Model discovery for dynamical systems with complex-valued product units

2026-05-26

AI Summary

This study addresses the challenge of directly discovering governing equations of dynamical systems from observed trajectories, moving beyond mere state prediction to uncover intrinsic system structure. To this end, the authors propose a data-driven approach based on complex-valued product unit networks, whose outputs are sparse linear combinations of complex monomials. This method automatically learns nonlinear terms—including those with fractional or negative exponents—without requiring a pre-specified library of candidate functions, thereby overcoming a key limitation of traditional approaches such as SINDy. It represents the first end-to-end framework capable of identifying governing equations in the form of complex monomials. Evaluated on four chaotic systems, the method recovers the true equations with 70%–90% success rates, and when applied to real human gait signals, it generates stable trajectories with test errors as low as 12%–14% of the signal amplitude range.

Abstract

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.

Problem

• model discovery
• dynamical systems
• governing equations
• complex-valued product units
• data-driven

Innovation

• complex-valued product units
• model discovery
• dynamical systems
• data-driven
• sparse identification