# Impilict Runge-Kutta based sparse identification of governing equations in biologically motivated systems

arXiv: https://arxiv.org/abs/2502.20319 · HTML: https://mlsanity.com/paper/2502.20319/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 95 references that are in the map.
- First posted on arXiv 27 Feb 2025.

## Builds on (most cited first, 27 of 27)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [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)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [Where are we now? A large benchmark study of recent symbolic regression methods](https://mlsanity.com/paper/1804.09331/) (2018)
- [Where Do We Stand with Implicit Neural Representations? A Technical and Performance Survey](https://mlsanity.com/paper/2411.03688/) (2024)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [Sparse Identification of Nonlinear Dynamical Systems via Reweighted ℓ1-regularized Least Squares](https://mlsanity.com/paper/2005.13232/) (2020)
- [DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimization](https://mlsanity.com/paper/2108.00069/) (2021)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [On sparse regression, Lp-regularization, and automated model discovery](https://mlsanity.com/paper/2310.06872/) (2023)
- [Deep-learning based discovery of partial differential equations in integral form from sparse and noisy data](https://mlsanity.com/paper/2011.11981/) (2020)
- [The Weak Form Is Stronger Than You Think](https://mlsanity.com/paper/2409.06751/) (2024)
- [Sparse Identification of Slow Timescale Dynamics](https://mlsanity.com/paper/2006.00940/) (2020)
- [Sparse dynamical system identification with simultaneous structural parameters and initial condition estimation](https://mlsanity.com/paper/2204.10472/) (2022)
- [Patient-specific, mechanistic models of tumor growth incorporating artificial intelligence and big data](https://mlsanity.com/paper/2308.14925/) (2023)
- [Scalable Sparse Regression for Model Discovery: The Fast Lane to Insight](https://mlsanity.com/paper/2405.09579/) (2024)
- [Spreading Processes with Mutations over Multi-layer Networks](https://mlsanity.com/paper/2210.05051/) (2022)
- [COVRECON: Combining Genome-scale Metabolic Network Reconstruction and Data-driven Inverse Modeling to Reveal Changes in Metabolic Interaction Networks](https://mlsanity.com/paper/2303.12526/) (2023)

## Abstract (arXiv)

Identifying governing equations in physical and biological systems from datasets remains a long-standing challenge across various scientific disciplines, providing mechanistic insights into complex system evolution. Common methods like sparse identification of nonlinear dynamics (SINDy) often rely on precise derivative estimations, making them vulnerable to data scarcity and noise. This study presents a novel data-driven framework by integrating high order implicit Runge-Kutta methods (IRKs) with the sparse identification, termed IRK-SINDy. The framework exhibits remarkable robustness to data scarcity and noise by leveraging the lower stepsize constraint of IRKs. Two methods for incorporating IRKs into sparse regression are introduced: one employs iterative schemes for numerically solving nonlinear algebraic system of equations, while the other utilizes deep neural networks to predict stage values of IRKs. The performance of IRK-SINDy is demonstrated through numerical experiments on benchmark problems with varied dynamical behaviors, including linear and nonlinear oscillators, the Lorenz system, and biologically relevant models like predator-prey dynamics, logistic growth, and the FitzHugh-Nagumo model. Results indicate that IRK-SINDy outperforms conventional SINDy and the RK4-SINDy framework, particularly under conditions of extreme data scarcity and noise, yielding interpretable and generalizable models.