Industrial data are often high-dimensional, nonlinear and multiple-modal. This paper develops a soft sensor model based on Gaussian mixture Variational Autoencoder (GMVAE) under the just-in-time learning (JITL) framework. To extract latent representations with multimode characteristics, GMVAE as a deep neural network model is utilized by considering Gaussian mixture models (GMM) in the latent space. After training the GMVAE model, each latent (or feature) variable can be described through a Gaussian mixture distribution. Subsequently, when a new sample arrives, a mixture symmetric Kullback-Leibler (MSKL) divergence is utilized to measure its similarity with historical data samples. MSKL divergence can measure similarity between two Gaussian mixture probability density functions. Based on the MSKL divergence, weighted input and output historical data are obtained, and then a local model is established. The effectiveness of the proposed soft sensor modeling method is validated through a numerical example along with simulation on the Tennessee Eastman benchmark process.

工业数据通常是高维，非线性和多模式的。本文在实时学习（JITL）框架下，基于高斯混合变分自编码器（GMVAE）开发了一种软传感器模型。为了提取具有多模式特征的潜在表示，通过考虑潜在空间中的高斯混合模型（GMM），将GMVAE作为一种深度神经网络模型。训练GMVAE模型后，可以通过高斯混合分布来描述每个潜在（或特征）变量。随后，当新样本到达时，将使用混合对称Kullback-Leibler（MSKL）散度来测量其与历史数据样本的相似性。MSKL散度可以测量两个高斯混合概率密度函数之间的相似性。根据MSKL的差异，获得加权的输入和输出历史数据，然后建立局部模型。通过数值示例以及对田纳西伊士曼基准过程的仿真，验证了所提出的软传感器建模方法的有效性。