# Phase Transitions, Distance Functions, and Implicit Neural Representations

arXiv: https://arxiv.org/abs/2106.07689 · HTML: https://mlsanity.com/paper/2106.07689/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 36 references that are in the map.
- First posted on arXiv 14 Jun 2021.

## Builds on (most cited first, 16 of 16)

- [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)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (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)
- [PIFu: Pixel-Aligned Implicit Function for High-Resolution Clothed Human Digitization](https://mlsanity.com/paper/1905.05172/) (2019)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision](https://mlsanity.com/paper/1912.07372/) (2019)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [Local Implicit Grid Representations for 3D Scenes](https://mlsanity.com/paper/2003.08981/) (2020)
- [Deep Local Shapes: Learning Local SDF Priors for Detailed 3D Reconstruction](https://mlsanity.com/paper/2003.10983/) (2020)
- [SALD: Sign Agnostic Learning with Derivatives](https://mlsanity.com/paper/2006.05400/) (2020)
- [Deep Geometric Prior for Surface Reconstruction](https://mlsanity.com/paper/1811.10943/) (2018)
- [Controlling Neural Level Sets](https://mlsanity.com/paper/1905.11911/) (2019)
- [Curriculum DeepSDF](https://mlsanity.com/paper/2003.08593/) (2020)
- [Neural Splines: Fitting 3D Surfaces with Infinitely-Wide Neural Networks](https://mlsanity.com/paper/2006.13782/) (2020)

## Built on by (most cited first, 30 of 38)

- [Volume Rendering of Neural Implicit Surfaces](https://mlsanity.com/paper/2106.12052/) (2021)
- [DiGS : Divergence guided shape implicit neural representation for unoriented point clouds](https://mlsanity.com/paper/2106.10811/) (2021)
- [ImFace: A Nonlinear 3D Morphable Face Model with Implicit Neural Representations](https://mlsanity.com/paper/2203.14510/) (2022)
- [Surface Reconstruction from Point Clouds: A Survey and a Benchmark](https://mlsanity.com/paper/2205.02413/) (2022)
- [Neural-Singular-Hessian: Implicit Neural Representation of Unoriented Point Clouds by Enforcing Singular Hessian](https://mlsanity.com/paper/2309.01793/) (2023)
- [Generalised Implicit Neural Representations](https://mlsanity.com/paper/2205.15674/) (2022)
- [Few 'Zero Level Set'-Shot Learning of Shape Signed Distance Functions in Feature Space](https://mlsanity.com/paper/2207.04161/) (2022)
- [StEik: Stabilizing the Optimization of Neural Signed Distance Functions and Finer Shape Representation](https://mlsanity.com/paper/2305.18414/) (2023)
- [VisCo Grids: Surface Reconstruction with Viscosity and Coarea Grids](https://mlsanity.com/paper/2303.14569/) (2023)
- [Frame Averaging for Equivariant Shape Space Learning](https://mlsanity.com/paper/2112.01741/) (2021)
- [Unsupervised Occupancy Learning from Sparse Point Cloud](https://mlsanity.com/paper/2404.02759/) (2024)
- [Neural-IMLS: Self-supervised Implicit Moving Least-Squares Network for Surface Reconstruction](https://mlsanity.com/paper/2109.04398/) (2021)
- [DeepCurrents: Learning Implicit Representations of Shapes with Boundaries](https://mlsanity.com/paper/2111.09383/) (2021)
- [Mixing-Denoising Generalizable Occupancy Networks](https://mlsanity.com/paper/2311.12125/) (2023)
- [NeurCADRecon: Neural Representation for Reconstructing CAD Surfaces by Enforcing Zero Gaussian Curvature](https://mlsanity.com/paper/2404.13420/) (2024)
- [3D Reconstruction with Fast Dipole Sums](https://mlsanity.com/paper/2405.16788/) (2024)
- [Robustifying Generalizable Implicit Shape Networks with a Tunable Non-Parametric Model](https://mlsanity.com/paper/2311.12967/) (2023)
- [ImFace++: A Sophisticated Nonlinear 3D Morphable Face Model with Implicit Neural Representations](https://mlsanity.com/paper/2312.04028/) (2023)
- [NeurCross: A Neural Approach to Computing Cross Fields for Quad Mesh Generation](https://mlsanity.com/paper/2405.13745/) (2024)
- [1-Lipschitz Neural Distance Fields](https://mlsanity.com/paper/2407.09505/) (2024)
- [UNIF: United Neural Implicit Functions for Clothed Human Reconstruction and Animation](https://mlsanity.com/paper/2207.09835/) (2022)
- [Progressive Learning for Physics-informed Neural Motion Planning](https://mlsanity.com/paper/2306.00616/) (2023)
- [p-Poisson surface reconstruction in curl-free flow from point clouds](https://mlsanity.com/paper/2310.20095/) (2023)
- [HotSpot: Signed Distance Function Optimization with an Asymptotically Sufficient Condition](https://mlsanity.com/paper/2411.14628/) (2024)
- [Neural PDE Solvers for Irregular Domains](https://mlsanity.com/paper/2211.03241/) (2022)
- [Implicit Gaussian process representation of vector fields over arbitrary latent manifolds](https://mlsanity.com/paper/2309.16746/) (2023)
- [DiffCD: A Symmetric Differentiable Chamfer Distance for Neural Implicit Surface Fitting](https://mlsanity.com/paper/2407.17058/) (2024)
- [Toward Robust Neural Reconstruction from Sparse Point Sets](https://mlsanity.com/paper/2412.16361/) (2024)
- [Self-Supervised Dual Contouring](https://mlsanity.com/paper/2405.18131/) (2024)
- [NeuVAS: Neural Implicit Surfaces for Variational Shape Modeling](https://mlsanity.com/paper/2506.13050/) (2025)

## Abstract (arXiv)

Representing surfaces as zero level sets of neural networks recently emerged as a powerful modeling paradigm, named Implicit Neural Representations (INRs), serving numerous downstream applications in geometric deep learning and 3D vision. Training INRs previously required choosing between occupancy and distance function representation and different losses with unknown limit behavior and/or bias. In this paper we draw inspiration from the theory of phase transitions of fluids and suggest a loss for training INRs that learns a density function that converges to a proper occupancy function, while its log transform converges to a distance function. Furthermore, we analyze the limit minimizer of this loss showing it satisfies the reconstruction constraints and has minimal surface perimeter, a desirable inductive bias for surface reconstruction. Training INRs with this new loss leads to state-of-the-art reconstructions on a standard benchmark.