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Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes
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Proposes probabilistic forecasting via stochastic interpolants: learns a drift by square-loss regression and an SDE initialized at the current state samples the conditional distribution of the next state, with a post-hoc tunable diffusion yielding a Föllmer process.
Problem
Probabilistic forecasting of dynamical systems: sampling the conditional distribution of the future state given the current state, from time-series data.
Approach
Use stochastic interpolants to build an SDE that maps a point mass at the current state to the conditional distribution of the next state; the drift is learned by square-loss regression, and the diffusion coefficient can be tuned after training, with a specific choice giving a Föllmer process that minimizes the KL estimation-error impact.
Key results
- On KTH at 250k steps, PFI reaches FVD 39.13 vs RIVER 41.88; on CLEVRER at 250k steps, 39.31 vs 48.96.
- Navier-Stokes forecasts reproduce the true enstrophy spectrum, including from 32x32 low-resolution inputs to 128x128 output.
- Learned drift can be reused with post-hoc diffusion tuning without retraining.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use PFI instead of RIVER | Video forecasting · KTH, 100k gradient steps | FVD: 46.69 -> 44.38 | Helped |
| Use PFI instead of RIVER | Video forecasting · KTH, 250k gradient steps | FVD: 41.88 -> 39.13 | Helped |
| Use PFI instead of RIVER | Video forecasting · CLEVRER, 100k gradient steps | FVD: 60.40 -> 54.7 | Helped |
| Use PFI instead of RIVER | Video forecasting · CLEVRER, 250k gradient steps | FVD: 48.96 -> 39.31 | Helped |
| Use PFI instead of RIVER | Video forecasting · KTH, 100k gradient steps | Shifted FVD: 13.24 -> 10.93 | Helped |
| Use PFI instead of RIVER | Video forecasting · KTH, 250k gradient steps | Shifted FVD: 8.43 -> 5.68 | Helped |
| Use PFI instead of RIVER | Video forecasting · CLEVRER, 100k gradient steps | Shifted FVD: 57.61 -> 51.91 | Helped |
| Use PFI instead of RIVER | Video forecasting · CLEVRER, 250k gradient steps | Shifted FVD: 46.17 -> 36.52 | Helped |
Limitations
- Experiments use g_s = sigma_s; the Föllmer diffusion choice is left for future work.
- Video experiments depend on VQGAN latent space, bounding FVD by the autoencoder's FVD.
- Assumes stationary Markovian time series for main theory.
Experimental setup
Datasets: KTH CLEVRER stochastically forced 2d Navier-Stokes (simulated) multi-modal jump-diffusion (synthetic)
Benchmarks: KTH CLEVRER 2d Navier-Stokes enstrophy spectrum
Metrics: FVD shifted FVD enstrophy spectrum KDE comparison
Baselines: RIVER flow matching deterministic forecasting
Builds on: stochastic interpolants flow matching föllmer process RIVER VQGAN
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