[Jan 27th 2022] Page online.

This work investigates the use of neural networks admitting high-order derivatives for modeling dynamic variations of smooth implicit surfaces. For this purpose, it extends the representation of differentiable neural implicit surfaces to higher dimensions, which opens up mechanisms that allow to exploit geometric transformations in many settings, from animation and surface evolution to shape morphing and design galleries.

The problem is modeled by a \(k-\)** parameter family** of surfaces \(S_c\), specified as a neural network function \(f : R^3 \times R^k \rightarrow R\), where \(S_c\) is the zero-level set of the implicit function \(f(\cdot, c) : R^3 \rightarrow R \), with \(c \in R^k\), with variations induced by the control variable \(c\).
In that context, restricted to each coordinate of \(R^k\), the underlying representation is a neural homotopy which is the solution of a general

Let \(S\) be a surface in \(R^3\), and \(g:R^3\rightarrow R\) be an unknown neural implicit function. To compute the parameter set of \(g\) such that \(g^{-1}(0)\) approximates \(S\), it is common to consider the ** Eikonal** problem.

We augment the domain \(R^3\) by a product space \(R^3\times R^k\), where \(p\in R^3\) denotes a point in the space and \(c\in R^k\) denotes a vector that control \(k\) directions of possible animations of the initial implicit surface. In this work we focus on the

Specifically, let \(f:R^3\times R \rightarrow R\) be a smooth neural function representing an

In the same way that we considered a neural implicit function as a solution of the Eikonal problem, we can consider the neural homotopy \(f: R^3\times R\rightarrow R\) to be a solution of a general PDE problem \(\mathcal{F}\big(\nabla^n f, \dots, \nabla^1 f, f, p, t\big)=0\).

**Neural Implicit Surfaces in Higher Dimension**

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

Please send feedback and questions to Tiago Novello.

```
@article{novello2022neural,
title = {Neural Implicit Surfaces in Higher Dimension},
author = {Novello, Tiago and da Silva, Vin\'icius and Schardong, Guilherme and Schirmer,
Luiz and Lopes, H\'elio and Velho, Luiz},
journal = {arXiv:2201.09636},
year = {2022},
month = jan
}
```

We would like to thank
Towaki Takikawa,
Joey Litalien,
Kangxue Yin,
Karsten Kreis,
Charles Loop,
Derek Nowrouzezahrai,
Alec Jacobson,
Morgan McGuire and
Sanja Fidler
for licensing the code of the paper Neural Geometric Level of Detail:
Real-time Rendering with Implicit 3D Surfaces and project page under the MIT License. This website is based on that page.

*We also thank the Stanford Computer Graphics Laboratory for the Bunny, Dragon, Armadillo, and Happy Buddha, acquired through the Stanford 3D scan repository.
*