Nonlinear Noise-Tightening Drift

Use this as a drift parameterization for a stochastic latent model, diffusion sampler, or contractive recurrent state update. Let a neural field output \(u_\theta(x,t)\), then define the drift as \(v_\theta(x,t)=u_\theta(x,t)-\lambda x-\beta x\odot\|x\|^2\) for vector states, or use a bounded radial version \(-\beta x\|x\|^2/(1+\|x\|^2/R^2)\) if exploding forces are a concern. Keep \(\lambda\) fixed as the certified baseline rate and tune \(\beta\geq0\). During training, estimate pairwise rates on perturbed minibatch states and penalize violations of \(r_\theta(x,y)\geq\lambda\), while monitoring stationary covariance from long stochastic rollouts. Pseudocode: sample \(x,y\); evaluate \(v_\theta(x),v_\theta(y)\); compute \(r_\theta\); add \(c\max(0,\lambda-r_\theta)^2\) to the loss; sample Euler-Maruyama trajectories and estimate covariance after burn-in. The paper provides the exact scalar cubic mechanism and strict variance reduction under additive noise; the multidimensional neural extension requires empirical verification because the learned residual may destroy monotonicity. First experiment: compare linear and cubic-augmented drifts at matched \(\lambda\) and identical noise \(D\) on a 1D mixture-restoration task and a 2D latent diffusion. The quantitative prediction is stationary variance below \(D/\lambda\), with the gap increasing monotonically with \(\beta\) for small \(\beta\), while the measured pairwise contraction rate rises above \(\lambda\) for samples at large radius.