Fourier-Collocation Loss for Quasiperiodic Latent States

Attach the method to a latent neural ODE used for sequence prediction. The exact integration point is the vector-field evaluation: in every training step, generate a fixed phase grid \(\Theta=\{\theta_\ell\}_{\ell=1}^{L^d}\), evaluate the Fourier embedding \(K_\alpha(\theta_\ell)\) and its analytic directional derivative, then evaluate the neural field on those latent states. Use FFTs when \(L^d\) is large; for a first MVP, explicitly construct the Fourier basis. Pseudocode: initialize Fourier coefficients \(\alpha\) and neural weights \(\phi\); for each minibatch compute the ordinary trajectory loss \(\mathcal L_{data}\); compute \(X=K_\alpha(\Theta)\); compute \(V=f_\phi(X)\); compute \(T=DK_\alpha(\Theta)\omega\) using coefficient multiplication by \(i(k\cdot\omega)\); set \(L_T=\mathrm{mean}\|V-T\|^2\); backpropagate \(L_{data}+\lambda_TL_T+\lambda_A\|\alpha\|^2\); periodically increase \(M\) from 2 to 8. The paper's mathematical contribution is the invariant-torus residual and normal-frame equation; autodiff estimates all neural Jacobians, while Fourier derivatives are exact. To prevent trivial constant embeddings, add a gauge loss enforcing phase variance and approximate orthogonality of \(\partial_{\theta_j}K\). First test on a synthetic two-frequency oscillator with noisy short trajectories, comparing a standard neural ODE against the same model with the torus loss. Evaluate long-horizon phase drift, one-step error, Lyapunov/transverse stability, and training samples required. Success means substantially lower long-horizon drift and stable oscillations after trajectory lengths much shorter than one full quasiperiodic beat period, without increasing inference cost.