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

TL;DR

This paper presents a generative learning framework using guided diffusion models and virtual observations to efficiently model and predict high-dimensional parametric PDE systems, demonstrated on fluid flow simulations.

Contribution

It introduces a novel framework combining gradient guidance and virtual observations for high-dimensional parametric PDE modeling, improving accuracy and computational efficiency.

Findings

Accurately predicts flow dynamics across different Reynolds numbers.

Reduces computational costs significantly compared to traditional methods.

Demonstrates robustness on unstructured and structured meshes.

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.