Aligning Exact-Likelihood Normalizing Flows with Physics-Aware Score Denoising

Normalizing flows offer expressive conditional generation and exact likelihood evaluation, making them attractive for scientific field reconstruction. However, practical reconstruction often relies on score-denoised outputs, whose interpretation depends on the density learned during training. Incorporating physical constraints changes this density, creating a potential mismatch between the training objective and the denoising rule. We introduce PAIM-Flow, a framework for aligning exact-likelihood normalizing flows with physics-aware score denoising. PAIM-Flow combines maximum likelihood with physical risk evaluated on fields generated through the inverse flow, while reserving denoising for inference. At the conditional population level, we characterize the density tilt induced by physical supervision and derive an objective-dependent score correction. This analysis clarifies the assumptions required for matched denoising and distinguishes the exact likelihood of the underlying flow from the distribution of denoised outputs. Experiments on Darcy pressure reconstruction, conditional Poisson problems, and sparse turbulent field reconstruction demonstrate substantial improvements in reconstruction accuracy over the evaluated baselines. Controlled ablations disentangle the effects of conditional modeling, physical supervision, and inference-time denoising, separating overall reconstruction gains from the contribution of score correction. Our work provides a principled foundation for connecting physics-aware training objectives with score-based inference in exact-likelihood generative models.