The use of machine learning in fluid dynamics is becoming more common to expedite the computation when solving forward and inverse problems of partial differential equations. Yet, a notable challenge with existing convolutional neural network (CNN)-based methods for data fidelity enhancement is their reliance on specific low-fidelity data patterns and distributions during the training phase. In addition, the CNN-based method essentially treats the flow reconstruction task as a computer vision task that prioritizes the element-wise precision which lacks a physical and mathematical explanation. This dependence can dramatically affect the models' effectiveness in real-world scenarios, especially when the low-fidelity input deviates from the training data or contains noise not accounted for during training. The introduction of diffusion models in this context shows promise for improving performance and generalizability. Unlike direct mapping from a specific low-fidelity to a high-fidelity distribution, diffusion models learn to transition from any low-fidelity distribution towards a high-fidelity one. Our proposed model - Physics-informed Residual Diffusion, demonstrates the capability to elevate the quality of data from both standard low-fidelity inputs, to low-fidelity inputs with injected Gaussian noise, and randomly collected samples. By integrating physics-based insights into the objective function, it further refines the accuracy and the fidelity of the inferred high-quality data. Experimental results have shown that our approach can effectively reconstruct high-quality outcomes for two-dimensional turbulent flows from a range of low-fidelity input conditions without requiring retraining.

翻译：摘要：在流体动力学中，机器学习正日益广泛用于加速求解偏微分方程的正反问题。然而，现有基于卷积神经网络（CNN）的数据保真度增强方法面临一个显著挑战：它们在训练阶段严重依赖于特定的低保真度数据模式与分布。此外，CNN方法本质上将流场重建任务视为计算机视觉任务，优先考虑逐元素精度，缺乏物理与数学可解释性。这种依赖性会严重影响模型在实际场景中的有效性，特别是当低保真输入偏离训练数据或包含训练中未考虑的噪声时。在此背景下引入扩散模型，有望提升性能与泛化能力。与直接从特定的低保真分布映射到高保真分布不同，扩散模型学习从任意低保真分布向高保真分布过渡。我们提出的模型——基于物理信息的残差扩散方法，能够将数据质量从标准低保真输入、注入高斯噪声的低保真输入以及随机采集的样本中提升至更优水平。通过将物理先验融入目标函数，该方法进一步提升了推断高保真数据的准确性与保真度。实验结果表明，本方法无需重新训练，即可在多种低保真输入条件下有效重建二维湍流的高质量结果。