Transfer Operator Learning with Fusion Frame

TL;DR

This work addresses the transferability challenge of neural operator models for PDEs by introducing a Fusion Frame–POD-DeepONet framework, which decomposes inputs into fusion-frame subspaces built from Fourier features and applies POD within each subspace to form a robust reduced representation for learning $G: X \to Y$. The method leverages a hybrid loss and fusion-frame reconstruction to achieve strong cross-domain generalization, enabling efficient transfer under input-distribution shifts and PDE-term variations with limited retraining. Empirical results on Darcy flow, Burgers’ equation, and elasticity show that FF-POD-DeepONet outperforms traditional POD-DeepONet, standard DeepONet, and Fourier Neural Operator on both source and transfer tasks, demonstrating improved generalization and practical impact for complex PDE simulations. Overall, the framework provides a mathematically grounded, scalable approach to cross-domain operator learning in scientific and engineering applications, with potential for extension to multi-physics and multi-scale problems.

Abstract

The challenge of applying learned knowledge from one domain to solve problems in another related but distinct domain, known as transfer learning, is fundamental in operator learning models that solve Partial Differential Equations (PDEs). These current models often struggle with generalization across different tasks and datasets, limiting their applicability in diverse scientific and engineering disciplines. This work presents a novel framework that enhances the transfer learning capabilities of operator learning models for solving Partial Differential Equations (PDEs) through the integration of fusion frame theory with the Proper Orthogonal Decomposition (POD)-enhanced Deep Operator Network (DeepONet). We introduce an innovative architecture that combines fusion frames with POD-DeepONet, demonstrating superior performance across various PDEs in our experimental analysis. Our framework addresses the critical challenge of transfer learning in operator learning models, paving the way for adaptable and efficient solutions across a wide range of scientific and engineering applications.

Figures (3)

• Figure 1: FF-POD-DeepONet: This figure illustrates the overall architecture of the Fusion Frame-enhanced POD-DeepONet for the source dataset. In the two-dimensional case, the input coefficients or boundary conditions $u$ are processed by multiple neural networks, with each network generating a set of outputs. Here, $u$ may represent certain physical parameters, such as material properties or boundary conditions. Meanwhile, the function space $v$ is processed through Fourier Feature Networks (FFNs) to produce multiple subspaces, each associated with a learnable weight parameter $w$. These subspaces can be considered as subspaces of the Fusion Frame, and they are processed accordingly. Each subspace $W_i$ undergoes dimension reduction and feature extraction using POD. The outputs from these subspaces are then multiplied by the outputs from the neural networks processing $u$, and the results are summed or averaged to obtain the final output. Finally, the query point $\xi$ allows for querying the value of $v$ at any point in the final result. This architecture, by combining multi-dimensional information and Fourier features, enhances the model's predictive capability and adaptability in complex physical scenarios.
• Figure 2: Fusion Frame POD DeepONet in TL: In this figure for target dataset, the red parts indicate the components that need to be relearned during transfer learning. These components include the MLP processing the input function $u$ and the learnable parameters f the function space.
• Figure 3: Fourier Feature Encoding

Theorems & Definitions (2)

• Definition 1: Frame
• Definition 2: Fusion Frame