  Neural implicit representations, which encode a surface as the level set of a
neural network applied to spatial coordinates, have proven to be remarkably
effective for optimizing, compressing, and generating 3D geometry. Although
these representations are easy to fit, it is not clear how to best evaluate
geometric queries on the shape, such as intersecting against a ray or finding a
closest point. The predominant approach is to encourage the network to have a
signed distance property. However, this property typically holds only
approximately, leading to robustness issues, and holds only at the conclusion
of training, inhibiting the use of queries in loss functions. Instead, this
work presents a new approach to perform queries directly on general neural
implicit functions for a wide range of existing architectures. Our key tool is
the application of range analysis to neural networks, using automatic
arithmetic rules to bound the output of a network over a region; we conduct a
study of range analysis on neural networks, and identify variants of affine
arithmetic which are highly effective. We use the resulting bounds to develop
geometric queries including ray casting, intersection testing, constructing
spatial hierarchies, fast mesh extraction, closest-point evaluation, evaluating
bulk properties, and more. Our queries can be efficiently evaluated on GPUs,
and offer concrete accuracy guarantees even on randomly-initialized networks,
enabling their use in training objectives and beyond. We also show a
preliminary application to inverse rendering.
