Perception of the full state is an essential technology to support the monitoring, analysis, and design of physical systems, one of whose challenges is to recover global field from sparse observations. Well-known for brilliant approximation ability, deep neural networks have been attractive to data-driven flow and heat field reconstruction studies. However, limited by network structure, existing researches mostly learn the reconstruction mapping in finite-dimensional space and has poor transferability to variable resolution of outputs. In this paper, we extend the new paradigm of neural operator and propose an end-to-end physical field reconstruction method with both excellent performance and mesh transferability named RecFNO. The proposed method aims to learn the mapping from sparse observations to flow and heat field in infinite-dimensional space, contributing to a more powerful nonlinear fitting capacity and resolution-invariant characteristic. Firstly, according to different usage scenarios, we develop three types of embeddings to model the sparse observation inputs: MLP, mask, and Voronoi embedding. The MLP embedding is propitious to more sparse input, while the others benefit from spatial information preservation and perform better with the increase of observation data. Then, we adopt stacked Fourier layers to reconstruct physical field in Fourier space that regularizes the overall recovered field by Fourier modes superposition. Benefiting from the operator in infinite-dimensional space, the proposed method obtains remarkable accuracy and better resolution transferability among meshes. The experiments conducted on fluid mechanics and thermology problems show that the proposed method outperforms existing POD-based and CNN-based methods in most cases and has the capacity to achieve zero-shot super-resolution.

翻译：全状态感知是支撑物理系统监测、分析与设计的一项关键技术，其核心挑战之一在于从稀疏观测中恢复全局场分布。深度神经网络凭借卓越的逼近能力，在数据驱动的流场与热场重构研究中备受青睐。然而受限于网络结构，现有研究大多在有限维空间中学习重构映射，对输出分辨率变化的迁移性较差。本文拓展了神经算子的新范式，提出一种兼具优异性能与网格迁移性的端到端物理场重构方法——RecFNO。该方法旨在学习从稀疏观测到无限维空间中流场与热场的映射，从而具备更强的非线性拟合能力与分辨率不变特性。首先，针对不同应用场景，我们开发了三种嵌入建模稀疏观测输入：MLP嵌入、掩码嵌入与Voronoi嵌入。MLP嵌入更适用于高度稀疏的输入，而另外两种嵌入有利于保留空间信息，随着观测数据增多表现更优。随后，我们采用堆叠的傅里叶层在傅里叶空间中重构物理场，通过傅里叶模态叠加正则化整体恢复场。受益于无限维空间中的算子特性，该方法在获得显著精度的同时，实现了网格间的分辨率迁移能力。在流体力学与热学问题上的实验表明，本方法在多数情况下优于现有的基于本征正交分解和卷积神经网络的方法，并具备零样本超分辨率重构能力。