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We study mechanism design for nonexcludable and excludable binary public project problems. We aim to maximize the expected number of consumers and the expected social welfare. For the nonexcludable public project model, we identify a sufficient condition on the prior distribution for the conservative equal costs mechanism to be the optimal strategy-proof and individually rational mechanism. For general distributions, we propose a dynamic program that solves for the optimal mechanism. For the excludable public project model, we identify a similar sufficient condition for the serial cost sharing mechanism to be optimal for $2$ and $3$ agents. We derive a numerical upper bound. Experiments show that for several common distributions, the serial cost sharing mechanism is close to optimality. The serial cost sharing mechanism is not optimal in general. We design better performing mechanisms via neural networks. Our approach involves several technical innovations that can be applied to mechanism design in general. We interpret the mechanisms as price-oriented rationing-free (PORF) mechanisms, which enables us to move the mechanism's complex (e.g., iterative) decision making off the network, to a separate program. We feed the prior distribution's analytical form into the cost function to provide quality gradients for training. We use supervision to manual mechanisms as a systematic way for initialization. Our approach of "supervision and then gradient descent" is effective for improving manual mechanisms' performances. It is also effective for fixing constraint violations for heuristic-based mechanisms that are infeasible.