Deep learning architectures for nonlinear operator functions and nonlinear inverse problems

We develop a theoretical analysis for special neural network architectures, termed operator recurrent networks, approximating nonlinear functions whose inputs are linear operators. Such commonly arise in solution algorithms inverse boundary value problems. Traditional networks treat input data as vectors, and thus they do not effectively capture the multiplicative structure associated with operators that correspond to such therefore introduce new family resembles standard architecture, but where acts multiplicatively on vectors. Motivated by compact appearing control of problems wave equation, we promote sparsity selected weight matrices network. After describing this study its representation properties well approximation properties.We furthermore show an explicit regularization can be introduced derived from mathematical mentioned problems, which leads certain guarantees generalization properties. observe improves estimates. Lastly, discuss how viewed deep learning analogue deterministic reconstructing unknown speed acoustic equation measurements.