Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations
This work addresses the computational expense of simulating parametric PDEs for applications like fluid dynamics, offering an efficient forecasting and reconstruction method, though it appears incremental as it builds on existing diffusion models.
The paper tackles the problem of modeling high-dimensional parametric systems described by PDEs by introducing a generative learning framework that uses guided diffusion models and virtual observations, resulting in accurate flow sequence generation across parameter settings with significantly reduced computational costs.
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.