One-shot learning for solution operators of partial differential equations

Anran Jiao, Haiyang He, Rishikesh Ranade, Jay Pathak, Lu Lu

Abstract:

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.