Spectral coefficient learning via operator networks for inverse problems of parametric partial differential equations

초록

This study presents a modified spectral coefficient learning framework for solving inverse problems of parametric partial differential equations (PDEs). Building on the Spectral Coefficient Learning via Operator Network (SCLON) framework, which originally targets forward PDE problems, the framework is adapted to infer unknown forcing terms from limited sensor data through a primarily unsupervised approach. Crucially, the proposed method relies on a spectral representation that automatically enforces boundary conditions, thereby obviating additional penalty loss terms and substantially enhancing training stability and computational efficiency. Extensive numerical experiments, encompassing both one-dimensional and twodimensional benchmark PDEs, demonstrate the proposed framework's robustness and accuracy, even in the absence of labeled training data. Moreover, the inclusion of a small number of supervised samples further refines the predictions, often outperforming standard operator learning models that rely solely on fully labeled datasets. Notably, the approach consistently maintains high accuracy in noisy environments, thus highlighting its robustness in realistic data-acquisition scenarios. Taken together, these results showcase the practical advantages of the hybrid unsupervised-supervised training strategy in terms of data efficiency, computational overhead, and noise resilience.

키워드