[論文レビュー] From Fourier to Koopman: Spectral Methods for Long-term Time Series Prediction

Henning Lange, Steven L. Brunton+1 more|arXiv (Cornell University)|Apr 1, 2020

Model Reduction and Neural Networks参考文献 63被引用数 49

ひとこと要約

The paper develops spectral forecasting methods for long-term prediction of quasi-periodic signals, using a Fourier-inspired linear oscillator approach and a nonlinear Koopman-based frequency decomposition, with uncertainty quantification and FFT-enabled computation.

ABSTRACT

We propose spectral methods for long-term forecasting of temporal signals stemming from linear and nonlinear quasi-periodic dynamical systems. For linear signals, we introduce an algorithm with similarities to the Fourier transform but which does not rely on periodicity assumptions, allowing for forecasting given potentially arbitrary sampling intervals. We then extend this algorithm to handle nonlinearities by leveraging Koopman theory. The resulting algorithm performs a spectral decomposition in a nonlinear, data-dependent basis. The optimization objective for both algorithms is highly non-convex. However, expressing the objective in the frequency domain allows us to compute global optima of the error surface in a scalable and efficient manner, partially by exploiting the computational properties of the Fast Fourier Transform. Because of their close relation to Bayesian Spectral Analysis, uncertainty quantification metrics are a natural byproduct of the spectral forecasting methods. We extensively benchmark these algorithms against other leading forecasting methods on a range of synthetic experiments as well as in the context of real-world power systems and fluid flows.

研究の動機と目的

• Motivate robust long-term forecasting for linear and nonlinear quasi-periodic systems and reduce error accumulation without precise dynamical models.
• Develop a Fourier-based linear oscillator method that works with arbitrary sampling intervals and no strict periodicity.
• Extend to nonlinear dynamics via Koopman theory to learn a nonlinear, data-driven oscillatory basis for forecasting.
• Provide uncertainty metrics tied to spectral analysis in a Bayesian-like framework.
• Benchmark the proposed methods against leading forecasting approaches on synthetic and real-world datasets (power systems and fluid flows).

提案手法

• Formulate a linear forecasting model with y_t = B y_{t-1} and x_t ≈ A y_t, with imaginary eigenvalues to reflect quasi-periodicity.
• Express the temporal loss in the frequency domain and exploit the FFT to obtain efficient global optima for frequency components.
• Apply coordinate descent over frequency variables to navigate non-convex loss surfaces, using the FFT to initialize and gradient descent to refine.
• Derive an analytic expression for the frequency update via residuals, enabling FFT-based evaluation of the error surface.
• Interpret the error surface symmetry between the residual Fourier transform and the squared error to connect with Bayesian Spectral Analysis concepts.
• Extend to Koopman forecasting by modeling x_t ≈ f(Ω(ω t)) with an invertible embedding ψ and a nonlinear, data-driven basis; optimize ω and φ jointly, using FFT-inspired strategies for the ω updates.
• Discuss connections to Dynamic Mode Decomposition and how the proposed approach yields unbiased frequency estimation under measurement noise.
• Highlight how spectral leakage and finite data influence the optimization, and how sequential frequency selection “explains away” leakage effects.

実験結果

リサーチクエスチョン

• RQ1 Can long-term forecasts be achieved for quasi-periodic systems without explicit dynamical models?
• RQ2 How can one efficiently estimate frequencies in a non-convex, time-series loss landscape using FFTs while breaking implicit periodicity?
• RQ3 Does a nonlinear, Koopman-based frequency decomposition improve forecasting for nonlinear dynamics compared to linear spectral methods?
• RQ4 What uncertainty quantification naturally arises from the spectral forecasting framework, and how does it relate to Bayesian spectral analysis?
• RQ5 How do the proposed methods compare to leading forecasting techniques (e.g., ARIMA, neural nets, DMD) on synthetic and real-world datasets?

主な発見

• A Fourier-inspired linear oscillator method can forecast quasi-periodic signals with arbitrary sampling, without assuming strict periodicity.
• Coordinate descent with FFT initialization provides a scalable route to global optima for frequency parameters despite non-convex loss landscapes.
• A nonlinear, Koopman-based frequency decomposition yields a data-dependent oscillatory basis that enables nonlinear forecasting while leveraging FFT efficiency.
• The framework yields uncertainty metrics naturally linked to Bayesian spectral analysis, arising from the spectral representation and measurement noise.
• Empirical benchmarks on synthetic data and real-world power systems and fluid flows show competitive long-term forecasting performance relative to leading methods.
• The approach relates to and complements Dynamic Mode Decomposition by refining the estimation of linear dynamics in a spectral context.