From latent dynamics to data geometry: Nonlinear diffusion modelling for protein structures

The role and impact of nonlinearity in generative diffusion models remain largely unexplored and theoretically underdeveloped. In this work, we introduce ProT-INDM, an implicit nonlinear diffusion framework that uses invertible normalizing flows to induce nonlinear data-space dynamics from tractable latent SDEs and employs probability-flow ODEs for exact likelihood computation. This design provides a principled and flexible methodology for modelling complex protein backbone distributions, forming the first tractable modelling scheme for training nonlinear diffusion models. This general framework bridges nonlinear SDEs with score-based generative modelling, providing both theoretical insights and empirical improvements over linear diffusion model baselines.