Abstract
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.
Abstract (translated)
机器学习在流体动力学中的应用变得越来越普遍，以加速求解偏微分方程的前向和反问题。然而，现有基于卷积神经网络（CNN）的数据质量增强方法的一个显著挑战是，在训练阶段依赖于特定的低质量数据模式和分布。此外，基于CNN的方法本质上将流体重建任务视为一个计算机视觉任务，这缺乏物理和数学解释。这种依赖可能导致在现实场景中模型效果的大幅下降，尤其是低质量输入与训练数据不一致或包含在训练过程中没有考虑到的噪声时。在這種情況下引入扩散模型具有提高性能和泛化能力的潜力。与直接从特定低质量到高质量分布的直接映射不同，扩散模型学会了从任何低质量分布转移到高质量分布。我们提出的模型——物理约束的残差扩散，展示了将数据质量从标准低质量输入提升到低质量输入带有注入高斯噪声和随机收集样本的能力。通过将物理基于性的见解整合到目标函数中，它进一步提高了预测高质量数据的准确性和准确性。实验结果表明，我们的方法可以有效地从一系列低质量输入条件下重构二维湍流的高质量结果，而无需重新训练。
URL
https://arxiv.org/abs/2404.08412
