Deep transfer operator learning for partial differential equations under conditional shift

TL;DR

This paper introduces a transfer learning framework for PDEs using DeepONet, enabling efficient task-specific operator learning under conditional shift by matching distributions in a reproducing kernel Hilbert space.

Contribution

The paper develops a novel transfer learning method for PDEs with DeepONet, incorporating conditional distribution matching to handle domain and model shifts.

Findings

Effective transfer learning across nonlinear PDEs under domain shifts

Improved accuracy and efficiency in task-specific PDE learning

Robustness to diverse geometric and dynamic changes

Abstract

Transfer learning (TL) enables the transfer of knowledge gained in learning to perform one task (source) to a related but different task (target), hence addressing the expense of data acquisition and labeling, potential computational power limitations, and dataset distribution mismatches. We propose a new TL framework for task-specific learning (functional regression in partial differential equations (PDEs)) under conditional shift based on the deep operator network (DeepONet). Task-specific operator learning is accomplished by fine-tuning task-specific layers of the target DeepONet using a hybrid loss function that allows for the matching of individual target samples while also preserving the global properties of the conditional distribution of target data. Inspired by the conditional embedding operator theory, we minimize the statistical distance between labeled target data and the surrogate prediction on unlabeled target data by embedding conditional distributions onto a reproducing kernel Hilbert space. We demonstrate the advantages of our approach for various TL scenarios involving nonlinear PDEs under diverse conditions due to shift in the geometric domain and model dynamics. Our TL framework enables fast and efficient learning of heterogeneous tasks despite significant differences between the source and target domains.