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

TL;DR

This paper introduces IRK-SINDy, a robust method combining implicit Runge-Kutta techniques with sparse identification to accurately discover governing equations in noisy, data-scarce biological and physical systems.

Contribution

It presents a novel IRK-SINDy framework that enhances sparse identification robustness using implicit Runge-Kutta methods and neural networks, outperforming existing methods under challenging data conditions.

Findings

IRK-SINDy outperforms traditional SINDy in noisy, data-scarce scenarios.

The framework accurately models complex biological systems like predator-prey and neural models.

Deep neural network implementation effectively predicts IRK stage values, improving identification accuracy.

Abstract

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.