# ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification

arXiv: https://arxiv.org/abs/2410.16528 · HTML: https://mlsanity.com/paper/2410.16528/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 71 references that are in the map.
- First posted on arXiv 21 Oct 2024.

## Builds on (most cited first, 14 of 14)

- [Training GANs with Optimism](https://mlsanity.com/paper/1711.00141/) (2017)
- [Adaptively Sparse Transformers](https://mlsanity.com/paper/1909.00015/) (2019)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Optimistic mirror descent in saddle-point problems: Going the extra (gradient) mile](https://mlsanity.com/paper/1807.02629/) (2018)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [The Limit Points of (Optimistic) Gradient Descent in Min-Max Optimization](https://mlsanity.com/paper/1807.03907/) (2018)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [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)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Interpreting and generalizing deep learning in physics-based problems with functional linear models](https://mlsanity.com/paper/2307.04569/) (2023)

## Built on by (most cited first, 11 of 11)

- [Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning](https://mlsanity.com/paper/2603.23854/) (2026)
- [SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networks](https://mlsanity.com/paper/2603.18548/) (2026)
- [Sparse identification of nonlinear dynamics with library optimization mechanism: Recursive long-term prediction perspective](https://mlsanity.com/paper/2507.18220/) (2025)
- [Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling](https://mlsanity.com/paper/2510.00233/) (2025)
- [Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systems](https://mlsanity.com/paper/2509.14219/) (2025)
- [MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamics](https://mlsanity.com/paper/2512.17273/) (2025)
- [FePySR: A Neural Feature Extraction Framework for Efficient and Scalable Symbolic Regression](https://mlsanity.com/paper/2605.12704/) (2026)
- [Optimizing Hard Thresholding for Sparse Model Discovery](https://mlsanity.com/paper/2504.20256/) (2025)
- [From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identification](https://mlsanity.com/paper/2512.14404/) (2025)
- [Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamics](https://mlsanity.com/paper/2603.05201/) (2026)
- [Strategic Governance of AI Models in Earth Science](https://mlsanity.com/paper/2610.10560/) (2026)

## Abstract (arXiv)

Identifying dynamical systems characterized by nonlinear parameters presents significant challenges in deriving mathematical models that enhance understanding of physics. Traditional methods, such as Sparse Identification of Nonlinear Dynamics (SINDy) and symbolic regression, can extract governing equations from observational data; however, they also come with distinct advantages and disadvantages. This paper introduces a novel method within the SINDy framework, termed ADAM-SINDy, which synthesizes the strengths of established approaches by employing the ADAM optimization algorithm. This facilitates the simultaneous optimization of nonlinear parameters and coefficients associated with nonlinear candidate functions, enabling precise parameter estimation without requiring prior knowledge of nonlinear characteristics such as trigonometric frequencies, exponential bandwidths, or polynomial exponents, thereby addressing a key limitation of SINDy. Through an integrated global optimization, ADAM-SINDy dynamically adjusts all unknown variables in response to data, resulting in an adaptive identification procedure that reduces the sensitivity to the library of candidate functions. The performance of the ADAM-SINDy methodology is demonstrated across a spectrum of dynamical systems, including benchmark coupled nonlinear ordinary differential equations such as oscillators, chaotic fluid flows, reaction kinetics, pharmacokinetics, as well as nonlinear partial differential equations (wildfire transport). The results demonstrate significant improvements in identifying parameterized dynamical systems and underscore the importance of concurrently optimizing all parameters, particularly those characterized by nonlinear parameters. These findings highlight the potential of ADAM-SINDy to extend the applicability of the SINDy framework in addressing more complex challenges in dynamical system identification.