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 on 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：诸如去噪扩散模型等生成模型正在快速提升其逼近高度复杂数据分布的能力。它们也日益被应用于科学机器学习领域，其中隐含数据分布的样本需要遵循特定的控制方程。我们提出了一种框架，在模型训练过程中将底层约束信息融入去噪扩散模型。该方法提升了生成样本与施加约束的一致性，且在不影响推理速度的情况下显著优于现有方法。此外，我们的研究结果表明，在训练过程中纳入此类约束可提供一种自然的正则化手段以防范过拟合。该框架易于实现且具有广泛适用性，可施加等式约束、不等式约束以及辅助优化目标。