Neural abstractions have been recently introduced as formal approximations of
complex, nonlinear dynamical models. They comprise a neural ODE and a certified
upper bound on the error between the abstract neural network and the concrete
dynamical model. So far neural abstractions have exclusively been obtained as
neural networks consisting entirely of $ReLU$ activation functions, resulting
in neural ODE models that have piecewise affine dynamics, and which can be
equivalently interpreted as linear hybrid automata. In this work, we observe
that the utility of an abstraction depends on its use: some scenarios might
require coarse abstractions that are easier to analyse, whereas others might
require more complex, refined abstractions. We therefore consider neural
abstractions of alternative shapes, namely either piecewise constant or
nonlinear non-polynomial (specifically, obtained via sigmoidal activations). We
employ formal inductive synthesis procedures to generate neural abstractions
that result in dynamical models with these semantics. Empirically, we
demonstrate the trade-off that these different neural abstraction templates
have vis-a-vis their precision and synthesis time, as well as the time required
for their safety verification (done via reachability computation). We improve
existing synthesis techniques to enable abstraction of higher-dimensional
models, and additionally discuss the abstraction of complex neural ODEs to
improve the efficiency of reachability analysis for these models.

神经抽象是复杂的非线性动力模型的形式近似，通过神经 ODE 和抽象神经网络与具体动力模型之间的误差的保证上界来表示。本文通过形式归纳合成过程生成动力学模型，并讨论复杂神经 ODE 的抽象以提高可达性分析的效率。