# Adaptive Mesh-Quantization for Neural PDE Solvers

arXiv: https://arxiv.org/abs/2511.18474 · HTML: https://mlsanity.com/paper/2511.18474/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 23 Nov 2025.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Quantization and Training of Neural Networks for Efficient Integer-Arithmetic-Only Inference](https://mlsanity.com/paper/1712.05877/) (2017)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [A White Paper on Neural Network Quantization](https://mlsanity.com/paper/2106.08295/) (2021)
- [HAQ: Hardware-Aware Automated Quantization with Mixed Precision](https://mlsanity.com/paper/1811.08886/) (2018)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [BitNet: Scaling 1-bit Transformers for Large Language Models](https://mlsanity.com/paper/2310.11453/) (2023)
- [Mixed Precision Quantization of ConvNets via Differentiable Neural Architecture Search](https://mlsanity.com/paper/1812.00090/) (2018)
- [Degree-Quant: Quantization-Aware Training for Graph Neural Networks](https://mlsanity.com/paper/2008.05000/) (2020)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Eagle: Large-Scale Learning of Turbulent Fluid Dynamics with Mesh Transformers](https://mlsanity.com/paper/2302.10803/) (2023)
- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [PaSca: a Graph Neural Architecture Search System under the Scalable Paradigm](https://mlsanity.com/paper/2203.00638/) (2022)
- [Resiliency of Deep Neural Networks under Quantization](https://mlsanity.com/paper/1511.06488/) (2015)
- [Neural Network Quantization with AI Model Efficiency Toolkit (AIMET)](https://mlsanity.com/paper/2201.08442/) (2022)
- [A Practical Mixed Precision Algorithm for Post-Training Quantization](https://mlsanity.com/paper/2302.05397/) (2023)
- [Efficient Neural PDE-Solvers using Quantization Aware Training](https://mlsanity.com/paper/2308.07350/) (2023)
- [rm A2Q: Aggregation-Aware Quantization for Graph Neural Networks](https://mlsanity.com/paper/2302.00193/) (2023)

## Built on by (most cited first, 2 of 2)

- [AVQ-Attention: Adaptive Vector-Quantized Attention](https://mlsanity.com/paper/2607.12789/) (2026)
- [LEIA: Learned Environment for Interactive Architected Materials](https://mlsanity.com/paper/2605.28368/) (2026)

## Abstract (arXiv)

Physical systems commonly exhibit spatially varying complexity, presenting a significant challenge for neural PDE solvers. While Graph Neural Networks can handle the irregular meshes required for complex geometries and boundary conditions, they still apply uniform computational effort across all nodes regardless of the underlying physics complexity. This leads to inefficient resource allocation where computationally simple regions receive the same treatment as complex phenomena. We address this challenge by introducing Adaptive Mesh Quantization: spatially adaptive quantization across mesh node, edge, and cluster features, dynamically adjusting the bit-width used by a quantized model. We propose an adaptive bit-width allocation strategy driven by a lightweight auxiliary model that identifies high-loss regions in the input mesh. This enables dynamic resource distribution in the main model, where regions of higher difficulty are allocated increased bit-width, optimizing computational resource utilization. We demonstrate our framework's effectiveness by integrating it with two state-of-the-art models, MP-PDE and GraphViT, to evaluate performance across multiple tasks: 2D Darcy flow, large-scale unsteady fluid dynamics in 2D, steady-state Navier-Stokes simulations in 3D, and a 2D hyper-elasticity problem. Our framework demonstrates consistent Pareto improvements over uniformly quantized baselines, yielding up to 50% improvements in performance at the same cost.