Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach

TL;DR

This paper introduces a novel Runge-Kutta inspired dictionary-based sparse regression method for discovering nonlinear dynamical systems from noisy, sparse data, producing interpretable models without derivative approximations.

Contribution

It combines numerical integration with dictionary learning to identify governing equations directly from noisy data, extending to models with rational nonlinearities and parameter variations.

Findings

Successfully identified diverse differential equations from noisy data

Effectively handled sparsely sampled and corrupted measurements

Extended method to models with rational nonlinearities and external inputs

Abstract

Discovering dynamical models to describe underlying dynamical behavior is essential to draw decisive conclusions and engineering studies, e.g., optimizing a process. Experimental data availability notwithstanding has increased significantly, but interpretable and explainable models in science and engineering yet remain incomprehensible. In this work, we blend machine learning and dictionary-based learning with numerical analysis tools to discover governing differential equations from noisy and sparsely-sampled measurement data. We utilize the fact that given a dictionary containing huge candidate nonlinear functions, dynamical models can often be described by a few appropriately chosen candidates. As a result, we obtain interpretable and parsimonious models which are prone to generalize better beyond the sampling regime. Additionally, we integrate a numerical integration framework with dictionary learning that yields differential equations without requiring or approximating derivative information at any stage. Hence, it is utterly effective in corrupted and sparsely-sampled data. We discuss its extension to governing equations, containing rational nonlinearities that typically appear in biological networks. Moreover, we generalized the method to governing equations that are subject to parameter variations and externally controlled inputs. We demonstrate the efficiency of the method to discover a number of diverse differential equations using noisy measurements, including a model describing neural dynamics, chaotic Lorenz model, Michaelis-Menten Kinetics, and a parameterized Hopf normal form.