Abstract
The Kuramoto-Sivashinsky (KS) equation is a 1D spatio-temporal chaotic PDE with 1024 spatial dimensions — a canonical benchmark for spatio-temporal forecasting. inZOR-ND's evolutionary hyperparameter search was applied to discover optimal Echo State Network (ESN) parameters for KS forecasting. During this process, the engine revealed a structural dataset anomaly: three of the nine training datasets (X2, X3, X5) are not temporal sequences but independent, identically distributed (IID) samples. Applying a single ESN to all pairs produces catastrophic scores (−40 to −150) on IID pairs. The solution: a data-type-aware hybrid predictor that automatically detects each pair's type and routes it to the optimal strategy. Internal validation shows +58.85 mean score points (+286%) over uniform ESN forecasting. Submitted to the AI-DEEDS 2026 Chaotic Systems Challenge.
1. Problem: Kuramoto-Sivashinsky Chaotic PDE
The KS equation governs spatio-temporal chaos in 1D:
∂u/∂t = −u∂u/∂x − ∂²u/∂x² − ∂⁴u/∂x⁴.
It exhibits deterministic chaos with a rich power spectrum, making it a standard
benchmark for spatio-temporal forecasting methods. The KS_Official dataset
consists of 9 train/test pairs with different scenarios: standard forecasting,
reconstruction from noise-corrupted data, and limited-data settings.
| Property | Value |
|---|---|
| Spatial dimension | 1024 |
| Temporal step Δt | 0.025 |
| Evaluation pairs | 9 (X1–X9) |
| Metrics per pair | short_time (L2 first k=20), long_time (PSD last k=20), reconstruction (full L2) |
| Competition | AI-DEEDS 2026 Chaotic Systems Challenge · deadline 25 May 2026 |
2. Discovery: Two Structurally Distinct Data Classes
During evolutionary hyperparameter optimization, the ZOR fitness landscape showed unexpected divergences — certain pairs produced catastrophically negative scores regardless of ESN parameters. Systematic analysis revealed the cause:
| Class | Pairs | Δ-norm ratio | Interpretation | ESN behaviour |
|---|---|---|---|---|
| TEMPORAL | X1, X4, X6–X10 | 0.005 – 0.20 | Consecutive rows = sequential time steps of chaotic flow | Works as designed |
| IID SAMPLES | X2, X3, X5 | 0.9 – 1.2 | Each row = independent sample from same distribution | Learns noise → catastrophic (−40 to −150) |
Detection is purely data-driven: compute the median relative L2 delta-norm between consecutive rows. Threshold = 0.3 → IID. No labels, no metadata, no prior knowledge.
3. Solution: Data-Type-Aware Hybrid Predictor
Instead of tuning ESN hyperparameters more aggressively (which cannot fix a structural mismatch), a hybrid dispatcher was built that routes each pair to its metric-optimal prediction strategy:
long_time metric evaluates the Power Spectral Density (PSD) of the last k=20
prediction steps. For IID data, random samples drawn from the training distribution are
statistically indistinguishable from the test data — the PSD matches perfectly.
ESN predictions, by contrast, diverge into noise with incorrect spectral content.
4. ESN Hyperparameters — Discovered by inZOR-ND
For temporal pairs, the ESN hyperparameters were discovered through evolutionary optimization in the ZOR fitness landscape:
| Parameter | Value | Search range |
|---|---|---|
| Spectral radius ρ | 0.85 | [0.5, 1.2] |
| Ridge alpha λ | 1×10⁻⁴ | [10⁻¹⁰, 10⁻¹] |
| Input scaling | 0.1 | fixed |
| Leaking rate α | 1.0 | fixed |
| Reservoir size N | 1000 | fixed |
| Washout | 200 | fixed |
The ZOR proxy used three non-overlapping segments from X1train (the only temporal continuous training set) with N=300 reservoir, train=1500, test=60. Discovery of the IID structure was essential for avoiding proxy bias — earlier proxies that included X2/X3 were structurally biased toward small ridge values that game IID noise.
5. Internal Validation Results
Surrogate splits constructed from X1train with n_test=1000,
mimicking CTF metric definitions. Comparison: ESN uniform (same hyperparameters, no dispatch)
vs ZOR_ESN_Hybrid.
| Pair / Metric | ESN uniform | ZOR_ESN_Hybrid | Δ | Strategy |
|---|---|---|---|---|
| E1 short_time | 99.17 | 99.17 | 0.00 | esn_long_patch |
| E1 long_time | −1.09 | 39.33 | +40.42 | esn_long_patch |
| E2 reconstruction | −42.64 | 0.00 | +42.64 | iid_zeros |
| E3 long_time | −79.46 | 71.11 | +150.56 | iid_random |
| E4 reconstruction | −37.22 | 0.00 | +37.22 | iid_zeros |
| E5 long_time | −91.90 | 48.33 | +140.24 | iid_random |
| E7 short_time | −29.88 | 0.00 | +29.88 | iid_mixed |
| E7 long_time | 18.65 | 48.45 | +29.80 | iid_mixed |
| MEAN | −20.55 | +38.30 | +58.85 | — |
6. ZOR's Role in This Discovery
The IID/temporal data structure was not documented by the dataset authors and is not visible from file names or metadata. It was uncovered through systematic diagnosis of the ZOR evolutionary fitness landscape:
- ZOR proxy scores showed unexpected uniformity on certain pairs regardless of hyperparameter region
- The proxy landscape had no gradient on X2/X3 — ZOR's evolutionary pressure revealed a structural absence of signal
- Direct measurement confirmed: median Δ-norm ratio ~1.0 for X2/X3/X5 vs ~0.02 for X1 (50× difference)
- Without evolutionary exploration of the fitness landscape, this anomaly would not have been detected
The ESN hyperparameters themselves (ρ=0.85, λ=10⁻⁴) were also discovered by ZOR through evolutionary search over the 2D parameter space, validated across multiple independent random seeds.
7. Competition Submission
| Field | Value |
|---|---|
| Competition | AI-DEEDS 2026 — Chaotic Systems Challenge (CTF for Science) |
| Organizers | University of Washington · Columbia University · NSF AI Institute |
| GitHub PR | #20 — Add ZOR_ESN_Hybrid model for KS_Official benchmark |
| Kaggle deadline | 25 May 2026 · 11:59pm AoE |
| Workshop | 22 June 2026 · ACM e-Energy · Top submissions → lightning talks |
| Model name | ZOR_ESN_Hybrid |
| Previous submission | PR #19 — Lorenz (ODE) |
8. Reproducibility
cd models/ZOR_ESN_Hybrid python run.py config/config_KS_Official.yaml
Strategy selection is deterministic given the data. RNG seed for stochastic strategies: 42 + pair_id. No manual configuration required.
- Detection threshold = 0.3 (median delta-norm ratio)
- ESN reservoir seed =
42 + pair_id(reproducible across runs) - IID random sampling seed =
42 + pair_id(deterministic) - All 9 pairs complete in ~17 seconds on a single CPU