Lie-Rinehart Vector-Field Module

Integrate this into a latent ODE, state-space RNN, or continuous-time sequence model. Let the latent state be x\in R^d, let a neural network produce K vector-field generators V_i(x)\in R^d, and let another network produce scalar gates a_i(x)\in R; define the drift F(x)=\sum_i a_i(x)V_i(x). The simplest MVP uses ordinary autodiff to compute Jacobian-vector products. For a stronger test, add a tangent head G_ heta(x,u) that predicts directional derivatives of scalar network outputs and vector-field components in direction u, avoiding explicit Jacobians. Training pseudocode: (1) sample x,u and scalar functions f,g, including random linear probes and internal gate outputs; (2) compute D_vf=G_f(x,v) or an autodiff JVP; (3) compute B(v,w)=J_wv-J_vw using two JVP calls, or use a bracket head B_ heta; (4) form residuals r_der=D_v(fg)-(D_vf)g-f(D_vg), r_LR=B(v,fw)-(D_vf)w-fB(v,w), and r_scale=D_{cv}f-cD_vf for random scalar c; (5) optimize the forecasting or reconstruction loss plus lambda_der||r_der||^2+lambda_LR||r_LR||^2+lambda_scale||r_scale||^2. With exact autodiff, the residuals mainly validate implementation; with a learned tangent head or low-rank Jacobian approximation, they provide training supervision. Estimate JVPs with forward-mode autodiff for the baseline and compare against the learned head. First experiment: use a 2D or 4D damped oscillator and Lorenz-63 trajectory prediction, comparing a standard MLP neural ODE, the same model with independently predicted JVPs, and the Lie-Rinehart-constrained version. Measure multi-step rollout error, derivative residuals, sensitivity to solver step size, and robustness to additive noise. Success means lower long-horizon error or stable integration at a larger solver step size at equal parameter count; target a 10-20% rollout-error reduction or a measurable increase in the largest stable solver step.