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We study the control of networked systems with the goal of optimizing both transient and steady-state performances while providing stability guarantees. Linear proportional-integral (PI) controllers are almost always used in practice, but the linear parameterization of the controller fundamentally limits its performance. Learning-based approaches are becoming popular in designing nonlinear controllers, but the lack of stability guarantees makes the learned controllers difficult to apply in practical applications. This paper bridges the gap between neural network-based controller design and the need for stability guarantees. Using equilibrium-independent passivity, a property present in a wide range of physical systems, we propose structured neural-PI controllers that have provable guarantees on the convergence of output to a desired agreement value. If communication between neighbours is available, we further extend the controller to distributedly achieve optimal resource allocation at the steady state. We explicitly characterize the stability conditions and engineer neural networks that satisfy them by design. Experiments on traffic and power networks demonstrate that the proposed approach can improve transient and steady-state performances compared to existing state-of-the-art, while unstructured neural networks lead to unstable behaviors.