# HAMLET: Graph Transformer Neural Operator for Partial Differential Equations

arXiv: https://arxiv.org/abs/2402.03541 · HTML: https://mlsanity.com/paper/2402.03541/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- Builds on 54 of its 68 references that are in the map.
- First posted on arXiv 5 Feb 2024.

## Builds on (most cited first, 30 of 54)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [A Generalization of Transformer Networks to Graphs](https://mlsanity.com/paper/2012.09699/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Graph Transformer Networks](https://mlsanity.com/paper/1911.06455/) (2019)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)

## Built on by (most cited first, 24 of 24)

- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Universal Physics Transformers: A Framework For Efficiently Scaling Neural Operators](https://mlsanity.com/paper/2402.12365/) (2024)
- [GeoTransolver: Learning Physics on Irregular Domains Using Multi-scale Geometry Aware Physics Attention Transformer](https://mlsanity.com/paper/2512.20399/) (2025)
- [Mamba Neural Operator: Who Wins? Transformers vs. State-Space Models for PDEs](https://mlsanity.com/paper/2410.02113/) (2024)
- [Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks](https://mlsanity.com/paper/2503.23167/) (2025)
- [FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning](https://mlsanity.com/paper/2509.12344/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Accurate and scalable deep Maxwell solvers using multilevel iterative methods](https://mlsanity.com/paper/2509.03622/) (2025)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [Method of Manufactured Learning for Solver-free Training of Neural Operators](https://mlsanity.com/paper/2511.12890/) (2025)
- [GITO: Graph-Informed Transformer Operator for Learning Complex Partial Differential Equations](https://mlsanity.com/paper/2506.13906/) (2025)
- [PDE Solvers Should Be Local: Fast, Stable Rollouts with Learned Local Stencils](https://mlsanity.com/paper/2509.26186/) (2025)
- [A Mathematical Explanation of Transformers](https://mlsanity.com/paper/2510.03989/) (2025)
- [Neural-Schwarz Tiling for Geometry-Universal PDE Solving at Scale](https://mlsanity.com/paper/2605.12343/) (2026)
- [ConDiff: A Challenging Dataset for Neural Solvers of Partial Differential Equations](https://mlsanity.com/paper/2406.04709/) (2024)
- [ATOM: A Pretrained Neural Operator for Multitask Molecular Dynamics](https://mlsanity.com/paper/2510.05482/) (2025)
- [Benchmarking Long Roll-outs of Auto-regressive Neural Operators for the Compressible Navier-Stokes Equations with Conserved Quantity Correction](https://mlsanity.com/paper/2601.22541/) (2026)
- [Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs](https://mlsanity.com/paper/2605.10451/) (2026)
- [Connecting the geometry and dynamics of many-body complex systems with message passing neural operators](https://mlsanity.com/paper/2502.15913/) (2025)
- [SpectraKAN: Conditioning Spectral Operators](https://mlsanity.com/paper/2602.05187/) (2026)
- [FlowForge: A Staged Local Rollout Engine for Flow-Field Prediction](https://mlsanity.com/paper/2604.18953/) (2026)
- [CATO: Charted Attention for Neural PDE Operators](https://mlsanity.com/paper/2605.09016/) (2026)
- [Neural Scaling Laws of Transformer Operator Network](https://mlsanity.com/paper/2609.33533/) (2026)
- [RINS: Residual-Image Neural Subspace Solvers for Large Sparse Linear Systems](https://mlsanity.com/paper/2610.02217/) (2026)

## Abstract (arXiv)

We present a novel graph transformer framework, HAMLET, designed to address the challenges in solving partial differential equations (PDEs) using neural networks. The framework uses graph transformers with modular input encoders to directly incorporate differential equation information into the solution process. This modularity enhances parameter correspondence control, making HAMLET adaptable to PDEs of arbitrary geometries and varied input formats. Notably, HAMLET scales effectively with increasing data complexity and noise, showcasing its robustness. HAMLET is not just tailored to a single type of physical simulation, but can be applied across various domains. Moreover, it boosts model resilience and performance, especially in scenarios with limited data. We demonstrate, through extensive experiments, that our framework is capable of outperforming current techniques for PDEs.