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Shell-model experiment: Coupling coefficients as dual numbers
Experiment: exp_ns_level9.sx | Domain: Shell-model ODE dynamics | Hierarchy Level: 9
Overview
The system is the six-shell cubic ODE with independent velocity-shell and vorticity-shell variables \(u_k\) and \(\omega_k\), stretching coefficients \(\lambda\), \(\lambda_2\), feedback coefficient \(\sigma\) and viscous damping \(\nu k^2\). It is not a Galerkin truncation of Navier–Stokes. The time to threshold crossing \(T\) is a finite-window excursion quantity that depends on the integration window, the excursion threshold and the solver. Level 8 of the differentiable hierarchy makes the run parameters (\(A\), \(\nu\), \(\text{dt}\)) dual numbers; this run (level 9) makes the coupling coefficients \(\lambda\), \(\lambda_2\) and \(\sigma\) dual numbers as well, giving the full gradient \(\nabla T = (\partial T / \partial A, \partial T / \partial \nu, \partial T / \partial \lambda, \partial T / \partial \lambda_2, \partial T / \partial \sigma)\) and the Hessian diagonal \(\partial^2 T / \partial x_i^2\) for each parameter by forward-mode automatic differentiation.
Results
Full Gradient (Sensitivity Ranking)
| Parameter | \(\partial T / \partial x\) | Sign | Reading |
|---|
| \(A\) (amplitude) | Largest magnitude | Negative | Larger amplitude → earlier threshold crossing |
| \(\nu\) (viscosity) | Second largest | Positive | More damping → later threshold crossing |
| \(\lambda\) (stretching) | Third | Negative | Larger \(\lambda\) → earlier threshold crossing |
| \(\lambda_2\) (quadratic) | Fourth | Negative | Larger \(\lambda_2\) → earlier threshold crossing |
| \(\sigma\) (feedback) | Smallest | Weakly negative | Feedback effect is secondary |
Hessian Diagonal (Curvature)
| Parameter | \(\partial^2 T / \partial x^2\) | Curvature | Reading |
|---|
| \(\lambda\) | Positive | Convex | Increasing \(\lambda\) shortens \(T\) at an increasing rate |
| \(\lambda_2\) | Positive | Convex | Quadratic term compounds super-linearly |
| \(\sigma\) | Negative | Concave | Feedback effect saturates at high coupling |
| \(\nu\) | Negative | Concave | Diminishing returns from added damping |
| \(A\) | Positive | Convex | Large amplitudes shorten \(T\) disproportionately |
Cross-Derivatives
| Cross-Derivative | Bounded within the window? | Reading |
|---|
| \(\partial^2 T / \partial \lambda \, \partial A\) | Yes | The \(A\)-dependence of \(T\) changes smoothly with \(\lambda\) |
| \(\partial^2 T / \partial \lambda_2 \, \partial A\) | Yes | The \(A\)-dependence of \(T\) changes smoothly with \(\lambda_2\) |
Observation. The Hessian diagonal is convex in the parameters that shorten \(T\) (\(\lambda\), \(\lambda_2\), \(A\)) and concave in those that lengthen it (\(\nu\), \(\sigma\)). The cross-derivatives with \(A\) are bounded within the window, and the sensitivity ranking places amplitude first. These are properties of the finite-window quantity \(T\) for this ODE.
Reproducibility
../simplex/build/sxc exp_ns_level9.sx -o build/exp_ns_level9.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_level9.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_level9 \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_level9
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