Fig. 2

PELS-VAE with regression model in Teacher–Student Architecture. \(\varvec{\zeta }\), \(\varvec{\xi }\), \(\varvec{\varphi }\) denote the network parameters of the encoder, decoder and the regression model respectively. The linear layer stands for: \({\varvec{y}}={\varvec{w}}^T{\varvec{x}}+{\varvec{b}}\). The components “layers” consist of a series of linear layers with each linear layer following a ReLU activation function. In the training process, the switch connects the encoder and the decoder. And the outputs of regression model, i.e., \(\varvec{\mu }_{\varvec{\varphi }}\), \(\varvec{\sigma }_{\varvec{\varphi }}\) are compared with the outputs of the encoder \(\varvec{\mu }\), \(\varvec{\sigma }\), respectively. After the three parts are well trained, the switch turns to the output of the regression model