Hyperplane-Gated Piecewise Neural Dynamics

Integrate this at the transition-function level of a neural ODE, continuous-time state-space model, or latent world model. Parameterize each switching function as s_r(x)=n_r^T x-c_r and each regional vector field as a small MLP f_k. For a fixed number R of hyperplanes, enumerate only the cells observed in the training data rather than constructing all O(R^d) possible cells. Compute signed distances s=Nx-c, assign a sign pattern, map that pattern to a regional expert, and return that expert's vector field. Train with one-step derivative loss plus multi-step rollout loss. During early training use a hysteresis band around each surface so routing does not oscillate from numerical noise; later anneal the band toward zero. Periodically refit hyperplanes using samples with high derivative residuals or abrupt changes, then jointly optimize hyperplanes and experts. The paper supplies the affine-arrangement construction and separation of geometry learning from smooth dynamics learning; N, c, cell assignments, and regional weights must be estimated from data. First cheap experiment: use a known 2D oscillator with a switch at x_1=0, comparing a smooth MLP, ReLU MLP, and this model. Measure derivative error versus signed distance to the true surface. The predicted signature is that the learned zero set should localize the switch within 10% of the state scale, while within-cell error decreases without requiring a very large network Lipschitz constant. A correct model should show a kink in error slope at the switching surface.