  The computational design of novel protein structures has the potential to
impact numerous scientific disciplines greatly. Toward this goal, we introduce
FoldFlow, a series of novel generative models of increasing modeling power
based on the flow-matching paradigm over $3\mathrm{D}$ rigid motions -- i.e.
the group $\text{SE}(3)$ -- enabling accurate modeling of protein backbones. We
first introduce FoldFlow-Base, a simulation-free approach to learning
deterministic continuous-time dynamics and matching invariant target
distributions on $\text{SE}(3)$. We next accelerate training by incorporating
Riemannian optimal transport to create FoldFlow-OT, leading to the construction
of both more simple and stable flows. Finally, we design FoldFlow-SFM, coupling
both Riemannian OT and simulation-free training to learn stochastic
continuous-time dynamics over $\text{SE}(3)$. Our family of FoldFlow,
generative models offers several key advantages over previous approaches to the
generative modeling of proteins: they are more stable and faster to train than
diffusion-based approaches, and our models enjoy the ability to map any
invariant source distribution to any invariant target distribution over
$\text{SE}(3)$. Empirically, we validate FoldFlow, on protein backbone
generation of up to $300$ amino acids leading to high-quality designable,
diverse, and novel samples.
