Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees
This addresses the challenge of computational inefficiency in operator learning for inverse problems in scientific computing and engineering, offering a fine-tunable framework with theoretical guarantees, though it appears incremental as it builds on existing operator learning paradigms.
The paper tackles the problem of solving inverse problems with adaptive discretizations by proposing the Coefficient-to-Basis Network (C2BNet), which enables efficient fine-tuning to new discretizations without retraining, reducing computational cost while maintaining high accuracy as validated by numerical experiments.
We propose a Coefficient-to-Basis Network (C2BNet), a novel framework for solving inverse problems within the operator learning paradigm. C2BNet efficiently adapts to different discretizations through fine-tuning, using a pre-trained model to significantly reduce computational cost while maintaining high accuracy. Unlike traditional approaches that require retraining from scratch for new discretizations, our method enables seamless adaptation without sacrificing predictive performance. Furthermore, we establish theoretical approximation and generalization error bounds for C2BNet by exploiting low-dimensional structures in the underlying datasets. Our analysis demonstrates that C2BNet adapts to low-dimensional structures without relying on explicit encoding mechanisms, highlighting its robustness and efficiency. To validate our theoretical findings, we conducted extensive numerical experiments that showcase the superior performance of C2BNet on several inverse problems. The results confirm that C2BNet effectively balances computational efficiency and accuracy, making it a promising tool to solve inverse problems in scientific computing and engineering applications.