A Probabilistic Digital Twin for Leak Localization in Water Distribution Networks Using Generative Deep Learning.

Localizing leakages in large water distribution systems is an important and ever-present problem. Due to the complexity originating from water pipeline networks, too few sensors, and noisy measurements, this is a highly challenging problem to solve. In this work, we present a methodology based on generative deep learning and Bayesian inference for leak localization with uncertainty quantification. A generative model, utilizing deep neural networks, serves as a probabilistic surrogate model that replaces the full equations, while at the same time also incorporating the uncertainty inherent in such models. By embedding this surrogate model into a Bayesian inference scheme, leaks are located by combining sensor observations with a model output approximating the true posterior distribution for possible leak locations. We show that our methodology enables producing fast, accurate, and trustworthy results. It showed a convincing performance on three problems with increasing complexity. For a simple test case, the Hanoi network, the average topological distance (ATD) between the predicted and true leak location ranged from 0.3 to 3 with a varying number of sensors and level of measurement noise. For two more complex test cases, the ATD ranged from 0.75 to 4 and from 1.5 to 10, respectively. Furthermore, accuracies upwards of 83%, 72%, and 42% were achieved for the three test cases, respectively. The computation times ranged from 0.1 to 13 s, depending on the size of the neural network employed. This work serves as an example of a digital twin for a sophisticated application of advanced mathematical and deep learning techniques in the area of leak detection.

Estimation of reinforced urn processes under left-truncation and right-censoring.

We propose a non-parametric estimator for bivariate left-truncated and right-censored observations that combines the expectation-maximization algorithm and the reinforced urn process. The resulting expectation-reinforcement algorithm allows for the inclusion of experts' knowledge in the form of a prior distribution, thus belonging to the class of Bayesian models. This can be relevant in applications where the data is incomplete, due to biases in the sampling process, as in the case of left-truncation and right-censoring. With this new approach, the distribution of the truncation variables is also recovered, granting further insight into those biases, and playing an important role in applications like prevalent cohort studies. The estimators are tested numerically using artificial and empirical datasets, and compared with other methodologies such as copula models and the Kaplan-Meier estimator.