# Learning the Solution Operator of Boundary Value Problems using Graph Neural Networks

arXiv: https://arxiv.org/abs/2206.14092 · HTML: https://mlsanity.com/paper/2206.14092/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 30 references that are in the map.
- First posted on arXiv 28 Jun 2022.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Combinatorial optimization and reasoning with graph neural networks](https://mlsanity.com/paper/2102.09544/) (2021)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Graph Element Networks: adaptive, structured computation and memory](https://mlsanity.com/paper/1904.09019/) (2019)
- [Boundary Graph Neural Networks for 3D Simulations](https://mlsanity.com/paper/2106.11299/) (2021)
- [Improving Molecular Graph Neural Network Explainability with Orthonormalization and Induced Sparsity](https://mlsanity.com/paper/2105.04854/) (2021)
- [FEA-Net: A Physics-guided Data-driven Model for Efficient Mechanical Response Prediction](https://mlsanity.com/paper/2002.01893/) (2020)

## Built on by (most cited first, 17 of 17)

- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [BENO: Boundary-embedded Neural Operators for Elliptic PDEs](https://mlsanity.com/paper/2401.09323/) (2024)
- [HAMLET: Graph Transformer Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2402.03541/) (2024)
- [Neural Control of Parametric Solutions for High-dimensional Evolution PDEs](https://mlsanity.com/paper/2302.00045/) (2023)
- [Multiscale Neural Operators for Solving Time-Independent PDEs](https://mlsanity.com/paper/2311.05964/) (2023)
- [Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equations](https://mlsanity.com/paper/2402.17853/) (2024)
- [Overtone: Cyclic Patch Modulation for Clean, Efficient, and Flexible Physics Emulators](https://mlsanity.com/paper/2507.09264/) (2025)
- [DADO -- Low-Cost Query Strategies for Deep Active Design Optimization](https://mlsanity.com/paper/2307.04536/) (2023)
- [Resolution invariant deep operator network for PDEs with complex geometries](https://mlsanity.com/paper/2402.00825/) (2024)
- [Towards a Machine-Learned Poisson Solver for Low-Temperature Plasma Simulations in Complex Geometries](https://mlsanity.com/paper/2306.07604/) (2023)
- [Approximation of Solution Operators for High-dimensional PDEs](https://mlsanity.com/paper/2401.10385/) (2024)
- [Fredholm Integral Equations Neural Operator (FIE-NO) for Data-Driven Boundary Value Problems](https://mlsanity.com/paper/2408.12389/) (2024)
- [Virtual domain extension for imposing boundary conditions in flow simulation using pre-trained local neural operator](https://mlsanity.com/paper/2504.09807/) (2025)
- [A Boundary Integral-based Neural Operator for Mesh Deformation](https://mlsanity.com/paper/2602.23703/) (2026)

## Abstract (arXiv)

As an alternative to classical numerical solvers for partial differential equations (PDEs) subject to boundary value constraints, there has been a surge of interest in investigating neural networks that can solve such problems efficiently. In this work, we design a general solution operator for two different time-independent PDEs using graph neural networks (GNNs) and spectral graph convolutions. We train the networks on simulated data from a finite elements solver on a variety of shapes and inhomogeneities. In contrast to previous works, we focus on the ability of the trained operator to generalize to previously unseen scenarios. Specifically, we test generalization to meshes with different shapes and superposition of solutions for a different number of inhomogeneities. We find that training on a diverse dataset with lots of variation in the finite element meshes is a key ingredient for achieving good generalization results in all cases. With this, we believe that GNNs can be used to learn solution operators that generalize over a range of properties and produce solutions much faster than a generic solver. Our dataset, which we make publicly available, can be used and extended to verify the robustness of these models under varying conditions.