Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems

2026-03-13

AI Summary

This work addresses the challenge of high-fidelity simulation of high-dimensional chaotic nonlinear dynamical systems, which is computationally prohibitive and difficult to capture with conventional deterministic surrogate models due to their inability to represent intrinsic uncertainties. The authors propose a probabilistic surrogate framework based on diffusion models, integrating a multi-scale graph Transformer with voxel-grid pooling and employing a multi-step autoregressive diffusion training strategy to enable long-term stable predictions. Furthermore, they introduce a retraining-free diffusion posterior sampling mechanism that facilitates uncertainty-aware dynamic sensor placement and efficient data assimilation. Demonstrated on two-dimensional homogeneous isotropic turbulence and backward-facing step flows, the method achieves accurate long-horizon forecasting, adaptive observation layout, and real-time state correction.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

High-fidelity numerical simulations of chaotic, high dimensional nonlinear dynamical systems are computationally expensive, necessitating the development of efficient surrogate models. Most surrogate models for such systems are deterministic, for example when neural operators are involved. However, deterministic models often fail to capture the intrinsic distributional uncertainty of chaotic systems. This work presents a surrogate modeling formulation that leverages generative machine learning, where a deep learning diffusion model is used to probabilistically forecast turbulent flows over long horizons. We introduce a multi-step autoregressive diffusion objective that significantly enhances long-rollout stability compared to standard single-step training. To handle complex, unstructured geometries, we utilize a multi-scale graph transformer architecture incorporating diffusion preconditioning and voxel-grid pooling. More importantly, our modeling framework provides a unified platform that also predicts spatiotemporally important locations for sensor placement, either via uncertainty estimates or through an error-estimation module. Finally, the observations of the ground truth state at these dynamically varying sensor locations are assimilated using diffusion posterior sampling requiring no retraining of the surrogate model. We present our methodology on two-dimensional homogeneous and isotropic turbulence and for a flow over a backwards-facing step, demonstrating its utility in forecasting, adaptive sensor placement, and data assimilation for high dimensional chaotic systems.

Problem

• nonlinear dynamical systems
• chaotic systems
• surrogate modeling
• uncertainty quantification
• data assimilation

Innovation

• diffusion posterior sampling
• autoregressive diffusion
• multi-scale graph transformer
• adaptive sensor placement
• data assimilation