GeoPert: Perturbation Prediction as Geometry-Constrained Distribution Modeling

Predicting transcriptomic responses to genetic perturbations is a fundamental objective of virtual cell modeling. However, because single-cell sequencing is inherently destructive, cellular states cannot be tracked longitudinally within the same physical cell. Instead, assays such as Perturb-seq yield snapshot measurements across cell populations, framing perturbation prediction as a conditional distribution generative modeling problem. Under this formulation, $\text{in silico}$ generation follows a two-level hierarchical generative process:$\text{1) Population-level sampling}$: A condition-specific cell distribution $P_c$ is drawn from a meta-distribution ${\mathcal{P}}_c$ ($P_c\sim {\mathcal{P}}_c$) that governs biological and technical variability across replicates, including incubation fluctuations and sequencing batch effects. $\text{2) Cell-level sampling}$: Individual cellular profiles are subsequently sampled from this realized population distribution ($x\sim P_c$). Because ${\mathcal{P}}_c$ defines an abstract distribution over probability measures, typically characterized via kernel mean embeddings in a latent RKHS, most existing frameworks collapse this hierarchy. By approaching in silico generation as a single-step marginal sampling task, they fail to account for replicate-level distributional shifts and are inherently incompatible with large-scale data augmentation schemes. To address these limitations, we propose GeoPert, a hierarchical virtual cell generative framework. GeoPert indirectly characterizes a cell population distribution in the latent RKHS by its energy distances, defined by the mean geodesic $\text{Fisher-Rao}$ distances over a statistical manifold, against a small set of prototypical anchor populations. For a specific condition, GeoPert first generate its post-perturbation population distribution signature through conditional diffusion, supported by a foundational diffusion model via Classifier-Free Guidance (CFG). Virtual cells of a post-perturbation population are then generated through a variational inference framework conditioned on the energy distance vector signature of the population. By decoupling the virtual cell generation into a two-stage hierarchical generative process, GeoPert naturally support large-scale data augmentation, by harnessing large scale synthetic cell populations in the transcriptomic space with matching energy distance vectors, which improves cell generation authenticity. Benchmarks on public Perturb-seq datasets show that GeoPert achieves state-of-the-art performance in single-cell perturbation response prediction and is capable of generalizing to unseen gene perturbations.