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.

翻译：本文提出一种基于深度学习的方法，用于恢复由隐式水平集函数表示的超曲面的符号距离函数（SDF）。借助构建神经网络的灵活性，我们通过定义辅助输出来表示SDF的梯度，从而使用一种增强型网络。该增强型网络具有三个优点：（i）精确捕获目标界面，（ii）梯度具有单位范数，（iii）通过单一网络近似两个输出。此外，与使用程函方程残差的传统损失项不同，本文设计了一种包含三个损失项的新型训练目标。第一项损失函数强制增强型网络的两个输出之间逐点匹配；第二项损失函数利用SDF的几何特性，通过梯度施加最短路径约束；第三项损失函数则对因梯度不连续性导致的SDF奇异性进行正则化。在二维和三维域中，针对一系列复杂和不规则界面的数值结果验证了所提方法的有效性和精度。我们还将所提方法的结果与物理信息神经网络方法及快速行军法进行了对比。