Chaining Spectral Pearls: Ellipsoidal Forecasting Beyond Trajectories for Time Series

Qilin Wang

Keywords: Time Series Forecasting, Optimal Transport, Interpretability, Normalizing Flows, Geometric Deep Learning, Dynamical Systems

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.

Abstract: Current long-term time-series forecasting (LTSF) benchmarks are dominated by noisy stochastic datasets and pointwise losses, so models that look strong on ETT-type tasks can behave unpredictably under deterministic chaos or controlled regime shifts. We argue that forecasters should be stress-tested on canonical chaotic systems and on synthetic benchmarks with precisely scripted non-stationarity, and that evaluation should focus on the geometry of predictive distributions, not just single trajectories. We present FERN (Forecasting with Ellipsoidal RepresentatioN), a geometry-aware forecaster that uses a bidirectional encoder and a per-patch local linear transport map, factored as translate--rotate--scale--rotate-back with explicit eigenvalues and eigenvectors. The network therefore "only'' learns to generate stable Jacobians, while users obtain spectral diagnostics of local stretching, volume change, and regime switches. Alongside MSE/MAE we report Wasserstein Distance (shape fidelity) and Effective Prediction Time (horizon stability). Across 21 synthetic systems (chaotic, stochastic, and switching) and cleaned ETT/Weather benchmarks, FERN is a strong all-round "safe'' model: it achieves the best or second-best MSE or SWD on 19/21 synthetic tasks, maintains geometric fidelity far beyond the Lyapunov horizon on Lorenz-63, and remains competitive on real-world LTSF. The codebase also releases our controlled-shock benchmark and data-cleaning protocol.

Primary Area: learning on time series and dynamical systems

Submission Number: 4624