# PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations

arXiv: https://arxiv.org/abs/2412.05994 · HTML: https://mlsanity.com/paper/2412.05994/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 70 references that are in the map.
- First posted on arXiv 8 Dec 2024.

## Builds on (most cited first, 30 of 42)

- [3D Gaussian Splatting for Real-Time Radiance Field Rendering](https://mlsanity.com/paper/2308.04079/) (2023)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [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)
- [Plenoxels: Radiance Fields without Neural Networks](https://mlsanity.com/paper/2112.05131/) (2021)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [TensoRF: Tensorial Radiance Fields](https://mlsanity.com/paper/2203.09517/) (2022)
- [Direct Voxel Grid Optimization: Super-fast Convergence for Radiance Fields Reconstruction](https://mlsanity.com/paper/2111.11215/) (2021)
- [Neural Sparse Voxel Fields](https://mlsanity.com/paper/2007.11571/) (2020)
- [PlenOctrees for Real-time Rendering of Neural Radiance Fields](https://mlsanity.com/paper/2103.14024/) (2021)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [K-Planes: Explicit Radiance Fields in Space, Time, and Appearance](https://mlsanity.com/paper/2301.10241/) (2023)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Point-NeRF: Point-based Neural Radiance Fields](https://mlsanity.com/paper/2201.08845/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [NeuRBF: A Neural Fields Representation with Adaptive Radial Basis Functions](https://mlsanity.com/paper/2309.15426/) (2023)
- [Don't Unroll Adjoint: Differentiating SSA-Form Programs](https://mlsanity.com/paper/1810.07951/) (2018)
- [Efficient Learning of Mesh-Based Physical Simulation with BSMS-GNN](https://mlsanity.com/paper/2210.02573/) (2022)

## Built on by (most cited first, 11 of 11)

- [MASIV: Toward Material-Agnostic System Identification from Videos](https://mlsanity.com/paper/2508.01112/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features](https://mlsanity.com/paper/2510.19399/) (2025)
- [Weak-PDE-Net: Discovering Open-Form PDEs via Differentiable Symbolic Networks and Weak Formulation](https://mlsanity.com/paper/2603.22951/) (2026)
- [Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.03806/) (2025)
- [On the Effect of Neural Field Reparameterization for 4DVAR](https://mlsanity.com/paper/2509.21751/) (2025)
- [Physics-Informed Mixture Models and Surrogate Models for Precision Additive Manufacturing](https://mlsanity.com/paper/2510.26586/) (2025)
- [PILIR: Physics-Informed Local Implicit Representation](https://mlsanity.com/paper/2605.00385/) (2026)
- [MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions](https://mlsanity.com/paper/2605.12368/) (2026)
- [Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEs](https://mlsanity.com/paper/2605.24774/) (2026)

## Abstract (arXiv)

The numerical approximation of partial differential equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and nonlinear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive bias of MLPs. However, they usually require high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting. In addition, the fixed positions of the mesh parameters restrict their flexibility, making accurate approximation of complex PDEs challenging. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/