# Efficient physics-informed neural networks using hash encoding

arXiv: https://arxiv.org/abs/2302.13397 · HTML: https://mlsanity.com/paper/2302.13397/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 40 references that are in the map.
- First posted on arXiv 26 Feb 2023.

## Builds on (most cited first, 23 of 23)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [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)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Neural Geometric Level of Detail: Real-time Rendering with Implicit 3D Shapes](https://mlsanity.com/paper/2101.10994/) (2021)
- [Local Implicit Grid Representations for 3D Scenes](https://mlsanity.com/paper/2003.08981/) (2020)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Modulated Periodic Activations for Generalizable Local Functional Representations](https://mlsanity.com/paper/2104.03960/) (2021)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Real-time Neural Radiance Caching for Path Tracing](https://mlsanity.com/paper/2106.12372/) (2021)
- [Neural Control Variates](https://mlsanity.com/paper/2006.01524/) (2020)
- [Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations](https://mlsanity.com/paper/2205.09332/) (2022)
- [Robust Camera Pose Refinement for Multi-Resolution Hash Encoding](https://mlsanity.com/paper/2302.01571/) (2023)
- [PINNup: Robust neural network wavefield solutions using frequency upscaling and neuron splitting](https://mlsanity.com/paper/2109.14536/) (2021)
- [Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh](https://mlsanity.com/paper/2211.01604/) (2022)

## Built on by (most cited first, 21 of 21)

- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective](https://mlsanity.com/paper/2502.00604/) (2025)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations](https://mlsanity.com/paper/2412.05994/) (2024)
- [Learnable Activation Functions in Physics-Informed Neural Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2411.15111/) (2024)
- [F-INR: Functional Tensor Decomposition for Implicit Neural Representations](https://mlsanity.com/paper/2503.21507/) (2025)
- [LatentPINNs: Generative physics-informed neural networks via a latent representation learning](https://mlsanity.com/paper/2305.07671/) (2023)
- [Physics-informed neural wavefields with Gabor basis functions](https://mlsanity.com/paper/2310.10602/) (2023)
- [Physics-informed waveform inversion using pretrained wavefield neural operators](https://mlsanity.com/paper/2509.08967/) (2025)
- [RED-DiffEq: Regularization by denoising diffusion models for solving inverse PDE problems with application to full waveform inversion](https://mlsanity.com/paper/2509.21659/) (2025)
- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [Gabor-Enhanced Physics-Informed Neural Networks for Fast Simulations of Acoustic Wavefields](https://mlsanity.com/paper/2502.17134/) (2025)
- [How Learnable Grids Recover Fine Detail in Low Dimensions: A Neural Tangent Kernel Analysis of Multigrid Parametric Encodings](https://mlsanity.com/paper/2504.13412/) (2025)
- [Enforcing hidden physics in physics-informed neural networks](https://mlsanity.com/paper/2511.14348/) (2025)
- [Characterizing and Optimizing the Spatial Kernel of Multi Resolution Hash Encodings](https://mlsanity.com/paper/2602.10495/) (2026)
- [FlashPDE: A Drop-In Fused Triton Operator Library for Neural PDE Solvers](https://mlsanity.com/paper/2607.18020/) (2026)
- [Mollifier Layers: Enabling Efficient High-Order Derivatives in Inverse PDE Learning](https://mlsanity.com/paper/2505.11682/) (2025)
- [Coordinate Encoding on Linear Grids for Physics-Informed Neural Networks](https://mlsanity.com/paper/2603.22700/) (2026)
- [A Green-Integral-Constrained Neural Solver with Stochastic Physics-Informed Regularization](https://mlsanity.com/paper/2604.21411/) (2026)
- [Hermite-NGP: Gradient-Augmented Hash Encoding for Learning PDEs](https://mlsanity.com/paper/2605.24774/) (2026)
- [Staged Depth Training: A Representation Curriculum for PINNs](https://mlsanity.com/paper/2609.30299/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have attracted a lot of attention in scientific computing as their functional representation of partial differential equation (PDE) solutions offers flexibility and accuracy features. However, their training cost has limited their practical use as a real alternative to classic numerical methods. Thus, we propose to incorporate multi-resolution hash encoding into PINNs to improve the training efficiency, as such encoding offers a locally-aware (at multi resolution) coordinate inputs to the neural network. Borrowed from the neural representation field community (NeRF), we investigate the robustness of calculating the derivatives of such hash encoded neural networks with respect to the input coordinates, which is often needed by the PINN loss terms. We propose to replace the automatic differentiation with finite-difference calculations of the derivatives to address the discontinuous nature of such derivatives. We also share the appropriate ranges for the hash encoding hyperparameters to obtain robust derivatives. We test the proposed method on three problems, including Burgers equation, Helmholtz equation, and Navier-Stokes equation. The proposed method admits about a 10-fold improvement in efficiency over the vanilla PINN implementation.