Deep learning is also suitable because it provides the capability of ingesting data from multiple sources, which better informs the model of what is happening on the ground. Our work aims to close this gap in the literature. Existing approaches for uncertainty quantification can be categorized into three lines. The second line tries to combine the stochastic processes and DNNs. The third line is based on model ensembling [24] which trains multiple DNNs with different initializations and use their predictions for uncertainty quantification.

False and nuisance alarms in industrial fault detection systems are often triggered by uncertainty, causing normal process variable fluctuations to be erroneously identified as faults. This paper introduces a novel encoder-based residual design that effectively decouples the stochastic and deterministic components of process variables without imposing detection delay. The proposed model employs two distinct encoders to factorize the latent space into two orthogonal spaces: one for the deterministic part and the other for the stochastic part. To ensure the identifiability of the desired spaces, constraints are applied during training. The deterministic space is constrained to be smooth to guarantee determinism, while the stochastic space is required to resemble standard Gaussian noise. Additionally, a decorrelation term enforces the independence of the learned representations. The efficacy of this approach is demonstrated through numerical examples and its application to the Tennessee Eastman process, highlighting its potential for robust fault detection. By focusing decision logic solely on deterministic factors, the proposed model significantly enhances prediction quality while achieving nearly zero false alarms and missed detections, paving the way for improved operational safety and integrity in industrial environments.

Stochastic encoders have been used in rate-distortion theory and neural compression because they can be easier to handle. However, in performance comparisons with deterministic encoders they often do worse, suggesting that noise in the encoding process may generally be a bad idea. It is poorly understood if and when stochastic encoders do better than deterministic encoders. In this paper we provide one illustrative example which shows that stochastic encoders can significantly outperform the best deterministic encoders. Our toy example suggests that stochastic encoders may be particularly useful in the regime of "perfect perceptual quality".

Recent state-of-the-art autoencoder based generative models have an encoder-decoder structure and learn a latent representation with a pre-defined distribution that can be sampled from. Implementing the encoder networks of these models in a stochastic manner provides a natural and common approach to avoid overfitting and enforce a smooth decoder function. However, we show that for stochastic encoders, simultaneously attempting to enforce a distribution constraint and minimising an output distortion leads to a reduction in generative and reconstruction quality. In addition, attempting to enforce a latent distribution constraint is not reasonable when performing disentanglement. Hence, we propose the variance-constrained autoencoder (VCAE), which only enforces a variance constraint on the latent distribution. Our experiments show that VCAE improves upon Wasserstein Autoencoder and the Variational Autoencoder in both reconstruction and generative quality on MNIST and CelebA. Moreover, we show that VCAE equipped with a total correlation penalty term performs equivalently to FactorVAE at learning disentangled representations on 3D-Shapes while being a more principled approach.