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Neural abstractions have been recently introduced as formal approximations ofcomplex, nonlinear dynamical models. They comprise a neural ODE and a certifiedupper bound on the error between the abstract neural network and the concretedynamical model. So far neural abstractions have exclusively been obtained asneural networks consisting entirely of $ReLU$ activation functions, resultingin neural ODE models that have piecewise affine dynamics, and which can beequivalently interpreted as linear hybrid automata. In this work, we observethat the utility of an abstraction depends on its use: some scenarios mightrequire coarse abstractions that are easier to analyse, whereas others mightrequire more complex, refined abstractions. We therefore consider neuralabstractions of alternative shapes, namely either piecewise constant ornonlinear non-polynomial (specifically, obtained via sigmoidal activations). Weemploy formal inductive synthesis procedures to generate neural abstractionsthat result in dynamical models with these semantics. Empirically, wedemonstrate the trade-off that these different neural abstraction templateshave vis-a-vis their precision and synthesis time, as well as the time requiredfor their safety verification (done via reachability computation). We improveexisting synthesis techniques to enable abstraction of higher-dimensionalmodels, and additionally discuss the abstraction of complex neural ODEs toimprove the efficiency of reachability analysis for these models.

On the Trade-off Between Efficiency and Precision of Neural Abstraction