The reconstruction of unsteady flow fields from limited measurements is a challenging and crucial task for many engineering applications. Machine learning models are gaining popularity for solving this problem due to their ability to learn complex patterns from data and to generalize across diverse conditions. Among these, diffusion models have emerged as being particularly powerful for generative tasks, producing high-quality samples by iteratively refining noisy inputs. In contrast to other methods, these generative models are capable of reconstructing the smallest scales of the fluid spectrum. In this work, we introduce a novel sampling method for diffusion models that enables the reconstruction of high-fidelity samples by guiding the reverse process using the available sparse data. Moreover, we enhance the reconstructions with available physics knowledge using a conflict-free update method during training. To evaluate the effectiveness of our method, we conduct experiments on 2 and 3-dimensional turbulent flow data. Our method consistently outperforms other diffusion-based methods in predicting the fluid's structure and in pixel-wise accuracy. This study underscores the remarkable potential of diffusion models in reconstructing flow field data, paving the way for leveraging them in fluid dynamics research and applications ranging from super-resolution to reconstructions of experiments.

翻译：从有限测量数据中重建非定常流场是许多工程应用中一项具有挑战性且至关重要的任务。机器学习模型因其能够从数据中学习复杂模式并在不同条件下泛化，正日益广泛地应用于解决此类问题。其中，扩散模型在生成任务中展现出尤为强大的能力，通过迭代优化含噪声输入来生成高质量样本。与其他方法相比，这类生成模型能够重建流体频谱中最小的尺度。在本研究中，我们提出了一种用于扩散模型的新型采样方法，该方法能够利用可用的稀疏数据引导反向过程，从而实现高保真样本的重建。此外，我们通过在训练过程中采用无冲突更新方法，利用现有物理知识对重建结果进行增强。为评估本方法的有效性，我们在二维和三维湍流数据上进行了实验。在预测流体结构及像素级精度方面，我们的方法始终优于其他基于扩散的方法。本研究凸显了扩散模型在重建流场数据方面的巨大潜力，为将其应用于从超分辨率到实验重建等流体动力学研究与应用领域开辟了道路。