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

TL;DR

This paper introduces a probabilistic surrogate modeling approach using diffusion models for complex nonlinear chaotic systems, enabling long-term forecasting, adaptive sensor placement, and data assimilation without retraining.

Contribution

It develops a multi-step autoregressive diffusion framework with a multi-scale graph transformer for stable long-horizon predictions and dynamic sensor placement in chaotic systems.

Findings

Enhanced long-rollout stability with multi-step diffusion training

Effective forecasting of turbulent flows and flow over a backward-facing step

Successful data assimilation at dynamically chosen sensor locations

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.