  We propose a data-driven mean-curvature solver for the level-set method. This
work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy
in [DOI: 10.1007/s10915-022-01952-2][1] and the hybrid inference system of
[DOI: 10.1016/j.jcp.2022.111291][2]. However, in contrast to [1,2], which built
resolution-dependent neural-network dictionaries, here we develop a pair of
models in $\mathbb{R}^3$, regardless of the mesh size. Our feedforward networks
ingest transformed level-set, gradient, and curvature data to fix numerical
mean-curvature approximations selectively for interface nodes. To reduce the
problem's complexity, we have used the Gaussian curvature to classify stencils
and fit our models separately to non-saddle and saddle patterns. Non-saddle
stencils are easier to handle because they exhibit a curvature error
distribution characterized by monotonicity and symmetry. While the latter has
allowed us to train only on half the mean-curvature spectrum, the former has
helped us blend the data-driven and the baseline estimations seamlessly near
flat regions. On the other hand, the saddle-pattern error structure is less
clear; thus, we have exploited no latent information beyond what is known. In
this regard, we have trained our models on not only spherical but also
sinusoidal and hyperbolic paraboloidal patches. Our approach to building their
data sets is systematic but gleans samples randomly while ensuring
well-balancedness. We have also resorted to standardization and dimensionality
reduction and integrated regularization to minimize outliers. In addition, we
leverage curvature rotation/reflection invariance to improve precision at
inference time. Several experiments confirm that our proposed system can yield
more accurate mean-curvature estimations than modern particle-based interface
reconstruction and level-set schemes around under-resolved regions.
