Branched Rough Residual Block

Integrate the block as a replacement for the recurrent update in a neural CDE, continuous-time RNN, or lightweight state-space model. The driver is an input sequence x_k\in\mathbb{R}^n observed at times t_k, and the hidden state is y_k\in\mathbb{R}^d. First normalize each channel and compute \Delta x_k=x_{k+1}-x_k. Construct a second-order tensor with symmetric products \Delta x_k^\alpha\Delta x_k^\beta and an optional order-sensitive area proxy A_k^{\alpha\beta}=0.5(\Delta x_k^\alpha\Delta x_{k+1}^\beta-\Delta x_k^\beta\Delta x_{k+1}^\alpha); set A_k=0 for the simplest baseline. Let F_1 be an MLP producing a d imes n matrix and F_2 an MLP producing a d imes n imes n tensor. Apply the update y_{k+1}=y_k+F_1(y_k)\Delta x_k+F_2(y_k):\Delta x_k^{(2)}, where ':' contracts the two driver-channel indices. Normalize \Delta x_k^{(2)} by its batch RMS to prevent second-order terms from dominating. Estimate the local control as \widehat{\omega}_k=\|\Delta x_k\|^p+\|\Delta x_k^{(2)}\|^{p/2}; if \widehat{\omega}_k exceeds a threshold, subdivide the interval or apply two half-steps, implementing the paper's local-expansion regime. Stack the block with shared transformation rules if using coordinate changes. The cheap first experiment should use d=64 on PhysioNet irregular time-series or synthetic fractional-Brownian/noisy streams, comparing a GRU and neural-CDE Euler block at matched parameters and FLOPs. Measure validation NLL or accuracy, gradient-norm variance, and sensitivity to random timestamp thinning. Success means lower error at equal compute, fewer exploding trajectories, and better degradation under irregular sampling.