✓✓ Beats tuned baseline 2026

KPZ latent evolution block

Implementation & benchmark of arXiv:2607.28106 — Three-Dimensional Kardar--Parisi--Zhang Scaling in Polariton Condensates

Usefulness7/10

Difficulty6/10

Novelty7/10

Architecture Stability Accuracy Generalization Ssm World model Training Regularization Pde Stochastic processes Statistical mechanics Dynamical systems Renormalization

Source paper: Three-Dimensional Kardar--Parisi--Zhang Scaling in Polariton Condensates arXiv:2607.28106 ⓘ · analyzed Aug 31, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace an unconstrained recurrent or neural-operator latent transition with a differentiable KPZ cell acting on a spatial latent field. The cell explicitly separates smoothing, nonequilibrium nonlinear steepening, and stochastic forcing, making it suitable for driven dissipative systems and long-horizon roughening that generic networks may fail to reproduce.

Formulas

$$\partial_t h(\mathbf r,t)=\nu\nabla^2h(\mathbf r,t)+\frac{\lambda}{2}\lVert\nabla h(\mathbf r,t)\rVert^2+\sqrt{D}\,\eta(\mathbf r,t),\qquad \langle\eta(\mathbf r,t)\eta(\mathbf r',t')\rangle=2\delta^{(3)}(\mathbf r-\mathbf r')\delta(t-t')$$

$$h^{k+1}=h^k+\Delta t\left[\nu\Delta_h h^k+\frac{\lambda}{2}\lVert\nabla_h h^k\rVert^2\right]+\sqrt{\frac{2D\Delta t}{\Delta x^3}}\,\xi^k,\qquad \xi^k_i\sim\mathcal N(0,1)$$

$$-\ln|g^{(1)}(0,\Delta t)|\propto|\Delta t|^{2\beta},\qquad -\ln|g^{(1)}(\Delta\mathbf r,0)|\propto|\Delta\mathbf r|^{2\chi},\qquad \beta=0.1845,\ \chi=0.3135$$

Mathematical statement

Let h(r,t) be a scalar latent field on a three-dimensional grid, nu > 0 the diffusion coefficient, lambda the nonequilibrium slope-nonlinearity coefficient, D >= 0 the noise intensity, and eta normalized space-time white noise satisfying E[eta(r,t)eta(r',t')] = 2 delta^3(r-r') delta(t-t'). The KPZ dynamics are partial_t h = nu nabla^2 h + (lambda/2)||nabla h||^2 + sqrt(D) eta. For grid spacing Delta x and time step Delta t, use Euler-Maruyama: h^(k+1) = h^k + Delta t[nu Delta_h h^k + (lambda/2)||nabla_h h^k||^2] + sqrt(2D Delta t / Delta x^3) xi^k, where xi^k has independent standard-normal entries. In the 3+1-dimensional intermediate-asymptotic regime, temporal and spatial coherence obey slopes 2 beta approximately 0.369 and 2 chi approximately 0.627, with beta = 0.1845 and chi = 0.3135.

Implementation notes

1. Integration point: use the block as the transition operator inside a latent neural operator, recurrent world model, or video-prediction model. Encode an observation into a 3-D latent tensor h, apply several KPZ substeps, and decode the resulting state. Parameterize nu = softplus(a) and D = softplus(b); allow lambda to have either sign. A small learned residual branch can model effects absent from the coarse-grained equation. 2. Pseudocode: h = encoder(x_t); for j in range(n_substeps): lap = laplacian(h); grad2 = sum(centered_gradient(h,d)**2 for d in xyz); noise = sqrt(2*D*dt/dx**3)*randn_like(h); h = h + dt*(nu*lap + 0.5*lambda*grad2) + noise; y = decoder(h). Start with periodic boundaries. For explicit integration enforce approximately dt*nu*2*d/dx**2 < 1 for dimension d = 3, and reduce dt if the nonlinear increment becomes too large. 3. Computed from the paper: the diffusion, gradient-square, and noise terms and the target exponents. Estimated empirically: nu, lambda, D, latent normalization, and the finite scaling window. If h represents a phase, evaluate g1(dr,dt) = average[exp(i(h(r+dr,t+dt)-h(r,t)))]; otherwise use normalized complex latent correlations. 4. First cheap experiment: generate synthetic 32^3 KPZ trajectories and train a KPZ block, a diffusion-only block with lambda = 0, and a ConvGRU baseline for 20–100-step rollouts. Fit log(-log|g1|) against log distance after transient removal and before finite-size saturation. The predicted signature is temporal slope 2*beta approximately 0.369 and spatial slope 2*chi approximately 0.627, each within 20 percent in the KPZ model. The diffusion-only model should show different slopes. Sweeping lambda should produce a crossover when the nonlinear-to-diffusive ratio |lambda|*h_rms/nu becomes order one.

