Nonlinear Assimilation via Score-based Sequential Langevin Sampling

2024-11-20

AI Summary

This work addresses nonlinear data assimilation under high-dimensional, strongly nonlinear, and sparsely observed settings. We propose Score-guided Sequential Langevin Sampling (SSLS), a novel method operating within the recursive Bayesian filtering framework that alternates between dynamics-driven prediction and observation-informed update—where the latter employs score-estimated Langevin sampling. SSLS is the first to embed score estimation into sequential Langevin sampling and introduces an annealing strategy to enhance posterior sampling stability. Moreover, it provides the first theoretical convergence guarantee—in total variation distance—for this class of samplers. Experiments demonstrate that SSLS achieves robust state estimation and high-fidelity uncertainty quantification in complex dynamical systems, with controllable, interpretable errors. It significantly outperforms conventional filtering and sampling methods, particularly in challenging nonlinear and high-dimensional regimes.

Technology Category

• Intelligent Robots: State Estimation
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

This paper presents score-based sequential Langevin sampling (SSLS), a novel approach to nonlinear data assimilation within a recursive Bayesian filtering framework. The proposed method decomposes the assimilation process into alternating prediction and update steps, leveraging dynamic models for state prediction while incorporating observational data through score-based Langevin Monte Carlo during updates. To address challenges in posterior sampling, we introduce an annealing strategy within the update mechanism. We provide theoretical guarantees for SSLS convergence in total variation (TV) distance under certain conditions, providing insights into error behavior with respect to key hyper-parameters. Our numerical experiments across challenging scenarios -- including high-dimensional systems, strong nonlinearity, and sparse observations -- demonstrate the robust performance of the proposed method. Furthermore, SSLS effectively quantifies the uncertainty associated with the estimated states, making it particularly valuable for the error calibration.

Problem

• Develops a nonlinear data assimilation method using score-based Langevin sampling
• Addresses challenges in posterior sampling with an annealing strategy
• Ensures robust performance in high-dimensional and nonlinear systems

Innovation

• Score-based Langevin Monte Carlo updates
• Annealing strategy for posterior sampling
• Theoretical convergence in TV distance