A Unified DeepONet Framework for Logarithmically Stable Infinite-Dimensional Inverse Problems
This work addresses the challenge of solving ill-posed inverse problems with logarithmic stability, providing a theoretical framework and practical method for stable reconstruction.
The paper develops a unified DeepONet framework for logarithmically stable infinite-dimensional inverse problems, using inverse acoustic scattering as a model. It provides quantitative a priori error bounds and demonstrates stable reconstructions under measurement noise in 2D and 3D numerical experiments.
We develop a unified DeepONet framework for logarithmically stable infinite-dimensional inverse problems, with inverse acoustic scattering as a model application. The framework is formulated at the operator level by separating the learned inverse map into measurement encoding, finite-dimensional neural approximation, and functional reconstruction components. For inverse maps satisfying a logarithmic stability estimate, we establish quantitative a priori error bounds giving separate estimates for the encoder error, the neural approximation error, and the reconstruction error, thereby characterizing the dependence on the encoder dimension, the network size, and the reconstruction dimension. For comparison, we also record the corresponding Lipschitz-stable estimate arising from the same error decomposition. The abstract theory is then specialized to the recovery of a medium contrast from fixed-frequency far-field measurements. Numerical experiments in two and three dimensions illustrate stable reconstructions under measurement noise.