Model discovery for dynamical systems with complex-valued product units

A new data-driven method can recover the governing equations of dynamical systems directly from observed data without requiring a predefined library of candidate functions, according to research posted to arXiv on 26 May 2026 [1][2]. The approach relies on complex-valued product-unit networks, where each unit represents a complex monomial and the network output is a sparse linear combination of those monomials [2]. Unlike established library-based techniques such as SINDy, the method learns relevant monomials — including those with fractional or negative exponents — straight from the data [2]. Machine learning, the broader field underpinning the work, encompasses statistical algorithms that learn from data and generalize to unseen examples without explicit programming [4]. Researchers tested the technique on four chaotic benchmark systems: Lorenz63, Lorenz84, the Four-Wing attractor, and a fractional variant of Lorenz63 [2]. The exact governing equations were recovered 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 [2]. The team also applied the model to real-world human-gait accelerometer signals [2]. The model produced stable trajectories with bounded prediction errors over a test horizon three times longer than the training interval [2]. The prediction errors corresponded to an RMSE of approximately 12–14% of the signal amplitude range [2]. The results indicate potential for high-dimensional systems where analytic equations are unavailable [2]. Discovering governing equations from observed trajectories offers deeper structural insight than merely predicting future states [2]. The work was shared through arXivLabs, a framework that lets collaborators develop and share new features on the arXiv platform [1].