# Invertible Fourier Surrogate for Periodic Sequence Modeling

- ID: 1892
- Canonical URL: https://synthcore.org/idea/1892/invertible-fourier-surrogate-for-periodic-sequence-modeling
- API JSON: https://synthcore.org/api/idea/1892.json
- API Markdown: https://synthcore.org/api/idea/1892.md
- Verification status: unverified
- Source: [arXiv:2608.06419](https://arxiv.org/abs/2608.06419)
- Category: architecture
- Solves: speedup, stability
- ML areas: world-model, ssm, inference-speedup, initialization
- Math tags: dynamical-systems, control-theory, harmonic-analysis, linear-algebra
- Ratings: usefulness 6/10; difficulty 6/10; novelty 4/10

## Idea description

Represent periodic input-output behavior using a compact real vector of Fourier coefficients and learn an invertible neural map from input coefficients to output coefficients. Inference then obtains the input representation for a desired periodic output by a single inverse pass instead of iterative optimization through a nonlinear forward model, while the Fourier representation reduces sequence dimensionality when high-rate signals are spectrally sparse.

## Mathematical statement

For a period-T signal, let U_hat(omega) and Y_hat(omega) be stacked real Fourier features containing the DC coefficient and real and imaginary parts of selected harmonics. The surrogate learns Y_hat = F_hat_theta(U_hat) using a regression loss J_r(theta). An invertible architecture enforces a bijection F_hat_theta from R^d to R^d with a tractable inverse, so a desired periodic output Y_hat_d gives U_hat_star = F_hat_theta inverse(Y_hat_d) directly. If the learned forward map has minimum singular value m greater than zero and forward modeling error is at most delta, inverse perturbations obey approximately ||U_hat_star - U_star||_2 <= delta / m. Thus inverse conditioning, rather than training loss alone, predicts feedforward sensitivity.

## Key formulas

- U_hat(omega) = [U_hat(0), Re(U_hat(omega)), Im(U_hat(omega)), ..., Re(U_hat(K omega)), Im(U_hat(K omega))]^T
- Y_hat(omega) = F_hat_theta(U_hat(omega)), and theta_star = argmin_theta J_r(theta) = argmin_theta sum_i ||Y_hat_i - F_hat_theta(U_hat_i)||_2^2
- U_hat_star(omega) = F_hat_theta inverse(Y_hat_d(omega))
- sigma_min(J_F(U_star)) = m > 0 implies ||U_hat_star - U_star||_2 approximately <= delta / m

## Implementation notes

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.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.