Uncertainty Quantification in hydraulic prediction Reframing Noise-Contrastive Estimation for Regression in Water Supply Systems

This work is funded by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the project/support UID/00481 – Centre for Mechanical Technology and Automation (TEMA). TEchMA 2025 New Frontiers in Mechanical Engineering Uncertainty Quantification in hydraulic prediction Reframing Noise-Contrastive Estimation for Regression in Water Supply Systems Abstract Water Supply Systems (WSS) are fundamental infrastructures for the general well-being of society, responsible for providing safe drinking water. These systems are faced with numerous challenges, such as aging components, climate change, and volatility in energy costs. Among many of its management facets, its primary task is to produce pump schedules that minimize costs, while respecting hydraulic constraints. This is often achieved with a pump scheduling optimization algorithm, which requires an accurate predictive component. In practice, a WSS is a critical system, hence its decisionmaking should be robust to avoid any malfunction. Subsequently, the prediction component of the pump scheduling algorithm should reflect this by being both accurate and robust. In the context of regression tasks, Machine Learning (ML) techniques are widely used in numerous domains. However, conventional ML models provide only point estimates, which contain no information regarding the confidence of its prediction. Uncertainty Quantification (UQ) methodologies have emerged as approaches that enhance ML predictions by providing additional information regarding the certainty of the prediction. For instance, Bayesian Neural Networks (BNN) (e.g., [1]) are popular UQ methodologies with stateof-the-art results. Beyond BNN models, there is also a body of research in Density Regression that focuses on directly estimating the Probability Density Function (PDF) of target variables. Although Density Regression are often applied to unsupervised learning or generative tasks, this work showcases the effectiveness of one such technique in a supervised learning regression task. This work proposes the use of a Density Regression technique, specifically Noise-Contrastive Estimation (NCE) [2], to quantify uncertainty in a water hydraulic prediction task. To the authors’ knowledge the NCE technique has not been applied to a regression task, making the proposed work novel. Preliminary results both on synthetic and real-world hydraulic prediction datasets demonstrate that the developed methodology is promising. These results also suggest that Density Regression techniques, such as NCE, are viable alternatives to other UQ methodologies, with evidence of competitive performance. Introduction Water Supply Systems (WSS) are critical infrastructures for providing safe-drinking water. Their management must simultaneously be economically efficient and robust enough to avoid interrupting the operation due to catastrophic malfunctions. In this context, effective Decision Support Systems often employ hydraulic models that infer only point estimates, thus they do not contain any information regarding the confidence of predictions. Uncertainty Quantification (UQ) models are essential for the development of Uncertainty-Aware control algorithms, subsequently being fundamental for safety-critical systems. Fig 1 / Schematic of the NCE algorithm The Estimation problem Given several xobservations from vector space X,it is aimed to estimate the marginal likelihood p(X), i.e., the probability density function.. This problem inherently exists in unsupervised learning domains, such as generative tasks. NoiseContrastive Estimation (NCE) algorithm [1] tackles this problem by applying a Machine Learning (ML) model to discriminate between real and fake inputs samples. (Fig. 1). NCE uses a sigmoid function to convert model's outputs (logits) into probabilities. Proposed Method This work leverages the NCE algorithm for regression tasks, with the main goal of quantifying the model's output uncertainty (Fig. 2). The proposed approach considers pairs of observations (x,y) and calculates the predictive posterior, i.e., p(y|x). Two main mechanisms are validated: •Output Discrimination: Injection of noise on the output of fake samples, to train the model to discriminate between fake and real outputs, given specific inputs; •Input Noise Robustness: Injection of noise on the inputs of both fake and real samples. It is argued that this helps the model to consider the noise in the inputs, when discriminating the real and fake outputs. It has been observed in experiments to enhance UQ performance significantly. Fig 2/ Schematic of the proposed UQ approach. Fig 3 a) Probability Density Function of a sample of the water tank OC, produced by the proposed model. b) Application of the proposed model and Monte Carlo sampling to propagate error over time, regarding water tank OC’s level . Tiago Pereira(a,b), António Andrade-Campos(a), Ramon Arbos(b) (a) Department of Mechanical Engineering, TEMA –Centre for Mechanical Technology and Automation, LASI –Intelligent Systems Associate Laboratory, University of Aveiro; (b) Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona Acknowledgements This work is supported by the doctoral grant (Ref. 2023.01763.BDANA) financed by the Portuguese Foundation for Science and Technology (FCT) and by the projects: UIDB/00481/2020 and UIDP/00481/2020 - Fundação para a Ciência e a Tecnologia, DOI 10.54499/UIDB/00481/2020 and DOI 10.54499/UIDP/00481/2020, and through the Regional Operational Program of the CenterRegion (CENTRO2030) within projectI-ReTiS-LeaksD&Op nº 17304 (CENTRO2030-FEDER-01177300) and through the Portuguese Foundation for Science and Technology (FCT), supported by the Recovery and Resilience Plan (PRR), within project I-ReTiS-Leaks (2024.07270.IACDC). This work is funded by national funds through FCT –Fundação para a Ciência e a Tecnologia, I.P., under the project/support UID/00481 –Centre for Mechanical Technology and Automation (TEMA). This work has also received support from the Catalan Government under Project 2021 SGR 00197 and also by the Spanish Government under MICINN projects PID2019-105434RB-C33 co-funded with the European Union ERDF funds and MCIN/AEI/10.13039 /501100011033 project TED2021-129134B-I00 cofunded with the European Union "NextGenerationEU"/PRTR funds. References [1] C. Blundell, J. Cornebise, K. Kavukcuoglu, and D. Wierstra, “Weight uncertainty in neural networks,” International Conference on Machine Learning, pp. 1613–1622, 2015. [2] M. Gutmann and A. Hyvärinen, “Noise-contrastive estimation: A new estimation principle for unnormalized statistical models,” Journal of Machine Learning Research, vol. 9, pp. 297–304, 2010. [Online]. Results Preliminary results in benchmark datasets shows promising improvements in UQ performance. The proposed approach was applied on a real-world water network case study. Given a current water tank level, water demands and pump status, the model predicts the rate of change of water tank levels and pump power. Figure 2.a) illustrates a predictive posterior distribution for water tank OC’s level at a representative sample. Furthermore, Monte Carlo sampling is applied over time to obtain the error propagation of the proposed model in water tank level prediction. Conclusion & Future Work This work presents a novel UQ approach, which uses a well-established method for the estimation problem and repurposes it for regression tasks. The proposed method is suitable for hydraulic behavior prediction as it as shown promising performance on benchmark datasets and multi-modal behavior in its predictive posterior distributions. Future work mainly includes quantitative validation. a) b)