[2505.17370] FRIREN: Beyond Trajectories -- A Spectral Lens on Time

Computer Science > Machine Learning arXiv:2505.17370 (cs) [Submitted on 23 May 2025 (v1), last revised 23 Mar 2026 (this version, v5)] Title:FRIREN: Beyond Trajectories -- A Spectral Lens on Time Authors:Qilin Wang View a PDF of the paper titled FRIREN: Beyond Trajectories -- A Spectral Lens on Time, by Qilin Wang View PDF Abstract:Long-term time-series forecasting (LTSF) models are often presented as general-purpose solutions that can be applied across domains, implicitly assuming that all data is pointwise predictable. Using chaotic systems such as Lorenz-63 as a case study, we argue that geometric structure - not pointwise prediction - is the right abstraction for a dynamic-agnostic foundational model. Minimizing the Wasserstein-2 distance (W2), which captures geometric changes, and providing a spectral view of dynamics are essential for long-horizon forecasting. Our model, FRIREN (Flow-inspired Representations via Interpretable Eigen-networks), implements an augmented normalizing-flow block that embeds data into a normally distributed latent representation. It then generates a W2-efficient optimal path that can be decomposed into rotation, scaling, inverse rotation, and translation. This architecture yields locally generated, geometry-preserving predictions that are independent of the underlying dynamics, and a global spectral representation that functions as a finite Koopman operator with a small modification. This enables practitioners to identify which modes grow, d...