Neural Implicit Surfaces in Higher Dimension

Tiago Novello, Vinícius da Silva, Hélio Lopes, Guilherme G. Schardong, Luiz Schirmer, Luiz Velho

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