MetaNO: How to Transfer Your Knowledge on Learning Hidden Physics

TL;DR

MetaNO introduces a meta-learning framework for neural operators that effectively transfers knowledge across different PDEs, improving efficiency and accuracy in complex physical system modeling.

Contribution

The paper presents a novel meta-learning method for neural operators that captures parameter fields in the first layer, enabling universal transfer across PDE tasks.

Findings

Effective transfer of solution operators between PDEs

Improved sampling efficiency in unseen tasks

Successful application to real-world material modeling

Abstract

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.