Title: Neural network gradient-based learning of black-box function interfaces. Abstract: Deep neural networks work well at approximating complicated functions when provided with data and trained by gradient descent methods. At the same time, there is a vast amount of existing functions that programmatically solve different tasks in a precise manner eliminating the need for training. In many cases, it is possible to decompose a task to a series of functions, of which for some we may prefer to use a neural network to learn the functionality, while for others the preferred method would be to use existing black-box functions. We propose a method for end-to-end training of a base neural network that integrates calls to existing black-box functions. We do so by approximating the black-box functionality with a differentiable neural network in a way that drives the base network to comply with the black-box function interface during the end-to-end optimization process. At inference time, we replace the differentiable estimator with its external black-box non-differentiable counterpart such that the base network output matches the input arguments of the black-box function. Using this ``Estimate and Replace'' paradigm, we train a neural network, end to end, to compute the input to black-box functionality while eliminating the need for intermediate labels. We show that by leveraging the existing precise black-box function during inference, the integrated model generalizes better than a fully differentiable model, and learns more efficiently compared to RL-based methods.