The industrially collected process data usually exhibit non-Gaussian and multi-mode characteristics. Due to sensor failures, irregular disturbances, and transmission problems, there are unavoidable outliers that make the data exhibit heavy-tailed characteristics. To this end, a variational auto-encoder regression method based on the mixture Laplacian distribution (MLVAER) is proposed, by introducing a type-II multivariate Laplacian distribution in the latent variable space for robust modeling, and further extending it to the mixture form to accommodate multi-mode processes, the corresponding reparameterization trick is finally proposed for the mixture form of this distribution for neural network gradient descent training. The model based on this distribution assumption has higher degrees of freedom than the model based on the traditional multivariate Laplace distribution assumption when the network structure is the same. Numerical simulation and experiments on two industrial examples demonstrate that the proposed algorithm reduces the root mean square error by over 15% compared to other algorithms.

Aiming to address soft sensing model degradation under changing working conditions, and to accommodate dynamic, nonlinear, and multimodal data characteristics, this paper proposes a nonlinear dynamic transfer soft sensor algorithm. The approach leverages time-delay data augmentation to capture dynamics and projects the augmented data into a latent space for constructing a nonlinear regression model. Two regular terms, distribution alignment regularity and first-order difference regularity, are introduced during data projection to address data distribution disparities. Laplace regularity is incorporated into the nonlinear regression model to ensure geometric structure preservation. The final optimization objective is formulated within the framework of partial least squares, and hyperparameters are determined using Bayesian optimization. The effectiveness of the proposed algorithm is demonstrated through experiments on three public datasets.

Accurately predicting runoff is crucial for managing water resources, preventing and mitigating floods, scheduling hydropower plant operations, and protecting the environment. The hydrological dynamic composite system that forms runoff is complex and random, and seemingly random behavior may be caused by nonlinear variables in a simple deterministic system, which poses a challenge to runoff prediction. In this paper, we construct parallel and multi-timescale reservoirs from a chaotic theory perspective to simulate the stochasticity of chaotic systems. We propose a multi-task-based "Decomposition-Integration-Prediction" (Multi-SDIPC) model for runoff prediction. To validate our research results, we use the Catchment Attributes and Meteorology for Large-Sample Studies (CAMELS) dataset and compare our proposed model with 10 baseline models. The results show that our model has an average NSE metric of 0.83 and exhibits higher accuracy, better generalization, and greater stability than the other models in multi-step forecasting. Based on our findings, we recommend wider application of the Multi-SDIPC model in different regions of the world for medium or long-term runoff prediction.