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.

中文翻译：

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使用引导扩散模型和虚拟观测对参数偏微分方程解进行生成学习


我们引入了一个生成学习框架，以使用梯度引导和虚拟观察对高维参数系统进行建模。我们考虑由偏微分方程 （PDE） 描述的系统，这些系统使用结构化或非结构化网格进行离散化。该框架集成了多级信息，以生成系统动力学的高保真时间序列。我们通过两个案例研究来展示我们框架的有效性和多功能性，分别是非结构化网格上的不可压缩、二维、低雷诺圆柱流和结构化网格上的不可压缩湍流通道流，两者都由雷诺数参数化。我们的结果说明了该框架的稳健性和在各种参数设置中生成准确流序的能力，显著降低了计算成本，从而实现了流动力学的高效预测和重建。