Coefficient-to-Basis Network: a fine-tunable operator learning framework for inverse problems with adaptive...

Area of Science:

• Applied mathematics
• Scientific computing
• Operator learning

Background:

• Inverse problems are fundamental in science and engineering.
• Traditional methods often require extensive retraining for new discretizations.
• Operator learning offers a data-driven approach to solving such problems.

Purpose of the Study:

• To introduce Coefficient-to-Basis Network (C2BNet), a novel framework for solving inverse problems.
• To enable efficient adaptation to varying discretizations with minimal computational overhead.
• To provide theoretical guarantees for approximation and generalization errors.

Main Methods:

• Development of the Coefficient-to-Basis Network (C2BNet) architecture.
• Fine-tuning a pre-trained model for adaptation to new discretizations.
• Theoretical analysis establishing approximation and generalization error bounds.
• Exploitation of low-dimensional data structures for efficient learning.

Main Results:

• C2BNet demonstrates efficient adaptation to different discretizations via fine-tuning.
• The framework significantly reduces computational cost while maintaining high accuracy.
• Theoretical bounds confirm C2BNet's ability to leverage low-dimensional structures.
• Numerical experiments validate superior performance on inverse problems.

Conclusions:

• C2BNet provides a robust and efficient solution for inverse problems.
• The method effectively balances computational efficiency and predictive accuracy.
• C2BNet is a promising tool for scientific computing and engineering applications.