Ellipsoidal Time Series Forecasting

We argue that long-horizon forecasting requires learning local Jacobians with explicit spectral structure, not only matching conditional means. Our method, Fern (\textbf{F}orecasting with \textbf{E}llipsoidal \textbf{R}epresentatio\textbf{N}s), invokes Brenier's theorem to directly parameterize the Jacobian as a symmetric positive semi-definite (SPD) factorization, treating forecasting as the optimal transport of probability mass from a fixed Gaussian source to data-dependent ellipsoids. This formulation avoids post-hoc eigendecomposition of dense Jacobians, enables efficient Householder-based orthogonal factors, and exposes interpretable diagnostics such as local stretching, spectral radius, and volume change. To rigorously evaluate robustness, we introduce controlled synthetic stress tests with nonstationary shocks, together with Wasserstein-based shape metrics and Effective Prediction Time. Fern demonstrates exceptional stability, outperforming baselines like DLinear and Koopa by over two orders of magnitude (up to $790\times$) on nonstationary settings where standard benchmarks fail to expose model brittleness.

Forecasting models are often compared on a small set of standard benchmark datasets reused across many papers. The trouble is that these datasets may not reveal how models behave when the world changes. A model can look strong on historical tables and still break when the system shifts, when a shock hits, or when small errors snowball into large ones. We take a different approach: forecasting should be tested more like a controlled scientific experiment. We build a benchmark of 21 simulated systems where the underlying dynamics are known exactly, and where chaos, sudden shocks, and regime changes can be introduced on purpose. This lets us ask not just “which model has the lowest average error?” but “which model is still useful when the world changes?” We also develop Fern, a forecasting model that predicts not just a number but a shape: an ellipse describing where future outcomes may stretch or shrink. On these stress tests, several popular methods fail sharply, sometimes catastrophically. Fern is often much more stable, with errors more than a hundred times smaller in the most brittle cases, and its ellipses help signal when forecasts should no longer be trusted. The broader point is that forecasting models should be judged by their failure modes, not only by average accuracy on quiet historical data.