Lie-Scheffers Macroscopic Recurrent Layer

1. Integration point: implement a recurrent population layer or neural ODE with n exchangeable latent nodes, but do not use an unconstrained MLP for the node vector field. Parameterize it as f_theta(x,mu,t)=sum_a u_{a,theta}(mu,t) X_{a,theta}(x), where X_a are selected analytic basis fields with verified or numerically checked Lie-bracket closure. Start with the scalar Riccati basis X=(1,x,x^2), or use d-dimensional affine/projective generators. 2. Pseudocode: initialize m fundamental states Q and invariant labels Xi_i from the initial node states; at each step compute reconstructed states x_i=Phi_theta(Q,Xi_i), compute mu=aggregate(x_i), calculate coefficients u_a=u_a_theta(mu,t), integrate the md-dimensional macro ODE Q'=F_macro(Q,mu,t), and reconstruct outputs. Fit Phi and coefficient networks jointly, while enforcing reconstruction of the initial states and bracket-closure penalties if the basis is learned. 3. Computed versus estimated: Phi and F_macro are computed from the chosen Lie-system construction; mu is computed from reconstructed states, while any unknown basis coefficients and superposition parameters are estimated during training. For large n, precompute or cache invariant-label statistics when the mean field is a polynomial or low-order moment. 4. First cheap experiment: use n=128 scalar Riccati nodes with shared coefficients driven by a small MLP of the mean x, compare a standard 128-state neural ODE against the m=3 reduced layer on synthetic trajectories. Measure maximum reconstruction error, rollout error, wall time, and memory as n increases from 16 to 1024. The quantitative prediction is that, with matched solver tolerances, reduced-versus-full trajectories differ only by integration and floating-point error, and macro integration cost remains approximately independent of n except for mean-field aggregation; measured state dimension should change from n to 3. A falsification test is reconstruction error that grows systematically with horizon despite tight tolerances, indicating that the learned vector field is not actually Lie-Scheffers closed.