MODEL TEST · SPARSE IDENTIFICATION OF NONLINEAR DYNAMICS

SINDy

SINDy was tested as an autonomous evolution model for POD coefficients: can a small set of sparse equations generate the reduced dynamics without replaying phase histories?

MIXED · PARKEDStrong r=2 velocity oscillator; higher-rank and surface-state closure remain weak

MAIN SUCCESS

A very compact U125 joint velocity oscillator

2POD statesjoint u/w model

88.72%fluctuation energycaptured by the two states

2nonzero termsacross both equations

0.665%frequency errortwo-cycle autonomous holdout

dz₁/dt* = +0.0857794 z₂³
dz₂/dt* = −0.1528301 z₁

The identified model behaves as a conservative nonlinear oscillator. It is sparse, bounded and robust to derivative estimation, but it represents only the dominant two-state velocity dynamics.

SINDy two-state velocity phase portrait and vector field — Selected U125 joint u/w rank-2 SINDy system: compact and dynamically coherent near the observed trajectory.

WHERE IT BREAKS

More spatial states do not automatically produce a better autonomous model

Velocity r=2

Meaningful success

Sparse, stable and phase coherent. Long rollouts remain bounded and the discovered support is robust to derivative method.

Velocity r=4–8

Identification becomes difficult

Higher POD rank improves spatial coverage, but models become dense, unstable or dynamically wrong. State dimension grows faster than reliable closure.

Surface Cd/Cl state

Autonomous closure fails

Surface POD reconstructs Cd/Cl very accurately with true coefficients, yet autonomous SINDy evolves those coefficients poorly. The bottleneck is dynamics/state closure, not POD compression.

SINDy surface force comparison — Representative surface-force test: POD coordinates are strong reconstruction coordinates, but the autonomous SINDy trajectory does not reproduce the pressure-dominated load waveform reliably.

SUMMARY

A useful discovery, not the current deployment path

What worked

The dominant joint velocity dynamics can collapse to an exceptionally sparse nonlinear oscillator. This is a genuine interpretable reduced-dynamics result.

What did not

Simply increasing rank or changing polynomial order does not yield a reliable high-fidelity autonomous flow model. A trigonometric library improves local fit but loses sparsity and off-attractor robustness.

Current decision

The SINDy track is parked. If revisited, the priority is state structure and closure—mode pairing, velocity-driven surface dynamics, memory/delay coordinates—not open-ended library hunting.