Model discovery for dynamical systems with complex-valued product units
For researchers in dynamical systems and model discovery, this method eliminates the need for predefined candidate functions, enabling discovery of equations with fractional or negative exponents directly from data.
The paper introduces a data-driven method using complex-valued product-unit networks to discover governing equations of dynamical systems without requiring a predefined library of candidate functions. It recovers exact equations in 70-90% of trials across chaotic benchmarks and achieves stable predictions on real-world gait data with RMSE of 12-14% of signal amplitude.
Discovering the governing equations of a dynamical system from observed trajectories provides deeper insight into its structure than mere prediction of future states. We present a data-driven approach to model discovery based on complex-valued product-unit networks, in which each unit represents a complex monomial and the network output is a sparse linear combination of such monomials. In contrast to established library-based methods such as SINDy, our approach does not require a predefined set of candidate functions: the relevant monomials, including those with fractional or negative exponents, are learned directly from data. Across four chaotic benchmark systems (Lorenz63, Lorenz84, the Four-Wing attractor, and a fractional variant of Lorenz63), we recover the exact governing equations in 90% of trials for the first three systems, and in 70-90% of trials for the fractional case, using at least 3000 training points. Applied to real-world human-gait accelerometer signals, the model produced stable trajectories with bounded prediction errors, corresponding to an RMSE of approximately 12-14% of the signal amplitude range over a test horizon three times longer than the training interval, demonstrating its potential for high-dimensional systems in which analytic equations are unavailable.