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

TL;DR

The paper tackles the challenge of controlling diffusion-model outputs for fields to realize user-specified topologies, enabling targeted analysis of simulation ensembles. It introduces topology guidance, which couples a field-conditioned INR (Functa/SIREN) with a diffusion model and uses differentiable energy terms derived from local derivatives (e.g., $oldsymbol{ ext{E}}_c=ig Vert f_{oldsymbol{ heta}}(oldsymbol{p};oldsymbol{\hat{z}}_t)ig Vert$, Jacobian eigenvalues, and stability measures) to steer denoising via $ ilde{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ abla}}}}}$. The contributions include a training-free mechanism to preserve specified critical points (location), type (sink/source/saddle), and stability, support for multiple points, and demonstration on a 2D fluid-flow ensemble showing high alignment (>$85 ext{%}$) with topology inputs while maintaining distribution fidelity (FD baseline $ ext{FD}=8.84$). This work enables topology-aware exploration and comparison of simulation ensembles, with potential extensions to higher resolution, 3D fields, or other feature types such as vorticity or Morse-Smale complexes, thus providing a flexible framework for guided generative analysis in computational science.

Abstract

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.

Figures (9)

• Figure 1: We present an overview of the model we use for topology guidance. (A) A diffusion model operates in a compact latent space, performing forward and denoising diffusion. (B) Mapping sampled latent vectors back to the data space using a pre-trained SIREN. The latent vector modulates the SIREN, which takes an arbitrary position in the flow field as input and outputs the vector field value at that point.
• Figure 2: Localized effect of guidance on vector field generation. Left: Sample without guidance. Right: Sample with guidance. The line graph shows the norm change at the user-specified location over denoising timesteps. Guidance is injected at time step 600. After prescribing a critical point through guidance, the norm at the specified location significantly decreases to near 0.
• Figure 3: Our method provides a general framework for drawing vector fields from a generative model containing a variety of topology specifications, namely critical points. The figure summarizes five configurations of vector field critical pointsHelmanVisualVectorTopo1991, along with examples, necessary constraints, and corresponding energy functions in our approach.
• Figure 4: Manipulating critical point types using an energy function inspired by the sigmoid function. By optimizing the energy function to control the eigenvalues signs of the Jacobian matrix at specified locations, a guide can generate a sink (left) or a source (right).
• Figure 5: An energy function that applied to norms, eigenvalues, and $\Delta$ can work independently. Moreover, it is also possible to combine these components to create a more comprehensive energy function.