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