Data-driven Uncertainty Analysis in Neural Networks with Applications to Manufacturing Process Monitoring
Artificial neural networks, including deep neural networks, play a central role in data-driven science due to their superior learning capacity and adaptability to different tasks and data structures. However, although quantitative uncertainty analysis is essential for training and deploying reliable data-driven models, the uncertainties in neural networks are often overlooked or underestimated in many studies, mainly due to the lack of a high-fidelity and computationally efficient uncertainty quantification approach. In this work, a novel uncertainty analysis scheme is developed. The Gaussian mixture model is used to characterize the probability distributions of uncertainties in arbitrary forms, which yields higher fidelity than the presumed distribution forms, like Gaussian, when the underlying uncertainty is multimodal, and is more compact and efficient than large-scale Monte Carlo sampling. The fidelity of the Gaussian mixture is refined through adaptive scheduling of the width of each Gaussian component based on the active assessment of the factors that could deteriorate the uncertainty representation quality, such as the nonlinearity of activation functions in the neural network.
Following this idea, an adaptive Gaussian mixture scheme of nonlinear uncertainty propagation is proposed to effectively propagate the probability distributions of uncertainties through layers in deep neural networks or through time in recurrent neural networks. An adaptive Gaussian mixture filter (AGMF) is then designed based on this uncertainty propagation scheme. By approximating the dynamics of a highly nonlinear system with a feedforward neural network, the adaptive Gaussian mixture refinement is applied at both the state prediction and Bayesian update steps to closely track the distribution of unmeasurable states. As a result, this new AGMF exhibits state-of-the-art accuracy with a reasonable computational cost on highly nonlinear state estimation problems subject to high magnitudes of uncertainties. Next, a probabilistic neural network with Gaussian-mixture-distributed parameters (GM-PNN) is developed. The adaptive Gaussian mixture scheme is extended to refine intermediate layer states and ensure the fidelity of both linear and nonlinear transformations within the network so that the predictive distribution of output target can be inferred directly without sampling or approximation of integration. The derivatives of the loss function with respect to all the probabilistic parameters in this network are derived explicitly, and therefore, the GM-PNN can be easily trained with any backpropagation method to address practical data-driven problems subject to uncertainties.
The GM-PNN is applied to two data-driven condition monitoring schemes of manufacturing processes. For tool wear monitoring in the turning process, a systematic feature normalization and selection scheme is proposed for the engineering of optimal feature sets extracted from sensor signals. The predictive tool wear models are established using two methods, one is a type-2 fuzzy network for interval-type uncertainty quantification and the other is the GM-PNN for probabilistic uncertainty quantification. For porosity monitoring in laser additive manufacturing processes, convolutional neural network (CNN) is used to directly learn patterns from melt-pool patterns to predict porosity. The classical CNN models without consideration of uncertainty are compared with the CNN models in which GM-PNN is embedded as an uncertainty quantification module. For both monitoring schemes, experimental results show that the GM-PNN not only achieves higher prediction accuracies of process conditions than the classical models but also provides more effective uncertainty quantification to facilitate the process-level decision-making in the manufacturing environment.
Based on the developed uncertainty analysis methods and their proven successes in practical applications, some directions for future studies are suggested. Closed-loop control systems may be synthesized by combining the AGMF with data-driven controller design. The AGMF can also be extended from a state estimator to the parameter estimation problems in data-driven models. In addition, the GM-PNN scheme may be expanded to directly build more complicated models like convolutional or recurrent neural networks.