A Unified DeepONet Framework for Logarithmically Stable Infinite-Dimensional Inverse Problems

Abstract

We develop a unified DeepONet framework for logarithmically stable inverse problems between infinite-dimensional function spaces, 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 that separate the encoder, finite-dimensional neural approximation, and reconstruction contributions. For prescribed encoder and reconstruction ranks, we obtain a network-size-dependent bound for the finite-dimensional neural approximation error, together with rank-dependent bounds for the encoding and reconstruction errors. For comparison, we also record the corresponding Lipschitz-stable estimate arising from the same error decomposition. The quantitative inverse-scattering analysis is then specialized, in three dimensions, to the recovery of a medium contrast from fixed-frequency far-field measurements. Numerical experiments in two and three dimensions, using both function-space and finite-dimensional priors, illustrate the reconstruction performance and empirical sensitivity to synthetic measurement noise.