  The recent surge of utilizing deep neural networks for geometric processing
and shape modeling has opened up exciting avenues. However, there is a
conspicuous lack of research efforts on using powerful neural representations
to extend the capabilities of parametric surfaces, which are the prevalent
surface representations in product design, CAD/CAM, and computer animation. We
present Neural Parametric Surfaces, the first piecewise neural surface
representation that allows coarse patch layouts of arbitrary $n$-sided surface
patches to model complex surface geometries with high precision, offering
greater flexibility over traditional parametric surfaces. By construction, this
new surface representation guarantees $G^0$ continuity between adjacent patches
and empirically achieves $G^1$ continuity, which cannot be attained by existing
neural patch-based methods. The key ingredient of our neural parametric surface
is a learnable feature complex $\mathcal{C}$ that is embedded in a
high-dimensional space $\mathbb{R}^D$ and topologically equivalent to the patch
layout of the surface; each face cell of the complex is defined by
interpolating feature vectors at its vertices. The learned feature complex is
mapped by an MLP-encoded function $f:\mathcal{C} \rightarrow \mathcal{S}$ to
produce the neural parametric surface $\mathcal{S}$. We present a surface
fitting algorithm that optimizes the feature complex $\mathcal{C}$ and trains
the neural mapping $f$ to reconstruct given target shapes with high accuracy.
We further show that the proposed representation along with a compact-size
neural net can learn a plausible shape space from a shape collection, which can
be used for shape interpolation or shape completion from noisy and incomplete
input data. Extensive experiments show that neural parametric surfaces offer
greater modeling capabilities than traditional parametric surfaces.
