Biological transport networks are highly optimized structures that ensure power-efficient distribution of fluids across various domains, including animal vasculature and plant venation. Theoretically, these networks can be described as space-embedded graphs, and rich structures that align well with observations emerge from optimizing their hydrodynamic energy dissipation. Studies on these models typically use regular grids and focus solely on edge width optimization. Here, we present a generalization of the hydrodynamic graph model that permits additional optimization of node positioning. We achieve this by defining sink regions, accounting for the energy dissipation of delivery within these areas, and optimizing by means of differentiable physics. In the context of leaf venation patterns, our method results in organic networks that adapt to irregularities of boundaries and node misalignment, as well as overall improved efficiency. We study the dependency of the emergent network structures on the capillary delivery conductivity and identify a transition in which the network collapses above a critical threshold. Our findings provide insights into the early formation of biological systems and the efficient construction of transport networks.

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• Cellular organization, physiology & dynamics
• Fitness
• Network flow optimization
• Network formation & growth
• Network phase transitions
• Transport in networks
• Biological networks
• Plants
• Scaling methods