Pyphysdisc: co-evolutionary symbolic regression with adaptive smoothing windows for autonomous physical law discovery from noisy data
International Journal of Dynamics and Control, vol.14, no.5, 2026 (ESCI, Scopus)
- Publication Type: Article / Article
- Volume: 14 Issue: 5
- Publication Date: 2026
- Doi Number: 10.1007/s40435-026-02122-0
- Journal Name: International Journal of Dynamics and Control
- Journal Indexes: Emerging Sources Citation Index (ESCI), Scopus
- Keywords: Co-evolution, Data-driven equation discovery, Genetic programming, Noise robustness, Numerical differentiation, Ordinary differential equations, Symbolic regression
- Hatay Mustafa Kemal University Affiliated: Yes
Abstract
Discovering differential equations from noisy data faces the derivative–noise dilemma: numerical differentiation amplifies noise, requiring a smoothing window (w) whose optimal value depends on unknown noise levels and dynamical timescales. Existing methods decouple smoothing from the equation search, making them fragile to misspecification. We introduce PyPhysDisc, a genetic-programming framework where each individual carries a candidate expression tree and a window gene encoding w. Consequently, signal processing and symbolic regression are jointly optimised under a single selection pressure. We provide quantitative evidence of this co-evolutionary coupling through Shannon entropy analysis: window-gene entropy drops by 82 % over 40 generations, with elite-population dominance reaching 1.0, confirming directed selection rather than drift. Benchmark experiments on chaotic (Lorenz), limit-cycle (Van der Pol), stiff (Duffing), and predator–prey (Lotka–Volterra) systems at up to 20 % multiplicative noise show that PyPhysDisc maintains high accuracy (R2≥0.89 for three systems) without manual tuning. On the Lorenz system, the co-evolutionary strategy achieves R2=0.961, within 2.2 percentage points of an oracle holding the true optimal window, while a naïve fixed-window baseline collapses to R2=0.279 and exhibits 22× higher variance. Furthermore, the framework discovers non-polynomial laws (-sinθ) without predefined basis libraries. By internalising the choice of w as a co-evolving genetic component, PyPhysDisc establishes a self-sufficient mechanism for robust physical law discovery.