FORT: Forward-Only Regression Training of Normalizing Flows

Simulation-free training frameworks have been at the forefront of the generative modelling revolution in continuous spaces, leading to neural dynamical systems that encompass modern large-scale diffusion and flow matching models. Despite the scalability of training, the generation of high-quality samples and their corresponding likelihood under the model requires expensive numerical simulation -- inhibiting adoption in numerous scientific applications such as equilibrium sampling of molecular systems. In this paper, we revisit classical normalizing flows as one-step generative models with exact likelihoods and propose a novel, scalable training objective that does not require computing the expensive change of variable formula used in conventional maximum likelihood training. We propose Forward-Only Regression Training (FORT), a simple $\ell_2$-regression objective that maps prior samples under our flow to specifically chosen targets. We demonstrate that FORT supports a wide class of targets, such as optimal transport targets and targets from pre-trained continuous-time normalizing flows (CNF). We further demonstrate that by using CNF targets, our one-step flows allow for larger-scale training that exceeds the performance and stability of maximum likelihood training, while unlocking a broader class of architectures that were previously challenging to train. Empirically, we elucidate that our trained flows can perform equilibrium conformation sampling in Cartesian coordinates of alanine dipeptide, alanine tripeptide, and alanine tetrapeptide.

@article{rehman2025_2506.01158,
  title={ FORT: Forward-Only Regression Training of Normalizing Flows },
  author={ Danyal Rehman and Oscar Davis and Jiarui Lu and Jian Tang and Michael Bronstein and Yoshua Bengio and Alexander Tong and Avishek Joey Bose },
  journal={arXiv preprint arXiv:2506.01158},
  year={ 2025 }
}