Binned Spectral Power Loss for Improved Prediction of Chaotic Systems

The paper derives the NTK-based Fourier-mode training dynamics: df̂θ(k)/dn ≈ −Θ(k) ∂L/∂f̂θ(k)* (their Eq. 24), and, for the BSP loss with Eθ(k) = 1/2 ||f̂θ(k)||^2 and LBSP = ∫ (1 − (Eθ+ε)/(Ev+ε))^2 dk, computes ∂LBSP/∂f̂θ(k)* = −(Ev(k) − Eθ(k))/(Ev(k)+ε)^2 f̂θ(k) (their Eq. 32), yielding df̂θ(k)/dn ≈ Θ(k) (Ev(k) − Eθ(k))/(Ev(k)+ε)^2 f̂θ(k). Under the scalar identity Θ(k) = Θ(k) I_D this is stated as their Eq. (34). The candidate solution independently reproduces exactly these steps (function-space NTK flow, Fourier diagonalization, Wirtinger derivative of Eθ, substitution), and reaches the same Eq. (34) along with consistent qualitative consequences (stationarity when Ev = Eθ; scaling without phase rotation). Note: in the paper’s intermediate derivation, a typographical omission of parentheses appears around (Ev − Eθ) in Eq. (32) as extracted, but the subsequent equations and the final Eq. (34) are consistent with the correct expression. Citations: NTK dynamics (Eq. 24) and assumptions , BSP energy and loss definitions plus derivative (Eqs. 30–32) , final mode-wise dynamics (Eqs. 33–34) , NTK diagonalization in Fourier basis (Eq. 23) .