Hypoelliptic transport-diffusion layer

Integrate the layer as a replacement for one local convolution, MLP-mixer block, or state-space transition in a model whose features are indexed by phase-space coordinates (x,y), with optional time conditioning. Let the input be a tensor f of shape [batch, spatial-x locations, spatial-y locations, channels], together with coordinate tensors x and y. First choose a positive step h. Implement D_h as a depthwise separable convolution over x axes only, using a normalized Gaussian kernel with variance 2h; on irregular coordinates, use K nearest x-neighbors with weights w_ij proportional to exp(-||x_i-x_j||^2/(4h)), normalized over j. Next implement characteristic transport by evaluating the diffused feature at y_i+h*x_i using bilinear interpolation or a differentiable grid sampler. Compute delta=f_shift-f, and update f_new=f+sigmoid(G(f))*delta, where G is a pointwise linear layer and the sigmoid gate supplies g in the formula. Optionally follow this with a pointwise channel-mixing MLP and LayerNorm. For a multiscale stack, use x receptive-field radius r_l, y radius r_l^3, and time or layer step proportional to r_l^2, directly matching delta_r. Pseudocode is: f_diff=DiffusionX(f,h); f_shift=InterpolateY(f_diff,y+h*x); delta=f_shift-f; f=f+sigmoid(G(f))*delta. The paper-derived quantities are the Kolmogorov operator, anisotropic dilation, Gaussian x diffusion, and x-directed y transport; h, kernel truncation radius, channel width, and gate initialization are empirical hyperparameters. The first experiment should use a small damped-particle or Lorenz-style phase-space prediction dataset. Compare a six-layer MLP or Transformer local mixer against an equal-parameter, equal-FLOP model replacing two mixers with kinetic layers. Measure one-step MSE, long-horizon rollout error, gradient norm variance, and performance as training-set size decreases. Success means lower rollout error and fewer unstable trajectories, especially on systems where y evolves through x. Include ablations removing the transport term, using isotropic diffusion, and replacing h*x by a learned unconstrained displacement.