Synergistic learning with multi-task DeepONet for efficient PDE problem solving.

Area of Science:

• Scientific Machine Learning
• Computational Science and Engineering
• Neural Operator Learning

Background:

• Multi-task learning (MTL) improves generalization in traditional machine learning by leveraging information across related tasks.
• Applying MTL to scientific problems governed by partial differential equations (PDEs) is challenging due to the need for task-specific modifications.
• Existing methods often require separate training for different physical processes or geometries.

Purpose of the Study:

• To develop a unified framework for solving diverse PDE-governed problems in science and engineering using MTL.
• To enhance the generalization capabilities of neural operators for problems with varying source terms and geometries.
• To reduce the overall computational cost associated with solving complex PDE tasks.

Main Methods:

• Introduction of a multi-task deep operator network (MT-DeepONet) integrating MTL principles.
• Modification of the branch network in DeepONet to handle various functional forms of parameterized coefficients in PDEs.
• Inclusion of a binary mask in the branch network and loss term to manage parameterized geometries, improving convergence and transfer learning.

Main Results:

• Demonstrated successful application on three benchmark problems: Fisher equation with varying source terms, 2D Darcy Flow with multiple geometries, and 3D heat transfer with parameterized geometries.
• Showcased improved transfer learning capabilities to new, unseen geometries.
• Validated the MT-DeepONet's ability to predict solutions for novel but similar geometric configurations.

Conclusions:

• The MT-DeepONet framework provides a novel and unified approach for solving a wide range of PDE problems in science and engineering.
• Synergistic learning through MTL significantly reduces the overall training cost for neural operators.
• The proposed modifications enable effective handling of diverse functional forms and geometries within a single training paradigm.