms and empirical performance](https://mlsanity.com/paper/1902.06547/) (2019)
- [Numerical differentiation of noisy data: A unifying multi-objective optimization framework](https://mlsanity.com/paper/2009.01911/) (2020)
- [A toolkit for data-driven discovery of governing equations in high-noise regimes](https://mlsanity.com/paper/2111.04870/) (2021)
- [Cardinality Minimization, Constraints, and Regularization: A Survey](https://mlsanity.com/paper/2106.09606/) (2021)
- [Sparse Methods for Automatic Relevance Determination](https://mlsanity.com/paper/2005.08741/) (2020)
- [Stability selection enables robust learning of partial differential equations from limited noisy data](https://mlsanity.com/paper/1907.07810/) (2019)
- [The Backbone Method for Ultra-High Dimensional Sparse Machine Learning](https://mlsanity.com/paper/2006.06592/) (2020)
- [Scalable Holistic Linear Regression](https://mlsanity.com/paper/1902.03272/) (2019)
- [Robust subset selection](https://mlsanity.com/paper/2005.08217/) (2020)

## Built on by (most cited first, 29 of 29)

- [Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamics](https://mlsanity.com/paper/2302.13271/) (2023)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Evolving Scientific Discovery by Unifying Data and Background Knowledge with AI Hilbert](https://mlsanity.com/paper/2308.09474/) (2023)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [OKRidge: Scalable Optimal k-Sparse Ridge Regression](https://mlsanity.com/paper/2304.06686/) (2023)
- [Learning Governing Equations of Unobserved States in Dynamical Systems](https://mlsanity.com/paper/2404.18572/) (2024)
- [Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discovery](https://mlsanity.com/paper/2308.10283/) (2023)
- [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024)
- [Mesh-free sparse identification of nonlinear dynamics](https://mlsanity.com/paper/2505.16058/) (2025)
- [Correlating Time Series with Interpretable Convolutional Kernels](https://mlsanity.com/paper/2409.01362/) (2024)
- [No Equations Needed: Learning System Dynamics Without Relying on Closed-Form ODEs](https://mlsanity.com/paper/2501.18563/) (2025)
- [Interpretable Time Series Autoregression for Periodicity Quantification](https://mlsanity.com/paper/2506.22895/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [A Physics-informed Machine Learning-based Control Method for Nonlinear Dynamic Systems with Highly Noisy Measurements](https://mlsanity.com/paper/2311.07613/) (2023)
- [Scalable First-order Method for Certifying Optimal k-Sparse GLMs](https://mlsanity.com/paper/2502.09502/) (2025)
- [Learning a Stochastic Differential Equation Model of Tropical Cyclone Intensification from Reanalysis and Observational Data](https://mlsanity.com/paper/2601.08116/) (2026)
- [Dynamics-aware identification of governing equations from sparse and noisy data](https://mlsanity.com/paper/2607.29036/) (2026)
- [Improved identification accuracy in equation learning via comprehensive boldsymbolR2-elimination and Bayesian model selection](https://mlsanity.com/paper/2311.13265/) (2023)
- [BackboneLearn: A Library for Scaling Mixed-Integer Optimization-Based Machine Learning](https://mlsanity.com/paper/2311.13695/) (2023)
- [Bridging the Gap Between Scientific Laws Derived by AI Systems and Canonical Knowledge via Abductive Inference with AI-Noether](https://mlsanity.com/paper/2509.23004/) (2025)
- [A joint optimization approach to identifying sparse dynamics using least squares kernel collocation](https://mlsanity.com/paper/2511.18555/) (2025)
- [When to Retrain after Drift: A Data-Only Test of Post-Drift Data Size Sufficiency](https://mlsanity.com/paper/2603.09024/) (2026)
- [Uncertainty-Aware Sparse Identification of Dynamical Systems via Bayesian Model Averaging](https://mlsanity.com/paper/2604.10854/) (2026)
- [From Sequential Nodes to GPU Batches: Parallel Branch and Bound for Optimal k-Sparse GLMs](https://mlsanity.com/paper/2605.22188/) (2026)
- [Modeling Dynamic Mixtures of Time-Delay Systems from Streaming Time Series](https://mlsanity.com/paper/2605.26191/) (2026)
- [AdaKoop: Efficient Modeling of Nonlinear Dynamics from Nonstationary Data Streams with Koopman Operator Regression](https://mlsanity.com/paper/2606.04930/) (2026)
- [On the post-hoc Evaluation of PDE Discovery: A Multifaceted Challenge of Scientific Advancement](https://mlsanity.com/paper/2607.23753/) (2026)

## Abstract (arXiv)

Discovering governing equations of complex dynamical systems directly from data is a central problem in scientific machine learning. In recent years, the sparse identification of nonlinear dynamics (SINDy) framework, powered by heuristic sparse regression methods, has become a dominant tool for learning parsimonious models. We propose an exact formulation of the SINDy problem using mixed-integer optimization (MIO) to solve the sparsity constrained regression problem to provable optimality in seconds. On a large number of canonical ordinary and partial differential equations, we illustrate the dramatic improvement of our approach in accurate model discovery while being more sample efficient, robust to noise, and flexible in accommodating physical constraints.