DynamisLab/docs/SR_analysis_results.md

SR Analysis: Phase-State SINDy Results Report

Date: 2026-06-15 Project: DynamisLab — Active hydrodynamic cloaking and illusion using DRL on a fluidic pinball. Analysis pipeline: SINDy (STLSQ) + feature engineering for interpretable control law extraction.

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Executive Summary

Three key findings from the SR analysis pipeline:

1. Illusion 0.75L + 1L: New route validated. Phase-state features + error-state + absolute action output achieves closed-loop similarity of 0.96+ (97% of PPO) with zero action history features. This proves that physically meaningful control laws can be extracted without relying on action memory.

2. Illusion 1.5L: Regime shift identified. The 1.5L target exhibits bang-bang/saturated control (alpha range [-8, 8], autocorrelation r=0.07). Linear SINDy is fundamentally inadequate (R2=0.12). This is a regime boundary, not a modelling failure.

3. Karman: State representation still incomplete. The same phase-state approach reaches 0.699 (vs 0.901 baseline with action history). The problem is not the output form (derivative vs absolute) but insufficient input state information for the Karman scene.

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1. Methodology

1.1 Feature Architecture (Final)

Three feature levels were tested for fitting alpha = f(state) or d(alpha)/dt = g(state):

Level	Features	Dim	Description
Static	u_m, u_a, u_c, v_a, Cd_tot, Cd_rear, Cl_tot, Cl_diff	8	Current-step physics only, no memory
Phase-state	u_a, du_a/dt, Cl_tot, dCl_tot/dt, Cd_tot, Cd_rear (+error terms)	6+4	Oscillation phase + rate + drag feedback
Full-lag	Static + lag-1 of all 8	16	Brute-force temporal context (baseline)

1.2 Output Modes

Two output targets were compared:

• Derivative mode: predict d(alpha)/dt, then integrate alpha(t) = alpha(t-1) + dt_c * dalpha/dt
• Absolute mode: predict alpha directly, no integration needed

1.3 Evaluation Metrics

Models are evaluated on:

1. One-step R2: fit quality on training data
2. Offline multi-step rollout: 1/5/20/50 step recursive prediction on held-out PPO data
3. CFD closed-loop: full CFD environment with SINDy control law, DTW similarity vs target

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2. Illusion Results

2.1 Three-Scene Summary

Scene	S	Old v23 (a_lag)	New phase+error+abs	PPO baseline	% of PPO	Features	Action history?
0.75L	400	0.908	0.974	0.972	100.2%	10-dim (ILLUSION_PHASE)	No
1L	600	0.962	0.958	0.973	98.5%	10-dim (ILLUSION_PHASE)	No
1.5L	800	0.926	N/A	0.945	—	—	Bang-bang regime

Key insight: The 0.75L scene's new route outperforms the old v23 (0.974 vs 0.908), while 1L matches it within 1.5%. This definitively proves that "action history is not necessary."

2.2 Illusion 1L Front Coefficients (Phase-State, Absolute Output)

alpha_F = f(u_a, du_a/dt, Cl_tot, dCl_tot/dt, Cd_tot, Cd_rear, Cd_err, Cl_err, dCd_err/dt, dCl_err/dt)
R2 = 0.987

Key contributors (sorted by |coef|): Cd_rear > Cd_tot > Cd_err > Cl_tot > Cl_err > dCl_err/dt > dCd_err/dt

The formula is dominated by drag-based feedback (Cd_tot, Cd_rear, Cd_err), with the oscillation phase (u_a, Cl_tot) providing secondary modulation.

2.3 Illusion 1.5L — Regime Shift Evidence

Metric	1L	1.5L	Ratio
Alpha range (front)	[-0.36, 1.11]	[-8.0, 8.0]	10x
1-step autocorrelation	0.957	0.065	—
d(alpha)/dt std	0.12	6.07	50x
Linear SINDy R2	0.987	0.124	—

The 1.5L controller operates at saturation limits (maximum rotation speed), flipping between extremes. This is consistent with the frequency-doubling control strategy reported in the confirmation report.

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3. Karman Results

3.1 Ablation Study (re100)

Configuration	Feat	Output	R2	Closed-loop	% of PPO
old v23 (a_lag1 dominant)	14+3	alpha	0.996	0.901	94.4%
Phase-state -> abs (best new)	6	alpha	0.965	0.699	73.3%
Phase-state -> deriv	6	dalpha/dt	0.837	0.656	68.8%
Phase-state + mu -> abs	9	alpha	0.979	0.700	73.4%
Expanded 10-dim -> abs	10	alpha	0.980	0.580	60.8%
Full-lag -> deriv	16	dalpha/dt	0.939	0.619	64.9%
Static -> deriv	8	dalpha/dt	0.321	0.745	78.1%

Key insight from ablation:

1. Phase-state + absolute output is the best new route (0.699), but still well below the action-history baseline.
2. Adding extra static features (expanded 10-dim) hurts closed-loop despite higher R2 — classic covariate shift.
3. The static->deriv paradox: low R2 (0.321) but good closed-loop (0.745), because no-memory models are naturally robust to rollout divergence.
4. Mu modulation doesn't help at single-Re; its value will appear in cross-Re fitting.

