# 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.

---

## 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.

---

## 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

---

## 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.

---

## 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

---

## 4. Roadmap to Next Steps

### 4.1 Illusion: Proceed to PySR

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

```python
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       │
                    └─────────────────────┘
```

---

## 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"|"absolute"; include_mu support |
| `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

---

## 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