# A deep learning approach to solve forward differential problems on graphs

arXiv: https://arxiv.org/abs/2210.03746 · HTML: https://mlsanity.com/paper/2210.03746/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 65 references that are in the map.
- First posted on arXiv 7 Oct 2022.

## Builds on (most cited first, 11 of 11)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Neural Networks Fail to Learn Periodic Functions and How to Fix It](https://mlsanity.com/paper/2006.08195/) (2020)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Learning continuous-time PDEs from sparse data with graph neural networks](https://mlsanity.com/paper/2006.08956/) (2020)
- [PDE Approaches to Graph Analysis](https://mlsanity.com/paper/1505.00185/) (2015)
- [Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone](https://mlsanity.com/paper/1312.6643/) (2013)
- [Optimal Control of Transient Flow in Natural Gas Networks](https://mlsanity.com/paper/1504.02505/) (2015)
- [Bilinear quantum systems on compact graphs: well-posedness and global exact controllability](https://mlsanity.com/paper/1710.06022/) (2017)
- [Hierarchical model reduction driven by machine learning for parametric advection-diffusion-reaction problems in the presence of noisy data](https://mlsanity.com/paper/2204.00538/) (2022)

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

- [QGPINNs: A Physics-Informed Neural Network Framework for Nonlocal Differential Equations on Quantum Graphs](https://mlsanity.com/paper/2608.28589/) (2026)

## Abstract (arXiv)

We propose a novel deep learning (DL) approach to solve one-dimensional non-linear elliptic, parabolic, and hyperbolic problems on graphs. A system of physics-informed neural network (PINN) models is used to solve the differential equations, by assigning each PINN model to a specific edge of the graph. Kirkhoff-Neumann (KN) nodal conditions are imposed in a weak form by adding a penalization term to the training loss function. Through the penalization term that imposes the KN conditions, PINN models associated with edges that share a node coordinate with each other to ensure continuity of the solution and of its directional derivatives computed along the respective edges. Using individual PINN models for each edge of the graph allows our approach to fulfill necessary requirements for parallelization by enabling different PINN models to be trained on distributed compute resources. Numerical results show that the system of PINN models accurately approximate the solutions of the differential problems across the entire graph for a broad set of graph topologies.