Monotone Jacobi Hybrid Neural ODE

(1) Integration point: replace the vector field in a small neural ODE, continuous-depth residual block, or state-space model with K smooth branches f_j(x;theta) and scalar scores H_j(x;theta). At each fixed solver step, select j_k=argmax_j H_j(x_k) and perform x_{k+1}=x_k+Delta t f_{j_k}(x_k). Start with K=2 and width-64 two-layer MLP branches. (2) Pseudocode: initialize x=x_0 and S=epsilon I. For k=0,...,N-1, compute every H_j(x), choose j=argmax_j H_j(x), compute f=f_j(x), obtain G=partial f_j(x)/partial x with automatic differentiation, update x=x+Delta t f, and update S=S+eta G^T G+epsilon I. Add lambda_S times the squared Frobenius norm of S-S^T if numerical asymmetry appears. Add a switching-margin term lambda_sw times max(0,delta-|H_(1)-H_(2)|)^2 away from deliberately targeted switches, where H_(1) and H_(2) are the largest and second-largest scores. (3) The paper contributes the max-of-branches construction, switching-surface coordinates, Lipschitz-flow motivation, and monotone positive-semidefinite tangent evolution. Estimate switch transversality empirically using |gradient_x(H_(1)-H_(2))^T f_j| at near-tie steps; clip the solver step or soften the switch when this value is below a threshold. Record the minimum eigenvalue of S, the maximum input-Jacobian norm, the number of branch switches, and the number of near-tie steps. (4) First experiment: train a four-step or eight-step neural ODE classifier on MNIST or CIFAR-10. Compare an unconstrained single-branch MLP vector field, a two-branch hard-switch model without S, and the proposed model at matched parameter count and FLOPs. Use identical AdamW schedules and five random seeds. Success means fewer NaN or exploding-gradient runs, lower variance of validation loss, and a smaller maximum input-Jacobian norm at equal accuracy. A stronger result would be stable training with a larger Euler step size or with twice as many branch switches.