Entropy Optimization for Flow Estimation in Water Distribution Systems under Data Scarcity

The primary objective of this study is to estimate the flow distributions in water networks with incomplete information by employing a modified version of the maximum entropy principle. Traditional probabilistic models often rely on complete datasets, but in many real-world water distribution systems, only partial information is available, such as network layout, direction of flow, and node-wise demands. This work develops an entropy-based inference model that avoids the traditional use of Lagrange multipliers by reducing the problem to anunconstrained optimization framework. Through analytical and numerical methods, this approach is shown to provide reliable estimates of flow rates across various network structures. Several illustrative examples are provided to demonstrate the applicability of the proposed method, which captures the most unbiased distribution of flow under limited data scenarios, and its consistency is validated by entropy-based performance measures.