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

Dibyajyoti Chakraborty, Hojin Kim, Romit Maulik

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.

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

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.
Paper Structure (34 sections, 2 theorems, 41 equations, 16 figures, 1 table)

This paper contains 34 sections, 2 theorems, 41 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related Works
  3. Methods
  4. Diffusion-based Generative Modeling
  5. Single and Multi-step Training
  6. Single-Step Denoising Loss.
  7. Multi-Step Autoregressive Loss.
  8. Conditional Generation
  9. Graph Diffusion Architecture
  10. Input Embedding and Noise Conditioning.
  11. Graph Transformer Block.
  12. Hierarchical Pooling and Unpooling.
  13. Output and Preconditioning.
  14. Sensor Placement
  15. Random Placement
  16. ...and 19 more sections

Key Result

Proposition 3.1

For a chaotic surrogate model with a one-step error $\epsilon \in \{\epsilon_{det}, \epsilon_{prob}\}$ and the Wasserstein-Lipschitz constant $L>1$, where $L \in \{L_{det}, L_{prob}\}$, the prediction error satisfies

Figures (16)

  • Figure 1: U velocity at different timesteps of forecast for single-step vs multi-step diffusion training along with the DNS ground truth.
  • Figure 2: Mean absolute error from DNS at different timesteps of forecast for single-step vs multi-step diffusion training.
  • Figure 3: U velocity at different timesteps of forecast for different sensor placement techniques. The sensor locations are shown as black dots.
  • Figure 4: Mean absolute error from DNS at different timesteps of forecast for different sensor placement techniques. Sensor placements improve the forecast. These values are for 50 sensor points with a minimum distance of 15 grid points between them.
  • Figure 5: Plots for the mean absolute error from DNS at different timesteps of forecast for different gap between($g$) and number of sensor points($s$). More sensor points and larger gaps between sensors lead to lower prediction error.
  • ...and 11 more figures

Theorems & Definitions (4)

  • Proposition 3.1
  • proof
  • Proposition 3.2: Monotone reduction with sensor count
  • proof