# GCN-FFNN: A Two-Stream Deep Model for Learning Solution to Partial Differential Equations

arXiv: https://arxiv.org/abs/2204.13744 · HTML: https://mlsanity.com/paper/2204.13744/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 33 references that are in the map.
- First posted on arXiv 28 Apr 2022.

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

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Image Generation from Scene Graphs](https://mlsanity.com/paper/1804.01622/) (2018)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Long-tail Relation Extraction via Knowledge Graph Embeddings and Graph Convolution Networks](https://mlsanity.com/paper/1903.01306/) (2019)
- [Deformable Shape Completion with Graph Convolutional Autoencoders](https://mlsanity.com/paper/1712.00268/) (2017)
- [Deep Graph Convolutional Networks for Wind Speed Prediction](https://mlsanity.com/paper/2101.10041/) (2021)
- [Multistream Graph Attention Networks for Wind Speed Forecasting](https://mlsanity.com/paper/2108.07063/) (2021)

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

- [Generating synthetic data for neural operators](https://mlsanity.com/paper/2401.02398/) (2024)

## Abstract (arXiv)

This paper introduces a novel two-stream deep model based on graph convolutional network (GCN) architecture and feed-forward neural networks (FFNN) for learning the solution of nonlinear partial differential equations (PDEs). The model aims at incorporating both graph and grid input representations using two streams corresponding to GCN and FFNN models, respectively. Each stream layer receives and processes its own input representation. As opposed to FFNN which receives a grid-like structure, the GCN stream layer operates on graph input data where the neighborhood information is incorporated through the adjacency matrix of the graph. In this way, the proposed GCN-FFNN model learns from two types of input representations, i.e. grid and graph data, obtained via the discretization of the PDE domain. The GCN-FFNN model is trained in two phases. In the first phase, the model parameters of each stream are trained separately. Both streams employ the same error function to adjust their parameters by enforcing the models to satisfy the given PDE as well as its initial and boundary conditions on grid or graph collocation (training) data. In the second phase, the learned parameters of two-stream layers are frozen and their learned representation solutions are fed to fully connected layers whose parameters are learned using the previously used error function. The learned GCN-FFNN model is tested on test data located both inside and outside the PDE domain. The obtained numerical results demonstrate the applicability and efficiency of the proposed GCN-FFNN model over individual GCN and FFNN models on 1D-Burgers, 1D-Schr"odinger, 2D-Burgers and 2D-Schr"odinger equations.