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.

本研究提出了一种生成学习框架，以建模高维参数系统，解决了传统方法在处理偏微分方程（PDEs）时的局限性。该框架结合了多层次信息，有效生成系统动态的高保真时间序列，显著降低计算成本，提高预测和再现流动动态的效率。

利用引导扩散模型和虚拟观测的参数偏微分方程解的生成学习