Verification

Beats tuned baseline

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Significant win ✓

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a periodic 3-D KPZ simulator and differentiable PyTorch KPZCell with explicit diffusion, gradient-square nonlinearity, and stochastic forcing. Noise variance matched 2Ddt/dx^3 within 1.2–1.4%, nonlinear drift matched the predicted linear dependence on lambda to numerical precision, and the checkerboard mode became unstable above the predicted explicit diffusion bound. This validates the mechanism, but no end-to-end learning win was tested.

Agent confidence: 9/10

Baseline: Diffusion-only rollout: final mean 0.003486 and RMS 0.169412; at the stability-bound factor 1.0, final checkerboard amplitude was 1.0.
Idea: Noise observed/predicted ratios were 0.125753/0.124320, 0.015317/0.015540, and 0.001969/0.001943 for dx 0.5, 1.0, and 2.0. Nonlinear drift relative error was below 6e-16 for lambda in [-2,2]. At 1.2 times the predicted stability bound, amplification was 1.4 and final amplitude grew to 14.758; KPZ rollout final mean was 0.030622 and RMS 0.169710.

Limitations: The 3+1-D exponents beta=0.1845 and chi=0.3135 were not estimated because that requires longer and larger ensembles. No encoder/decoder, ConvGRU, training-loss, FLOP, or long-horizon forecasting comparison was performed.

How to run: python3 kpz_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2607.28106) Sep 2, 2026

Beats the tuned baseline by 52.1% (p=0.0081, wins 8 / 8 paired seeds; metric: lower is better).

Benchmark: Custom_tracks/kpz_field_forecast
Model: kpz_cell_shared
Paired seeds: 8
Baseline mean: 0.0011
Idea mean: 0.0005
Effect (Δ): -0.0006 (−52.1%; negative = idea better)
Wins: 8 / 8 paired seeds
p-value: 0.0081 (permutation test, 20 000 shuffles)
Smallest detectable effect: ±1.2%
Mechanism: Confirmed ✓
Practical effect: Helps
Baseline tuning: swept over 3 configs

Limitations:

The custom task is a small one-step deterministic 8^3 synthetic PDE benchmark, not a long-horizon rollout or a real physical dataset. The implementation uses a fixed nonlinear coefficient per configuration, no stochastic forcing during supervised training, no learned residual branch, and does not estimate the 3+1-D KPZ roughening exponents.

How to run: python3 run_bench.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Aug 31, 2026 latest Worked ✓

Agent confidence: 8/10

Built a custom periodic 3-D KPZ field forecasting track because no built-in track had PDE/spatial-field structure. Using the same trained KPZ-cell architecture and equal-budget paired protocol, the nonlinear KPZ variant achieved significantly lower MSE than diffusion-only: delta_mean=-0.00058644 with permutation p=0.0081. The trained-model mechanism signature was confirmed: observed idea slope 0.03117 versus predicted 0.03000, a 3.9% relative error.

Baseline: Best diffusion-only configuration lr=0.006, lambda=0; 8-seed mean test MSE=0.0011265054, std=0.0000157286.
Idea: Best KPZ configuration lr=0.006, lambda=2.4; 8-seed mean test MSE=0.0005400674, std=0.0000024417; all 8 paired seeds improved; relative reduction about 52.0%.
Mechanism signature: Observed mean-increment slope 0.0311705 versus predicted dt*lambda/2=0.0300000; relative error 3.90%; confirmed=true
Paired delta: -0.0005864380
Permutation p value: 0.0081

Benchmark result