Topology Guidance: Controlling the Outputs of Generative Models via Vector Field Topology

For domains that involve numerical simulation, it can be computationally expensive to run an ensemble of simulations spanning a parameter space of interest to a user. To this end, an attractive surrogate for simulation is the generative modeling of fields produced by an ensemble, allowing one to synthesize fields in a computationally cheap, yet accurate, manner. However, for the purposes of visual analysis, a limitation of generative models is their lack of control, as it is unclear what one should expect when sampling a field from a model. In this paper we study how to make generative models of fields more controllable, so that users can specify features of interest, in particular topological features, that they wish to see in the output. We propose topology guidance, a method for guiding the sampling process of a generative model, specifically a diffusion model, such that a topological description specified as input is satisfied in the generated output. Central to our method, we couple a coordinate-based neural network used to represent fields, with a diffusion model used for generation. We show how to use topologically-relevant signals provided by the coordinate-based network to help guide the denoising process of a diffusion model. This enables us to faithfully represent a user's specified topology, while ensuring that the output field remains within the generative data distribution. Specifically, we study 2D vector field topology, evaluating our method over an ensemble of fluid flows, where we show that generated vector fields faithfully adhere to the location, and type, of critical points over the spatial domain. We further show the benefits of our method in aiding the comparison of ensembles, allowing one to explore commonalities and differences in distributions along prescribed topological features.