Title: Chaining Spectral Pearls: Ellipsoidal Forecasting Beyond Trajectories for Time Series
Primary area: learning on time series and dynamical systems
Keywords: Time Series Forecasting, Optimal Transport, Interpretability, Normalizing Flows, Geometric Deep Learning, Dynamical Systems

Abstract: Current long-term time-series forecasting (LTSF) practice---pointwise metrics on mostly stochastic data-masks brittleness under deterministic chaos. We argue for stress-testing on canonical chaotic systems and for predicting future \emph{geometry} rather than exact trajectories. We present \Fern (Forecasting with Ellipsoidal RepresentatioN), a geometry-aware forecaster that applies per-patch local linear transport with explicit spectral factors (eigenvectors/eigenvalues), yielding structure-preserving forecasts and actionable diagnostics of stability, modes, and regime shifts. Alongside MSE/MAE, we report Sliced Wasserstein Distance (shape fidelity) and Effective Prediction Time (horizon stability). On Lorenz63, Rössler, and Chua, \Fern delivers substantially lower error and improved stability versus strong LTSF baselines, while remaining competitive on ETT and Weather. By coupling simple local transport with explicit spectral interpretability, \Fern offers a practical path to robust, transparent long-horizon forecasting.

TL;DR: We introduce FERN, a forecaster that predicts a time series' future geometry using a chain of spectrally-interpretable ellipsoids, achieving robust performance on LTSF tasks.