# Geometry-Informed Neural Networks

arXiv: https://arxiv.org/abs/2402.14009 · HTML: https://mlsanity.com/paper/2402.14009/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 67 of its 113 references that are in the map.
- First posted on arXiv 21 Feb 2024.

## Builds on (most cited first, 30 of 67)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Machine Learning for Combinatorial Optimization: a Methodological Tour d'Horizon](https://mlsanity.com/paper/1811.06128/) (2018)
- [Neural Combinatorial Optimization with Reinforcement Learning](https://mlsanity.com/paper/1611.09940/) (2016)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [SAL: Sign Agnostic Learning of Shapes from Raw Data](https://mlsanity.com/paper/1911.10414/) (2019)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Neural Unsigned Distance Fields for Implicit Function Learning](https://mlsanity.com/paper/2010.13938/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [WIRE: Wavelet Implicit Neural Representations](https://mlsanity.com/paper/2301.05187/) (2023)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Modulated Periodic Activations for Generalizable Local Functional Representations](https://mlsanity.com/paper/2104.03960/) (2021)

## Built on by (most cited first, 12 of 12)

- [Scalable Discrete Diffusion Samplers: Combinatorial Optimization and Statistical Physics](https://mlsanity.com/paper/2502.08696/) (2025)
- [Neural topology optimization: the good, the bad, and the ugly](https://mlsanity.com/paper/2407.13954/) (2024)
- [STITCH: Surface reconstrucTion using Implicit neural representations with Topology Constraints and persistent Homology](https://mlsanity.com/paper/2412.18696/) (2024)
- [Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?](https://mlsanity.com/paper/2501.18527/) (2025)
- [Diverse Topology Optimization using Modulated Neural Fields](https://mlsanity.com/paper/2502.13174/) (2025)
- [Optimality-Informed Neural Networks for Solving Parametric Optimization Problems](https://mlsanity.com/paper/2512.20270/) (2025)
- [Minimizing Structural Vibrations via Guided Flow Matching Design Optimization](https://mlsanity.com/paper/2506.15263/) (2025)
- [Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds](https://mlsanity.com/paper/2601.00834/) (2025)
- [Gauss-Newton Natural Gradient Descent for Shape Learning](https://mlsanity.com/paper/2602.00099/) (2026)
- [GIBLy: Improving 3D Semantic Segmentation through an Architecture-Agnostic Lightweight Geometric Inductive Bias Layer](https://mlsanity.com/paper/2605.24243/) (2026)
- [Neural Representation of Minimal Surfaces](https://mlsanity.com/paper/2607.23437/) (2026)
- [Fourier-Latent Diffusion for Constrained Generation of Triply Periodic Minimal Surfaces](https://mlsanity.com/paper/2608.02151/) (2026)

## Abstract (arXiv)

Geometry is a ubiquitous tool in computer graphics, design, and engineering. However, the lack of large shape datasets limits the application of state-of-the-art supervised learning methods and motivates the exploration of alternative learning strategies. To this end, we introduce geometry-informed neural networks (GINNs) -- a framework for training shape-generative neural fields without data by leveraging user-specified design requirements in the form of objectives and constraints. By adding diversity as an explicit constraint, GINNs avoid mode-collapse and can generate multiple diverse solutions, often required in geometry tasks. Experimentally, we apply GINNs to several problems spanning physics, geometry, and engineering design, showing control over geometrical and topological properties, such as surface smoothness or the number of holes. These results demonstrate the potential of training shape-generative models without data, paving the way for new generative design approaches without large datasets.