  Gradient-based meta-learning methods have primarily been applied to classical
machine learning tasks such as image classification. Recently, PDE-solving deep
learning methods, such as neural operators, are starting to make an important
impact on learning and predicting the response of a complex physical system
directly from observational data. Since the data acquisition in this context is
commonly challenging and costly, the call of utilization and transfer of
existing knowledge to new and unseen physical systems is even more acute.
Herein, we propose a novel meta-learning approach for neural operators, which
can be seen as transferring the knowledge of solution operators between
governing (unknown) PDEs with varying parameter fields. Our approach is a
provably universal solution operator for multiple PDE solving tasks, with a key
theoretical observation that underlying parameter fields can be captured in the
first layer of neural operator models, in contrast to typical final-layer
transfer in existing meta-learning methods. As applications, we demonstrate the
efficacy of our proposed approach on PDE-based datasets and a real-world
material modeling problem, illustrating that our method can handle complex and
nonlinear physical response learning tasks while greatly improving the sampling
efficiency in unseen tasks.
