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FRIREN: Beyond Trajectories -- A Spectral Lens on Time

Qilin Wang

TL;DR

FRIREN reframes long-horizon forecasting as a geometry-preserving problem, leveraging an augmented normalizing-flow to embed data into a Gaussian latent and generate a $W_2$-efficient path decomposable into rotation, scaling, inverse rotation, and translation. This yields locally geometry-preserving predictions while a global, Koopman-inspired view provides a spectral interpretation of modes, enabling interpretable identification of growing, decaying, and oscillating dynamics. The work introduces a geometry-aware evaluation protocol (including $W_2$, SWD, and EPT) and demonstrates superior performance on chaotic systems (Lorenz63, Rossler, Chua) with strong results on standard benchmarks like ETT and Weather, while clarifying the model’s scope relative to NF/OT/Koopman methods. By fusing generative flows with spectral analysis, FRIREN offers both accuracy and interpretability for long-horizon forecasting and sets a new direction for LTSF model design. The approach emphasizes actionable interpretability via local eigenstructure and a finite Koopman-like operator, enabling practitioners to diagnose and reason about forecast reliability and regime shifts in chaotic time series.

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, 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.

FRIREN: Beyond Trajectories -- A Spectral Lens on Time

TL;DR

FRIREN reframes long-horizon forecasting as a geometry-preserving problem, leveraging an augmented normalizing-flow to embed data into a Gaussian latent and generate a -efficient path decomposable into rotation, scaling, inverse rotation, and translation. This yields locally geometry-preserving predictions while a global, Koopman-inspired view provides a spectral interpretation of modes, enabling interpretable identification of growing, decaying, and oscillating dynamics. The work introduces a geometry-aware evaluation protocol (including , SWD, and EPT) and demonstrates superior performance on chaotic systems (Lorenz63, Rossler, Chua) with strong results on standard benchmarks like ETT and Weather, while clarifying the model’s scope relative to NF/OT/Koopman methods. By fusing generative flows with spectral analysis, FRIREN offers both accuracy and interpretability for long-horizon forecasting and sets a new direction for LTSF model design. The approach emphasizes actionable interpretability via local eigenstructure and a finite Koopman-like operator, enabling practitioners to diagnose and reason about forecast reliability and regime shifts in chaotic time series.

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, 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.

Paper Structure

This paper contains 55 sections, 22 equations, 1 figure, 14 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. New Evaluation Protocol
  3. Distributional metrics
  4. What pointwise metrics miss
  5. Wasserstein Distance (WD)
  6. Stress testing on Chaotic Series
  7. Why not real-world benchmarks? Why chaotic systems?
  8. Takens' Embedding Theorem, DLinear, Patching, and Cautions
  9. You Can't be Too Wrong
  10. Methods
  11. Pearls threaded along a trajectory
  12. Macro Structure: an Expansion of ANF
  13. Ellipsoids as Optimal Transport
  14. Experiments
  15. Results Summary
  16. ...and 40 more sections

Figures (1)

  • Figure 1: An illustration of the Fern forecasting mechanism. (a) An input time series processed by the feature extractor. (b) An isotropic Gaussian is transformed into an anisotropic latent distribution based on input encodings. (c) This geometric transformation is used to generate a patchwise forecast, where predictions are guided by locally generated ellipsoids that represent the learned linear map.