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Shell-model experiment: Sweep-rate acceleration as a dual number
Experiment: exp_ns_level10.sx | Domain: Shell-model ODE dynamics | Hierarchy Level: 10
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. The oscillating-viscosity sweep on this system has a resonance at period ~10,000 steps; how fast \(\nu\) is swept determines whether the trajectory passes through or dwells in that resonance. This run makes the sweep acceleration \(d^2\nu / dt^2\) a dual number, giving \(\partial(\text{enstrophy response}) / \partial(\text{acceleration})\) and \(\partial T / \partial(\text{acceleration})\) by forward-mode automatic differentiation, so that the sweep schedule can be adjusted from the gradient.
Results
| Measurement | Value | What was measured |
|---|
| Meta-gradient sign change | At resonance period | \(\partial T / \partial(\text{accel})\) switches from positive to negative |
| Acceleration maximising \(T\) | 0.1 | Longest time to threshold crossing within the window |
| Positive acceleration | Sweeps past resonance | Fast sweep spends little time in the resonance zone |
| Negative acceleration | Lingers near resonance | Slow-down raises the enstrophy peak |
Chirp Sweep
A chirp sweep (linearly increasing frequency) of \(\nu\) shows the resonance structure. As the sweep frequency passes through the natural frequency at period ~10,000, the shell-enstrophy response peaks sharply and then decays. The meta-gradient changes sign at this crossing:
| Sweep Phase | Meta-Gradient | Gradient step |
|---|
| Below resonance (slow sweep) | Positive | Increase acceleration |
| At resonance | Zero (sign change) | Transition point |
| Above resonance (fast sweep) | Negative | Decrease acceleration |
Observation. The sign change of the meta-gradient at the resonance is the result of this run: the gradient tells a controller to accelerate through the resonance zone and decelerate outside it, and an acceleration of 0.1 gives the longest \(T\) within the window. This completes the ten-level differentiable hierarchy on this ODE, from system state (L0) to sweep control (L10).
The Hierarchy
| Levels | What They Measure |
|---|
| L0–L3 | System state and convergence score |
| L4–L7 | Stability margin and perturbation probing |
| L8 | Run parameters as dual numbers |
| L9 | Coupling coefficients as dual numbers |
| L10 | Sweep trajectory through parameter space |
Reproducibility
../simplex/build/sxc exp_ns_level10.sx -o build/exp_ns_level10.ll
OPENSSL_PREFIX=$(brew --prefix openssl)
clang -O2 build/exp_ns_level10.ll \
../simplex/runtime/standalone_runtime.c \
-o build/exp_ns_level10 \
-lm -lssl -lcrypto -L${OPENSSL_PREFIX}/lib
./build/exp_ns_level10
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