Continuous relaxation-based flow optimisation in water distribution networks

Introduction Water distribution networks (WDNs) are commonly optimised using mixed-integer nonlinear programmes that combine hydraulic constraints with discrete decisions. Although these methods provide interpretable solutions, they are non-differentiable and difficult to integrate with gradient-based learning.Methods We propose the Continuous Relaxation-based Flow model (CRFLOW), a differentiable framework that incorporates flow conservation. CRFLOW represents path selection through edge-wise flows and uses entropy regularisation to obtain smooth allocations. Edge costs are parameterised and updated during optimisation, allowing adaptive reshaping of the cost landscape, while flow conservation is promoted through the penalty. We evaluate whether coherent cost-driven flow structures emerge under continuous relaxation, remain stable underperturbed initialisation and recover embedded low-cost corridors without discrete path constraints.Results and Key Findings Results show stable convergence and consistent structural behaviour, supporting the conceptual validity of the framework.