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 (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R^{2}\ge 0.89$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>≥</mo> <mn>0.89</mn> </mrow> </math></EquationSource> </InlineEquation> for three systems) without manual tuning. On the Lorenz system, the co-evolutionary strategy achieves <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R^{2}=0.961$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.961</mn> </mrow> </math></EquationSource> </InlineEquation>, within 2.2 percentage points of an oracle holding the true optimal window, while a naïve fixed-window baseline collapses to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$R^{2}=0.279$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>R</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0.279</mn> </mrow> </math></EquationSource> </InlineEquation> and exhibits <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$22\times $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>22</mn> <mo>×</mo> </mrow> </math></EquationSource> </InlineEquation> higher variance. Furthermore, the framework discovers non-polynomial laws (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$-\!\sin \theta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mspace width="-0.166667em" /> <mo>sin</mo> <mi>θ</mi> </mrow> </math></EquationSource> </InlineEquation>) 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.

Pyphysdisc: co-evolutionary symbolic regression with adaptive smoothing windows for autonomous physical law discovery from noisy data

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发表日期：

2026年5月5日

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Ali Tozar

摘要

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 ( $R^{2}\ge 0.89$ $R^{2} \geq 0.89$ for three systems) without manual tuning. On the Lorenz system, the co-evolutionary strategy achieves $R^{2}=0.961$ $R^{2} = 0.961$ , within 2.2 percentage points of an oracle holding the true optimal window, while a naïve fixed-window baseline collapses to $R^{2}=0.279$ $R^{2} = 0.279$ and exhibits $22\times $ $22 \times$ higher variance. Furthermore, the framework discovers non-polynomial laws ( $-\!\sin \theta $ $- sin θ$ ) 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.