Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations

TL;DR

A generative learning framework to model high-dimensional parametric systems using gradient guidance and virtual observations is introduced, significantly reducing computational costs allowing for efficient forecasting and reconstruction of flow dynamics.

Abstract

We introduce a generative learning framework to model high-dimensional parametric systems using gradient guidance and virtual observations. We consider systems described by Partial Differential Equations (PDEs) discretized with structured or unstructured grids. The framework integrates multi-level information to generate high fidelity time sequences of the system dynamics. We demonstrate the effectiveness and versatility of our framework with two case studies in incompressible, two dimensional, low Reynolds cylinder flow on an unstructured mesh and incompressible turbulent channel flow on a structured mesh, both parameterized by the Reynolds number. Our results illustrate the framework's robustness and ability to generate accurate flow sequences across various parameter settings, significantly reducing computational costs allowing for efficient forecasting and reconstruction of flow dynamics.

Figures (22)

• Figure 1: A general overview of the unstructured mesh (11644 cells) for cylinder flow.
• Figure 2: The reduced graph (1024 nodes) from the original mesh in Figure \ref{['fig:fpc_mesh']} using the GNN auto-encoder.
• Figure 3: An overview of a micro-level state (${\bm{u}}$), its corresponding macro-level state (${\bm{z}}$) and decoded back to its micro-level ($D_{{\boldsymbol{\theta}}^*_\mathrm{CNN}}({\bm{z}})$): velocity magnitude (top row, left), pressure (top row, right), macro-level state components 1 to 4 (middle row, from left to right), and decoded velocity magnitude (bottom row, left), decoded pressure (bottom row, right). For visualization purposes, the colorbar is omitted.
• Figure 4: Velocity magnitude of the viscous flow at three test Reynolds numbers, as generated by the diffusion model and compared with results from DNS.
• Figure 5: Initial velocity magnitude of the viscous flow at three test Reynolds numbers, as generated by the diffusion model and compared with results from DNS.