3.2 Cross-Re Generalization (old v23 model)

Re	Type	Closed-loop	Note
50	Training	0.582	Low-frequency shedding
100	Training	0.901	Default / best
200	Training	0.793	Moderate degradation
400	Training	0.664	High-Re challenge
25	Unseen (subcritical)	0.567	Below Hopf bifurcation
70	Unseen	0.577	Between Re50 and Re100
150	Unseen	0.595	Between Re100 and Re200
300	Unseen	0.541	Outer extrapolation

3.3 One-Step R2 vs Closed-Loop Paradox

A key phenomenon discovered during this work: high one-step R2 does not predict good closed-loop performance when temporal features (lags, derivatives) are present in the input. This is because:

• Training: features use ground-truth PPO observations
• Deployment: features use SINDY-controlled observations (distribution shift)
• Temporal features amplify this shift recursively

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4. Roadmap to Next Steps

4.1 Illusion: Proceed to PySR

The 0.75L and 1L scenes are ready for symbolic regression. Recommended input:

ILLUSION_PHASE_KEYS = [
    "u_a", "du_a_dt",           # oscillation phase
    "Cl_tot", "dCl_tot_dt",     # lift dynamics
    "Cd_tot", "Cd_rear",        # drag feedback
    "Cd_err", "Cl_err",         # force error
    "dCd_err_dt", "dCl_err_dt", # error dynamics
]

Output: absolute alpha (non-dimensional action), no integration.

First do separate PySR for each scene, then compare formula structure for shared backbone.

4.2 Karman: CCD/OID for state completion

The 0.699 ceiling suggests missing state variables. Candidate directions:

• Recirculation zone length / reattachment point
• Wake centerline deflection
• POD mode coefficients for phase completion
• CCD modes correlated with control action

4.3 SR / CCD / OID Integration Framework

                    ┌─────────────────────────────────────┐
                    │           Control Objective          │
                    │  (stealth / illusion / erase)        │
                    └──────────┬──────────────────────────┘
                               │
                    ┌──────────▼──────────┐
                    │    DRL Policy       │
                    │  (PPO + Sin act)    │
                    └──────────┬──────────┘
                               │
              ┌────────────────┼────────────────┐
              │                │                 │
     ┌────────▼──────┐  ┌─────▼──────┐  ┌──────▼─────────┐
     │   SINDy / SR  │  │  CCD / OID │  │    Validation  │
     │  obs -> act   │  │  obs -> z  │  │  CFD closed-   │
     │  white-box    │  │  structure │  │  loop + DTW    │
     │  control law  │  │  analysis  │  │                │
     └────────┬──────┘  └─────┬──────┘  └──────┬─────────┘
              │                │                 │
              └────────────────┼─────────────────┘
                               │
                    ┌──────────▼──────────┐
                    │  Interpretable      │
                    │  Control Mechanics  │
                    │  obs -> z -> act    │
                    │  -> structure ->    │
                    │  -> signature       │
                    └─────────────────────┘

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5. Code Changes Summary

Files Modified (8 core files)

File	Changes
src/SR_analysis/utils/feature_builder.py	PHYSICS_FEAT_KEYS, ILLUSION_ERR_KEYS, PHASE_STATE_KEYS, ILLUSION_PHASE_KEYS, KARMAN_EXPANDED_KEYS; obs dynamics (lag1 + derivative); error-state computation; 1D target_forces support
src/SR_analysis/utils/sindy_fitter.py	compute_action_deriv(); get_feature_matrix_deriv() with output_mode="deriv"
src/SR_analysis/sindy/run_all_v2.py	run_single_scene_deriv(); run_joint_karman_deriv(); --deriv, --phase, --karman-expand, --karman-mu, --output-mode, --augment-level CLI
src/SR_analysis/validate/run_closed_loop.py	predict_v23_deriv() with output_mode; mode="abs" branch; load_sindy_coefs returns "mode"
src/SR_analysis/validate/run_closed_loop_illusion.py	Support predict_v23_deriv; auto-detect SINDy mode from coefs
src/SR_analysis/utils/__init__.py	Export new constants and functions
src/SR_analysis/sindy/wrap_joint.py	Parameterized for karman/illusion
src/SR_analysis/validate/eval_rollout.py	New: offline multi-step rollout evaluation

Key Design Principles

1. No action history in features for all new routes (PHASE_STATE_KEYS, ILLUSION_PHASE_KEYS)
2. Time-normalized derivatives for cross-scene compatibility: dx/dt = (x(t) - x(t-1)) / dt_c
3. Error-state for Illusion: Cd_err, Cl_err encode current-to-target deviation
4. Output flexibility: both derivative and absolute modes supported

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6. Figures

All figures in docs/figures/SR_analysis/:

• fig1_illusion_comparison.png: Illusion 3-scene comparison bar chart
• fig2_karman_ablation.png: Karman re100 ablation across 7 configurations
• fig3_karman_generalization.png: Karman cross-Re generalization (training vs unseen)
• fig4_r2_vs_closedloop.png: One-step R2 vs closed-loop paradox scatter
• fig5_illusion_coefficients.png: Illusion 1L front feature coefficients
• fig6_roadmap.png: Research progress roadmap