# An Operator-Consistent Graph Neural Network for Learning Diffusion Dynamics on Irregular Meshes

arXiv: https://arxiv.org/abs/2512.11860 · HTML: https://mlsanity.com/paper/2512.11860/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 20 references that are in the map.
- First posted on arXiv 5 Dec 2025.

## Builds on (most cited first, 8 of 8)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [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)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Energy Dissipation Preserving Physics Informed Neural Network for Allen-Cahn Equations](https://mlsanity.com/paper/2411.08760/) (2024)
- [P-Tensors: a General Formalism for Constructing Higher Order Message Passing Networks](https://mlsanity.com/paper/2306.10767/) (2023)
- [Sequential learning based PINNs to overcome temporal domain complexities in unsteady flow past flapping wings](https://mlsanity.com/paper/2503.15679/) (2025)

## Built on by (most cited first, 1 of 1)

- [Conservative Hybrid Graph Networks for Process Systems with Learned Routing](https://mlsanity.com/paper/2608.28896/) (2026)

## Abstract (arXiv)

Classical numerical methods solve partial differential equations (PDEs) efficiently on regular meshes, but many of them become unstable on irregular domains. In practice, multiphysics interactions such as diffusion, damage, and healing often take place on irregular meshes. We develop an operator-consistent graph neural network (OCGNN-PINN) that approximates PDE evolution under physics-informed constraints. It couples node-edge message passing with a consistency loss enforcing the gradient-divergence relation through the graph incidence matrix, ensuring that discrete node and edge dynamics remain structurally coupled during temporal rollout. We evaluate the model on diffusion processes over physically driven evolving meshes and real-world scanned surfaces. The results show improved temporal stability and prediction accuracy compared with graph convolutional and multilayer perceptron baselines, approaching the performance of Crank-Nicolson solvers on unstructured domains.