  This paper proposes a deep-learning-based method for recovering a signed
distance function (SDF) of a given hypersurface represented by an implicit
level set function. Using the flexibility of constructing a neural network, we
use an augmented network by defining an auxiliary output to represent the
gradient of the SDF. There are three advantages of the augmented network; (i)
the target interface is accurately captured, (ii) the gradient has a unit norm,
and (iii) two outputs are approximated by a single network. Moreover, unlike a
conventional loss term which uses a residual of the eikonal equation, a novel
training objective consisting of three loss terms is designed. The first loss
function enforces a pointwise matching between two outputs of the augmented
network. The second loss function leveraged by a geometric characteristic of
the SDF imposes the shortest path obtained by the gradient. The third loss
function regularizes a singularity of the SDF caused by discontinuities of the
gradient. Numerical results across a wide range of complex and irregular
interfaces in two and three-dimensional domains confirm the effectiveness and
accuracy of the proposed method. We also compare the results of the proposed
method with physics-informed neural networks approaches and the fast marching
method.
