FRIREN: Beyond Trajectories -- A Spectral Lens on Time

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, decay, or oscillate, both locally and system-wide. FRIREN achieves an MSE of 11.4, MAE of 1.6, and SWD of 0.96 on Lorenz-63 in a 336-in, 336-out, dt=0.01 setting, surpassing TimeMixer (MSE 27.3, MAE 2.8, SWD 2.1). The model maintains effective prediction for 274 out of 336 steps, approximately 2.5 Lyapunov times. On Rossler (96-in, 336-out), FRIREN achieves an MSE of 0.0349, MAE of 0.0953, and SWD of 0.0170, outperforming TimeMixer's MSE of 4.3988, MAE of 0.886, and SWD of 3.2065. FRIREN is also competitive on standard LTSF datasets such as ETT and Weather. By connecting modern generative flows with classical spectral analysis, FRIREN makes long-term forecasting both accurate and interpretable, setting a new benchmark for LTSF model design.