Invertible Fourier Surrogate for Periodic Sequence Modeling

First, build a RealNVP, coupling-flow, or invertible residual network whose input and output dimensions are equal, and use it as a periodic block predictor for a neural state-space model or actuator-to-observation surrogate. Convert every length-T trajectory with an FFT, retain the DC component plus K harmonics, and concatenate real and imaginary parts so the network operates on real vectors. Second, train with the supervised loss norm(F_theta(U_hat_i) - Y_hat_i)^2, optionally adding a mild penalty against extreme log-determinants or poor Jacobian conditioning. Given a target periodic output Y_hat_d, compute U_hat_star with one inverse pass and reconstruct the time-domain input by inverse FFT. Estimate the smallest singular value m using automatic differentiation and a few power iterations on held-out points; reject targets for which m is below a chosen threshold. The paper supplies the fixed-initial-condition lifting of a dynamical relation into a static Fourier map and direct inversion of an invertible surrogate. The implementation must estimate spectral truncation error, Jacobian conditioning, and physical tracking error. The first cheap experiment should use a Duffing oscillator or nonlinear pendulum with periodic forcing, comparing an MLP plus 50-step gradient-based inversion against the invertible Fourier surrogate at equal parameter count. The predicted signature is inverse input error scaling approximately as 1 divided by sigma_min, with sharp degradation as sigma_min approaches zero. Direct inversion should also have nearly constant inference cost in the number of optimization iterations, while iterative inversion cost grows with the number of gradient steps.