In this paper a structural correction in neural network architecture for the long-time integration of conservative dynamical systems is used to overcome uncontrolled invariant drift existing in standard physics-informed formulations. The Structurally Corrected IP-PINN is used to solve the Lotka–Volterra predator–prey system where the multiplicative architectural correction to the invariant manifold demonstrates from a statistical significance perspective improvements beyond both baseline and soft-constraint methods (Welch’s t-test, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p < 10^{-11}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>11</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>, Cohen’s <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$d = 5.03$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>5.03</mn> </mrow> </math></EquationSource> </InlineEquation>). Also our structurally corrected neural network (SC-PINN) reaches a 72.2% reduction in invariant drift (mean <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$9.54\times 10^{-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>9.54</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> vs. <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$3.43\times 10^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3.43</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>) contrasted to the baseline, and a 66.9% improvement over soft penalty techniques.

Structurally corrected invariant-preserving neural networks for conservative dynamical systems: a statistically validated approach

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2026年5月26日

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Fatima Ouaar

摘要

In this paper a structural correction in neural network architecture for the long-time integration of conservative dynamical systems is used to overcome uncontrolled invariant drift existing in standard physics-informed formulations. The Structurally Corrected IP-PINN is used to solve the Lotka–Volterra predator–prey system where the multiplicative architectural correction to the invariant manifold demonstrates from a statistical significance perspective improvements beyond both baseline and soft-constraint methods (Welch’s t-test, $p < 10^{-11}$ $p < 10^{- 11}$ , Cohen’s $d = 5.03$ $d = 5.03$ ). Also our structurally corrected neural network (SC-PINN) reaches a 72.2% reduction in invariant drift (mean $9.54\times 10^{-2}$ $9.54 \times 10^{- 2}$ vs. $3.43\times 10^{-1}$ $3.43 \times 10^{- 1}$ ) contrasted to the baseline, and a 66.9% improvement over soft penalty techniques.