In science, engineering and mathematics we often seek models of real-life dynamics to better our understanding and help predict behaviour. Deriving these models requires complex mathematics and sometimes results in model formulations which are too complicated to be accurate. Machine learning has recently been applied to this problem to derive such models in the form of partial differential equations. This technique can be used to discover the underlying dynamics within experimental data without the need for intricate analysis.
Partial differential equations (PDEs) are powerful tools for describing a variety of physical phenomena including electrodynamics, fluid dynamics and quantum mechanics. It is critical to develop techniques to model such systems in terms of PDEs so that underlying mechanisms of physical processes can be studied, and their behaviour predicted. Typically, models can be synthesised by first principles analysis of a system’s dynamics, however, most real-life systems are too complex or intricate to derive accurate governing equations. It is important to develop methods to address this issue since fields such as control engineering, predictive modelling and simulation require the correct equations to accurately formulate control algorithms or graphical solutions for these systems. Recently, neural networks and other machine learning techniques have been utilised to formulate PDE models for given data-sets. These methods have been enabled through the availability of inexpensive and fast computational simulations and by ubiquity of large data-sets.
Throughout this project, Python code was used to explore and optimise the sparse regression technique. Initial investigation of the approach showed that it could only discover a specific class of PDEs from data and relied on several assumptions to accurately discover these models. Through data preprocessing and alterations to the original code, this method was extended to work for higher time derivatives and thus discover a wider range of dynamics. Additionally, an alternate approach was also proposed that would theoretically enable the discover of any PDE (this is currently future work).
With advancements in machine learning and the field of data-driven discovery of partial differential equations, I believe we will soon be able to accurately learn the dynamics of experimental datasets more effectively. This is important in improving our understanding of dynamical systems and to aid in predictive modelling/ simulation. These technologies would support engineers in creating control algorithms for industrial processes, psychologists in computationally simulating neurons or environmental scientists in predicting bushfire spread.
The University of Adelaide