Flat-Connection Neural Dynamics

(1) Integration point: implement this as a recurrent state-space model or neural ODE for sequences. Let the hidden state be h_t∈R^d and let an MLP output two matrices A_t(h_t,u_t,z) and A_x(h_t,u_t,z)∈sl(N), represented by unconstrained matrices projected to traceless form X↦X−tr(X)I/N. Use z values sampled from a fixed interval, such as z∈{−1,−0.5,0.5,1}. For a real-valued implementation, use real traceless matrices; for a complex spectral parameter, represent real and imaginary parts as separate channels.

(2) Pseudocode:
inputs: sequence u[0:T], initial h0, collocation points (t_k,x_k), spectral samples z_j
for each training batch:
run the recurrent model and compute prediction loss L_pred
at each collocation point evaluate At=A_t(h,u,z) and Ax=A_x(h,u,z)
estimate ∂tAx and ∂xAt with automatic differentiation or centered finite differences
F = ∂tAx − ∂xAt + At@Ax − Ax@At
L_flat = mean(||F||_F^2)
L = L_pred + λ_flat L_flat + λ_reg mean(||A_t||_F^2+||A_x||_F^2)
update network parameters with AdamW

(3) Compute F and its derivatives from the curvature formula; estimate derivatives empirically with automatic differentiation. The invariant bilinear form can be the Frobenius inner product ⟨X,Y⟩=tr(XᵀY). Start with N=2 or N=4 and use 8–32 collocation points per sequence. Do not require exact flatness: tune λ_flat on validation data. A cheaper discrete variant uses a plaquette residual: compose exp(εA_t), exp(εA_x), exp(−εA_t), exp(−εA_x) around a small grid cell and penalize deviation from the identity matrix.

(4) First experiment: use a small character-level Penn Treebank task or a synthetic advection/world-model task with sequence lengths 128, 512, and 2048. Compare a GRU, a diagonal SSM, and the proposed sl(N)-connection recurrent model at matched parameter count and FLOPs. Measure validation loss, gradient norm, hidden-state norm, and error after rolling the model beyond the training horizon. The expected signal is lower long-horizon error and less hidden-state explosion at equal short-horizon loss; a meaningful result would be at least 20% lower extrapolation error or substantially slower degradation as sequence length increases.