  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.
