Discovery of discretized differential equations from data: Benchmarking and application to a plasma system

This study presents and evaluates Phi Method, a novel data-driven algorithm designed to discover discretized differential equations governing dynamical systems from data. Phi Method employs a constrained regression on a library of candidate terms to develop reduced-order models (ROMs) capable of accurate predictions of systems' state. To validate the approach, we first benchmark Phi Method against canonical dynamical systems governed by ordinary differential equations, highlighting the strengths and limitations of our approach. The method is then applied to a 2D fluid flow problem to verify its performance in learning governing partial differential equations (PDEs). The fluid flow test case also underlines the method's ability to generalize from transient training data and examines the characteristics of the learned local operator in both basic and parametric Phi Method implementations. The approach is finally applied to a 1D azimuthal plasma discharge problem, where data are now generated from a kinetic particle-in-cell simulation that does not explicitly solve the governing fluid-like equations. This application aims to demonstrate Phi Method's ability to uncover underlying dynamics from kinetic data in terms of optimally discretized PDEs, as well as the parametric dependencies in the discharge behavior. Comparisons with another ROM technique—the optimized dynamic mode decomposition—for the plasma test case emphasize Phi Method's advantages, mainly rooting in its ability to capture local dynamics with interpretable coefficients in the learned operator. The results establish Phi Method as a versatile tool for developing data-driven ROMs across a wide range of scenarios.