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Recently, deep neural networks (DNNs) have been increasingly used in the context of scientific computing, particularly in solving PDE-related problems. In this talk, we first constructed a series novel neural network architectures inspired by classical linear algebra algorithms, including the hierarchical matrices, the hierarchical nested bases and BCR's nonstandard wavelet form. The new architectures inherit the multiscale structure of these classical algorithms, thus called multiscale neural network. Then we apply the neural networks to solve classical PDE-based inverse problems, for example, electrical impedance tomography (EIT) and optical tomography (OT). The key of the application on the inverse problem is how to represent the properties of the inverse map by neural network.