An Analysis of Regularization and Fokker-Planck Residuals in Diffusion Models for Image Generation

arXiv cs.CV / 4/17/2026

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Key Points

The paper studies diffusion image generation models trained with the denoising score matching (DSM) objective and finds that they can violate the Fokker–Planck (FP) equation governing true data density dynamics.
It shows that directly penalizing FP deviations in the training objective reduces FP residuals but can incur substantial computational overhead, and that strict FP adherence does not always improve sample quality.
The authors evaluate multiple lightweight (simpler) regularizers that target FP residuals and empirically measure their impact on both FP residual magnitude and generation quality.
The results indicate that FP regularization can deliver much of the benefit at significantly lower computational cost, with better performance often coming from weaker regularization rather than strict enforcement.
The work includes released code for reproducing and extending the analysis (GitHub link provided in the paper).

Abstract

Recent work has shown that diffusion models trained with the denoising score matching (DSM) objective often violate the Fokker--Planck (FP) equation that governs the evolution of the true data density. Directly penalizing these deviations in the objective function reduces their magnitude but introduces a significant computational overhead. It is also observed that enforcing strict adherence to the FP equation does not necessarily lead to improvements in the quality of the generated samples, as often the best results are obtained with weaker FP regularization. In this paper, we investigate whether simpler penalty terms can provide similar benefits. We empirically analyze several lightweight regularizers, study their effect on FP residuals and generation quality, and show that the benefits of FP regularization are available at substantially lower computational cost. Our code is available at https://github.com/OnnoNiemann/fp_diffusion_analysis.