Random-Travel-Time Temporal Layer

(1) Integrate this immediately before a temporal mixer in an SSM, GRU, TCN, or lightweight RNN. Given a sequence tensor h of shape [batch, length, channels], create a transport layer that shifts h backward along the time axis by random travel times and averages the shifted tensors; use linear interpolation so that 1/(v Delta) need not be integral. The velocity distribution P_V is a fixed truncated distribution or a learned positive distribution, parameterized by mean and variance, and is shared across channels initially. (2) Pseudocode: sample m velocities v[j] from P_V; for each j compute delay[j]=1/(v[j]*Delta); shifted[j]=linear_interpolate(h, indices-delay[j]); z=sum_j w[j]*shifted[j]; feed z to the original temporal mixer. With probability p_correct, use this stochastic branch; otherwise compute z=linear_interpolate(h, indices-1/(mean_velocity*Delta)), the mean-velocity proxy. Add an optional consistency penalty L_cons=mean(||z_stochastic-stopgrad(z_proxy)||^2) only when velocity variance is below a threshold. (3) Compute from the paper's mathematics the reciprocal delay 1/v and the characteristic shift; estimate no PDE quantities. Empirically estimate whether the proxy is safe by tracking r=||z_stochastic-z_proxy||/(||z_stochastic||+epsilon) on minibatches and increase m or p_correct when r is large. (4) First experiment: train a small GRU or diagonal SSM on permuted-delay sequential MNIST, Copying Memory, and a synthetic advection dataset where targets are generated with velocities sampled from a known interval. Compare fixed-delay, mean-velocity proxy, and m=4 stochastic-delay layers at equal parameter count and report validation loss, robustness under a shifted velocity distribution, and wall-clock cost. Success is lower error under velocity shifts at no more than 20% training overhead, with the proxy retaining most of the stochastic layer's accuracy when the velocity variance is small.