Most formal verification approaches for neural networks conservatively enclose the output set of the neural network. However, computing a tight output enclosure is in general computationally hard. We sidestep this challenge with a novel neural network architecture that realizes mixed-monotone functions as the difference of monotone subnetworks. By utilizing the theory on mixed-monotone systems, this reduces the computation of a tight output enclosure from complicated set propagation to two simple forward passes of opposite corners of an input interval. Moreover, because the size of the resulting enclosure is a closed-form function of the weights of the neural network, we can directly minimize it during training. We prove a lower bound on the size of any such enclosure, given by the total variation of the represented function generalized to several inputs, which makes it possible to compare the tightness of the enclosure against the optimum. Our evaluation on regression and dynamical-system benchmarks demonstrates that our training reduces the enclosure size to within 0.5% of the lower bound without sacrificing prediction accuracy. Computing our enclosure is orders of magnitude faster than CROWN and $α$-CROWN and on deep networks it is also tighter than zonotope propagation and CROWN, while its size exceeds that of $α$-CROWN by only 5.1%.

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