  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.
