Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes

Recent advances in probabilistic modeling, including diffusion models and flow matching, have improved high-dimensional data generation. However, applying these to complex dynamical systems like Navier-Stokes or video sequences remains challenging due to the difficulty in conditional sampling and uncertainty quantification. Traditional methods often rely on linear approximations or low-dimensional assumptions, limiting their applicability. The integration of stochastic interpolants and Schrödinger bridge concepts offers a promising avenue, enabling flexible, theoretically grounded high-dimensional sampling. Prior works such as score-based diffusion (Ho et al., 2020) and neural flow models (Lipman et al., 2022) have laid foundational frameworks, but lack explicit post-hoc tuning and path measure optimality guarantees. This work builds upon these foundations, addressing the gap in scalable, high-fidelity probabilistic forecasting for complex systems.

The core challenge is to develop a generative model capable of accurately sampling the conditional distribution of future states given current observations in high-dimensional, nonlinear, and potentially stochastic systems. Existing deterministic predictors fail to capture uncertainty, especially in chaotic regimes. Probabilistic models like VAEs or GANs struggle with multi-modality and stability in high dimensions. The difficulty lies in designing a mathematically rigorous, computationally feasible approach that can adaptively tune parameters post-training, handle multi-modal distributions, and provide theoretical guarantees of optimality. Achieving this in systems like Navier-Stokes or video sequences, where data is high-dimensional and dynamics are complex, remains an open problem.

This work introduces a novel probabilistic forecasting framework that combines stochastic interpolants with Föllmer processes, leveraging entropy-regularized optimal transport principles. Key innovations include: 1) constructing a stochastic interpolant that bridges point mass and target distribution; 2) learning drift functions via square loss regression, ensuring efficiency; 3) tuning diffusion coefficients after training to optimize path measure KL divergence; 4) deriving a theoretical guarantee that the resulting process is a Föllmer process, minimizing entropy relative to the Wiener process. This approach allows flexible, high-dimensional conditional sampling with strong theoretical backing, surpassing prior diffusion or flow-matching models in stability and interpretability.

• �� Define a stochastic interpolant \( I_s = \alpha_s x_0 + eta_s x_1 + \sigma_s W_s \), bridging current and future states.
• �� Collect samples \((x_0, x_1)\) from data, estimate drift \(b_s(x, x_0)\) via squared loss regression.
• �� Formulate an SDE \( dX_s = b_s(X_s, x_0) ds + \sigma_s dW_s \), ensuring \(X_0 = x_0\) and \(X_1 \sim ho_c(\cdot|x_0)\).
• �� Extend the model with tunable diffusion \(g_s\), deriving \(b_s^{g}(x, x_0)\) using the score \(

abla \log ho_s\).

• �� Optimize \(g_s\) via KL divergence minimization, leading to the Föllmer process with theoretical guarantees.
• �� Implement algorithms for training (regression-based drift estimation) and sampling (SDE integration), enabling iterative, autoregressive forecasting.

The model was validated on Navier-Stokes turbulence data and video datasets (KTH, CLEVRER). Training involved learning drift functions from simulated trajectories, with hyperparameters tuned for stability. Performance metrics included energy spectrum accuracy, prediction error, and sample diversity. Results showed over 20% error reduction in spectral reconstruction, 15% improvement in video prediction accuracy, and successful autoregressive iteration without retraining. Ablation studies confirmed the importance of post-hoc diffusion tuning and the theoretical KL guarantees. The experiments demonstrated robustness in high-dimensional, multi-modal scenarios, validating the framework's scalability and effectiveness.

The approach accurately reconstructed the enstrophy spectrum in Navier-Stokes simulations, outperforming classical models by 20%. In video prediction, it achieved 15% lower error and generated diverse, realistic future frames. The KL divergence minimized by the diffusion tuning confirmed theoretical predictions, and autoregressive iterations maintained high fidelity beyond initial training data. These results highlight the method's capacity to handle complex, high-dimensional probabilistic forecasting with theoretical rigor and practical robustness.

Applicable to climate modeling, fluid dynamics, and multimedia synthesis, the framework enables accurate, diverse future state sampling from limited data. Its ability to incorporate uncertainty and multi-modality makes it ideal for real-time prediction, risk assessment, and scenario analysis. The method's flexibility allows integration with existing simulation pipelines, providing a powerful tool for scientific discovery and industrial applications. Future extensions could include adaptive parameter tuning and multi-scale modeling for even broader impact.

The method requires substantial training data for high accuracy, especially in non-stationary or sparse regimes. Post-training diffusion tuning adds computational overhead, limiting real-time deployment. Its reliance on known system dynamics constrains applicability to unknown or highly nonlinear systems. Further research is needed to improve data efficiency, extend to non-stationary environments, and reduce computational costs for broader practical use.