  Learning and solving governing equations of a physical system, represented by
partial differential equations (PDEs), from data is a central challenge in a
variety of areas of science and engineering. Traditional numerical methods for
solving PDEs can be computationally expensive for complex systems and require
the complete PDEs of the physical system. On the other hand, current
data-driven machine learning methods require a large amount of data to learn a
surrogate model of the PDE solution operator, which could be impractical. Here,
we propose the first solution operator learning method that only requires one
PDE solution, i.e., one-shot learning. By leveraging the principle of locality
of PDEs, we consider small local domains instead of the entire computational
domain and define a local solution operator. The local solution operator is
then trained using a neural network, and utilized to predict the solution of a
new input function via mesh-based fixed-point iteration (FPI), meshfree
local-solution-operator informed neural network (LOINN) or
local-solution-operator informed neural network with correction (cLOINN). We
test our method on diverse PDEs, including linear or nonlinear PDEs, PDEs
defined on complex geometries, and PDE systems, demonstrating the effectiveness
and generalization capabilities of our method across these varied scenarios.
