Automating the Discovery of Partial Differential Equations in Dynamical Systems

Identifying partial differential equations (PDEs) from data is crucial for understanding the governing mechanisms of natural phenomena, yet it remains a challenging task. We present an extension to the ARGOS framework, ARGOS-RAL, which leverages sparse regression with the recurrent adaptive lasso to identify PDEs from limited prior knowledge automatically. Our method automates calculating partial derivatives, constructing a candidate library, and estimating a sparse model. We rigorously evaluate the performance of ARGOS-RAL in identifying canonical PDEs under various noise levels and sample sizes, demonstrating its robustness in handling noisy and non-uniformly distributed data. We also test the algorithm's performance on datasets consisting solely of random noise to simulate scenarios with severely compromised data quality. Our results show that ARGOS-RAL effectively and reliably identifies the underlying PDEs from data, outperforming the sequential threshold ridge regression method in most cases. We highlight the potential of combining statistical methods, machine learning, and dynamical systems theory to automatically discover governing equations from collected data, streamlining the scientific modeling process.

Automatically identifying ordinary differential equations from data

Discovering nonlinear differential equations that describe system dynamics from empirical data is a fundamental challenge in contemporary science. Here, we propose a methodology to identify dynamical laws by integrating denoising techniques to smooth the signal, sparse regression to identify the relevant parameters, and bootstrap confidence intervals to quantify the uncertainty of the estimates. We evaluate our method on well-known ordinary differential equations with an ensemble of random initial conditions, time series of increasing length, and varying signal-to-noise ratios. Our algorithm consistently identifies three-dimensional systems, given moderately-sized time series and high levels of signal quality relative to background noise. By accurately discovering dynamical systems automatically, our methodology has the potential to impact the understanding of complex systems, especially in fields where data are abundant, but developing mathematical models demands considerable effort.