# Exploring Differential Geometry in Neural Implicits

arXiv: https://arxiv.org/abs/2201.09263 · HTML: https://mlsanity.com/paper/2201.09263/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 45 references that are in the map.
- First posted on arXiv 23 Jan 2022.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Convolutional Occupancy Networks](https://mlsanity.com/paper/2003.04618/) (2020)
- [Open3D: A Modern Library for 3D Data Processing](https://mlsanity.com/paper/1801.09847/) (2018)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [BSP-Net: Generating Compact Meshes via Binary Space Partitioning](https://mlsanity.com/paper/1911.06971/) (2019)
- [CvxNet: Learnable Convex Decomposition](https://mlsanity.com/paper/1909.05736/) (2019)
- [Seeing Implicit Neural Representations as Fourier Series](https://mlsanity.com/paper/2109.00249/) (2021)
- [Local Deep Implicit Functions for 3D Shape](https://mlsanity.com/paper/1912.06126/) (2019)

## Built on by (most cited first, 26 of 26)

- [ORCa: Glossy Objects as Radiance Field Cameras](https://mlsanity.com/paper/2212.04531/) (2022)
- [Tuning the Frequencies: Robust Training for Sinusoidal Neural Networks](https://mlsanity.com/paper/2407.21121/) (2024)
- [Neural Implicit Surface Evolution](https://mlsanity.com/paper/2201.09636/) (2022)
- [DebSDF: Delving into the Details and Bias of Neural Indoor Scene Reconstruction](https://mlsanity.com/paper/2308.15536/) (2023)
- [Neural Geometry Processing via Spherical Neural Surfaces](https://mlsanity.com/paper/2407.07755/) (2024)
- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [Robust One-Class Classification with Signed Distance Function using 1-Lipschitz Neural Networks](https://mlsanity.com/paper/2303.01978/) (2023)
- [Neural Implicit Morphing of Face Images](https://mlsanity.com/paper/2308.13888/) (2023)
- [DUDF: Differentiable Unsigned Distance Fields with Hyperbolic Scaling](https://mlsanity.com/paper/2402.08876/) (2024)
- [NAISR: A 3D Neural Additive Model for Interpretable Shape Representation](https://mlsanity.com/paper/2303.09234/) (2023)
- [Understanding Sinusoidal Neural Networks](https://mlsanity.com/paper/2212.01833/) (2022)
- [Neural Implicit Shape Editing using Boundary Sensitivity](https://mlsanity.com/paper/2304.12951/) (2023)
- [Geometric implicit neural representations for signed distance functions](https://mlsanity.com/paper/2511.07206/) (2025)
- [SASNet: Spatially-Adaptive Sinusoidal Networks for INRs](https://mlsanity.com/paper/2503.09750/) (2025)
- [FLOWING: Implicit Neural Flows for Structure-Preserving Morphing](https://mlsanity.com/paper/2510.09537/) (2025)
- [A Survey of AI Methods for Geometry Preparation and Mesh Generation in Engineering Simulation](https://mlsanity.com/paper/2512.23719/) (2025)
- [Enhancing Surface Neural Implicits with Curvature-Guided Sampling and Uncertainty-Augmented Representations](https://mlsanity.com/paper/2306.02099/) (2023)
- [MARF: The Medial Atom Ray Field Object Representation](https://mlsanity.com/paper/2307.00037/) (2023)
- [Implicit Neural Representation of Tileable Material Textures](https://mlsanity.com/paper/2402.02208/) (2024)
- [SDFs from Unoriented Point Clouds using Neural Variational Heat Distances](https://mlsanity.com/paper/2504.11212/) (2025)
- [High Resolution UDF Meshing via Iterative Networks](https://mlsanity.com/paper/2509.17212/) (2025)
- [Adaptive Training of INRs via Pruning and Densification](https://mlsanity.com/paper/2510.23943/) (2025)
- [ImplicitTerrainV2: Wavelet-Guided Spatially Adaptive Neural Terrain Representation](https://mlsanity.com/paper/2605.22556/) (2026)
- [Recent Advances and Trends in Learning-based 3D Representations](https://mlsanity.com/paper/2606.04871/) (2026)
- [An overview of 3D Vision-Language Models](https://mlsanity.com/paper/2609.05583/) (2026)
- [M-plicits: Neural Implicit Surfaces via Nested Multiscale Residuals](https://mlsanity.com/paper/2609.28684/) (2026)

## Abstract (arXiv)

We introduce a neural implicit framework that exploits the differentiable properties of neural networks and the discrete geometry of point-sampled surfaces to approximate them as the level sets of neural implicit functions. To train a neural implicit function, we propose a loss functional that approximates a signed distance function, and allows terms with high-order derivatives, such as the alignment between the principal directions of curvature, to learn more geometric details. During training, we consider a non-uniform sampling strategy based on the curvatures of the point-sampled surface to prioritize points with more geometric details. This sampling implies faster learning while preserving geometric accuracy when compared with previous approaches. We also use the analytical derivatives of a neural implicit function to estimate the differential measures of the underlying point-sampled surface.