This work investigates the use of smooth neural networks for modeling dynamic variations of implicit surfaces under the level set equation (LSE). For this, it extends the representation of neural implicit surfaces to the space-time $\mathbb{R}^3\times \mathbb{R}$, which opens up mechanisms for continuous geometric transformations. Examples include evolving an initial surface towards general vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of initial conditions. The network training considers two constraints. A data term is responsible for fitting the initial condition to the corresponding time instant, usually $\mathbb{R}^3 \times \{0\}$. Then, a LSE term forces the network to approximate the underlying geometric evolution given by the LSE, without any supervision. The network can also be initialized based on previously trained initial conditions, resulting in faster convergence compared to the standard approach.

翻译：本文研究了在水平集方程（LSE）下，利用光滑神经网络对隐式曲面的动态变化进行建模的方法。为此，我们将神经隐式曲面的表示扩展到时空域 $\mathbb{R}^3\times \mathbb{R}$，从而为连续几何变换提供了机制。示例包括：使初始曲面沿一般向量场演化、利用平均曲率方程进行平滑与锐化、以及初始条件的插值。网络训练考虑了两个约束条件。数据项负责将初始条件拟合到对应的时间瞬间，通常为 $\mathbb{R}^3 \times \{0\}$。随后，LSE项迫使网络在没有监督的情况下逼近由LSE描述的基本几何演化。网络还可基于先前训练好的初始条件进行初始化，相比标准方法，收敛速度更快。