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We propose spectral methods for long-term forecasting of temporal signalsstemming from linear and nonlinear quasi-periodic dynamical systems. For linearsignals, we introduce an algorithm with similarities to the Fourier transformbut which does not rely on periodicity assumptions, allowing for forecastinggiven potentially arbitrary sampling intervals. We then extend this algorithmto handle nonlinearities by leveraging Koopman theory. The resulting algorithmperforms a spectral decomposition in a nonlinear, data-dependent basis. Theoptimization objective for both algorithms is highly non-convex. However,expressing the objective in the frequency domain allows us to compute globaloptima of the error surface in a scalable and efficient manner, partially byexploiting the computational properties of the Fast Fourier Transform. Becauseof their close relation to Bayesian Spectral Analysis, uncertaintyquantification metrics are a natural byproduct of the spectral forecastingmethods. We extensively benchmark these algorithms against other leadingforecasting methods on a range of synthetic experiments as well as in thecontext of real-world power systems and fluid flows.

From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction