Universal Learning of Stochastic Dynamics for Exact Belief Propagation using Bernstein Normalizing Flows

2025-09-18

AI Summary

Analytical belief propagation remains intractable for unknown nonlinear stochastic systems due to the lack of closed-form state distribution evolution. Method: We propose a data-driven general dynamic modeling framework that uniquely integrates the expressive power of normalizing flows with the analytical integrability of Bernstein polynomials. The framework parameterizes nonlinear dynamics via neural networks and approximates the transition kernel using Bernstein polynomials, enabling exact closed-form belief propagation; it is trained end-to-end via variational inference. Contribution/Results: Evaluated on highly nonlinear systems with non-additive, non-Gaussian noise, our approach significantly improves long-horizon distributional prediction accuracy and reduces propagation error. It overcomes the fundamental analytical intractability bottleneck faced by existing data-driven methods in non-Gaussian, nonlinear settings—enabling rigorous, differentiable, and interpretable probabilistic state estimation without Monte Carlo approximation.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Predicting the distribution of future states in a stochastic system, known as belief propagation, is fundamental to reasoning under uncertainty. However, nonlinear dynamics often make analytical belief propagation intractable, requiring approximate methods. When the system model is unknown and must be learned from data, a key question arises: can we learn a model that (i) universally approximates general nonlinear stochastic dynamics, and (ii) supports analytical belief propagation? This paper establishes the theoretical foundations for a class of models that satisfy both properties. The proposed approach combines the expressiveness of normalizing flows for density estimation with the analytical tractability of Bernstein polynomials. Empirical results show the efficacy of our learned model over state-of-the-art data-driven methods for belief propagation, especially for highly non-linear systems with non-additive, non-Gaussian noise.

Problem

• Learning universally approximating models for nonlinear stochastic dynamics
• Enabling analytical belief propagation from learned system models
• Combining normalizing flows and Bernstein polynomials for tractability

Innovation

• Combining normalizing flows with Bernstein polynomials
• Enabling analytical belief propagation for nonlinear dynamics
• Learning universal stochastic dynamics from data