2025 AIChE Annual Meeting

(126e) Machine Learning Driven Discovery of Governing Equations of Physical Phenomena

Authors

M. M. Faruque Hasan - Presenter, National University of Singapore
The underlying physics of many industrial systems are usually unknown, partially known, or based on some assumptions. In chemical engineering domains like reaction kinetics and transport phenomena, understanding this physics directly from data can enable us to overcome the limitations of theoretical assumptions. Methods have been proposed to discover these underlying physics [e.g., 1, 2] mostly using sparse regression to determine the unknown parameters of a system of governing equations. However, most existing methods are intractable when performing predictions for higher-order ordinary differential equations (ODE) with non-constant coefficients. Additionally, numerical estimation of derivatives from noisy data often leads to impractical results.

To address these issues, we propose a two-stage parametric estimation methodology to estimate the general order governing equation coefficients. In the first stage, we estimate a functional approximation of the data using an analytical function representing an approximate solution of the governing equation. This can be solved using data-driven optimization technique, such as genetic algorithm [3]. The analytical form of the approximate solution alleviates the stability issues of interval-based functional approximation for noisy data. In the second stage, we compute a gradient matrix based on the analytical form of the derivatives at different data points. Using this gradient matrix, solving a simple linear model to determine the null space of the gradient matrix provides sparse and efficient parametric estimation of the governing equation. We demonstrate the efficacy of the framework through a set of examples. For instance, we efficiently estimate sparse coefficients for the dynamic equations of a general spring-mass system, which can be transferred to the modeling of any harmonic oscillator system. The inherent property of linear models provides sparse solutions to a system of equations that can be used to promote sparsity in complex models.

References:

[1] Brunton, S. L., Proctor, J. L., Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. PNAS, 113(15), 3932-3937.

[2] Lejarza, F., Baldea, M. (2022). Data-driven discovery of the governing equations of dynamical systems via moving horizon optimization. Scientific Reports, 12(1), 11836.

[3] Lambora, Annu, Kunal Gupta, and Kriti Chopra. "Genetic algorithm-A literature review." 2019 international conference on machine learning, big data, cloud and parallel computing (COMITCon). IEEE, 2019.