MercerFlow: Flow Matching in a Kernel-Induced Latent Space for Probabilistic Forecasting

Probabilistic flow matching for time series forecasting usually absorbs local correlation with a sequential architecture—an RNN, S4, or a Transformer—rather than a global MLP that maps an entire history to an entire horizon. That sequential backbone costs GPU memory and time per epoch. A cheaper alternative is latent flow matching: embed the time series via an invertible map to a single latent vector and learn the flow there, so a tabular MLP can treat the series as one coordinate vector. However, choosing that space remains challenging: a Fourier basis of the forecast window does not diagonalise a Gaussian-process prior on that window, a truncated signature is difficult to invert, and a learned encoder adds its own training cost. We introduce MercerFlow, which learns conditional flow matching in the orthonormal eigenbasis of a Gaussian-process kernel matrix: the latent covariance is diagonal under that kernel and the map is linearly invertible. Across six datasets, mean CRPS is lower than for sequential-architecture flow matching. On the long-window sets the MLP uses 20–60$\times$ less training memory and 3–7$\times$ less time per epoch (up to 19$\times$ on Solar).