FlowTSFM: Test-Time Scalable Time Series Forecasting with Recurrent Quantile Flows

Encoder-based time series foundation models (TSFMs) stack independently parameterized Transformer layers whose intermediate states carry no predictive meaning. We introduce FlowTSFM, which recasts encoder depth as transport over predictive quantiles. A single weight-shared Transformer block is integrated as a depth-conditioned residual update, and a shared decoder reads out a full quantile forecast at every step. A quantile-flow objective, inspired by flow matching, combines pinball supervision with a path term that moves the decoded quantiles along a straight line from an analytic Gaussian source to the terminal forecast. We prove that this position-matching term has the same zero-loss trajectories as regressing decoded velocities onto the constant flow-matching velocity. With 38.8M parameters, FlowTSFM is competitive with state-of-the-art TSFMs on GIFT-Eval and TIME and comes within 3–5% of the 119.5M-parameter Chronos-2. Its decoded updates consistently point toward the final forecast (CosMean 0.92, versus 0.35 for layer-wise probes of Chronos-2), and in our setup training without the path term diverged. A complete $5\times 5$ training/inference depth study shows that fixed-depth training specializes the recurrence to its budget. Training with randomized depth removes this specialization: a single checkpoint improves with inference depth and matches or exceeds depth-specialized models across budgets.