DiffFind: Discovering Differential Equations from Time Series

Given one or more time sequences, how can we extract their governing equations? Single and co-evolving time sequences appear in numerous settings, including medicine (neuroscience - EEG signals, cardiology - EKG), epidemiology (covid/flu spreading over time), physics (astrophysics, material science), marketing (sales and competition modeling; market penetration), and numerous more. Linear differential equations will fail, since the underlying equations are often non-linear (SIR model for virus/product spread; Lotka-Volterra for product/species competition, Van der Pol for heartbeat modeling). We propose DiffFind and we use genetic algorithms to find suitable, parsimonious, differential equations. Thanks to our careful design decisions, DiffFind has the following properties - it is: (a) Effective, discovering the correct model when applied on real and synthetic nonlinear dynamical systems, (b) Explainable, gives succinct differential equations, and (c) Hands-off, requiring no manual hyperparameter specification. DiffFind outperforms traditional methods (like auto-regression), includes as special case and thus outperforms a recent baseline (‘SINDy’), and wins first or second place for all 5 real and synthetic datasets we tried, often achieving excellent, zero or near-zero RMSE of 0.005.