The precise simulation of turbulent flows is of immense importance in a
variety of scientific and engineering fields, including climate science,
freshwater science, and the development of energy-efficient manufacturing
processes. Within the realm of turbulent flow simulation, direct numerical
simulation (DNS) is widely considered to be the most reliable approach, but it
is prohibitively expensive for long-term simulation at fine spatial scales.
Given the pressing need for efficient simulation, there is an increasing
interest in building machine learning models for turbulence, either by
reconstructing DNS from alternative low-fidelity simulations or by predicting
DNS based on the patterns learned from historical data. However, standard
machine learning techniques remain limited in capturing complex spatio-temporal
characteristics of turbulent flows, resulting in limited performance and
generalizability. This paper presents a novel physics-enhanced neural operator
(PENO) that incorporates physical knowledge of partial differential equations
(PDEs) to accurately model flow dynamics. The model is further refined by a
self-augmentation mechanism to reduce the accumulated error in long-term
simulations. The proposed method is evaluated through its performance on two
distinct sets of 3D turbulent flow data, showcasing the model's capability to
reconstruct high-resolution DNS data, maintain the inherent physical properties
of flow transport, and generate flow simulations across various resolutions.
Additionally, experimental results on multiple 2D vorticity flow series,
generated by different PDEs, highlight the transferability and generalizability
of the proposed method. This confirms its applicability to a wide range of
real-world scenarios in which extensive simulations are needed under diverse
settings.

精确模拟湍流流动对于气候科学、淡水科学和能源高效制造等众多科学和工程领域具有巨大重要性。本文提出了一种物理增强型神经算子（PENO）的模型，通过将偏微分方程（PDEs）的物理知识纳入模型，准确建模流动动力学，进而通过自增机制来减少长期模拟中所积累的误差。该方法在两组不同的 3D 湍流流动数据上进行了评估，展示了该模型重构高分辨率的 DNS 数据、保持流动传输的固有物理特性，并能在不同分辨率下生成流动模拟。此外，针对多个由不同 PDEs 生成的 2D 涡度流序列的实验结果突出了所提出方法的可转移性和普适性，证实了其在多种需要进行大量模拟的真实场景下的适用性。