Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework to inform denoising diffusion models of underlying constraints on such generated samples during model training. Our approach improves the alignment of the generated samples with the imposed constraints and significantly outperforms existing methods without affecting inference speed. Additionally, our findings suggest that incorporating such constraints during training provides a natural regularization against overfitting. Our framework is easy to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives.

翻译：去噪扩散模型等生成模型正迅速提升其逼近高度复杂数据分布的能力。这些模型在科学机器学习中也日益受到重视，因为从隐含数据分布中采样的样本需要遵循特定的控制方程。我们提出了一种框架，在模型训练期间使去噪扩散模型能够感知生成样本所遵循的底层约束。该方法提升了生成样本与既定约束的一致性，在保持推理速度不变的前提下显著优于现有方法。此外，我们的研究表明，在训练过程中引入此类约束可自然提供防止过拟合的正则化效果。本框架易于实现，且能灵活适用于等式与不等式约束以及辅助优化目标的施加。