# Automated discovery of finite volume schemes using Graph Neural Networks

arXiv: https://arxiv.org/abs/2508.19052 · HTML: https://mlsanity.com/paper/2508.19052/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 27 of its 45 references that are in the map.
- First posted on arXiv 26 Aug 2025.

## Builds on (most cited first, 27 of 27)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [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)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Language Modeling with Gated Convolutional Networks](https://mlsanity.com/paper/1612.08083/) (2016)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [Graph networks as learnable physics engines for inference and control](https://mlsanity.com/paper/1806.01242/) (2018)
- [Neural Execution of Graph Algorithms](https://mlsanity.com/paper/1910.10593/) (2019)
- [Neural Algorithmic Reasoning](https://mlsanity.com/paper/2105.02761/) (2021)
- [Flexible Neural Representation for Physics Prediction](https://mlsanity.com/paper/1806.08047/) (2018)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [Graph Neural Networks are Dynamic Programmers](https://mlsanity.com/paper/2203.15544/) (2022)
- [What Has a Foundation Model Found? Using Inductive Bias to Probe for World Models](https://mlsanity.com/paper/2507.06952/) (2025)
- [Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks](https://mlsanity.com/paper/2201.06780/) (2022)
- [Graph neural networks extrapolate out-of-distribution for shortest paths](https://mlsanity.com/paper/2503.19173/) (2025)
- [Multi-Grid Graph Neural Networks with Self-Attention for Computational Mechanics](https://mlsanity.com/paper/2409.11899/) (2024)
- [MeshMask: Physics-Based Simulations with Masked Graph Neural Networks](https://mlsanity.com/paper/2501.08738/) (2025)
- [High precision PINNs in unbounded domains: application to singularity formulation in PDEs](https://mlsanity.com/paper/2506.19243/) (2025)
- [Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp](https://mlsanity.com/paper/2310.05169/) (2023)

## Abstract (arXiv)

Graph Neural Networks (GNNs) have deeply modified the landscape of numerical simulations by demonstrating strong capabilities in approximating solutions of physical systems. However, their ability to extrapolate beyond their training domain (textite.g. larger or structurally different graphs) remains uncertain. In this work, we establish that GNNs can serve purposes beyond their traditional role, and be exploited to generate numerical schemes, in conjunction with symbolic regression. First, we show numerically and theoretically that a GNN trained on a dataset consisting solely of two-node graphs can extrapolate a first-order Finite Volume (FV) scheme for the heat equation on out-of-distribution, unstructured meshes. Specifically, if a GNN achieves a loss varepsilon on such a dataset, it implements the FV scheme with an error of O(varepsilon). Using symbolic regression, we show that the network effectively rediscovers the exact analytical formulation of the standard first-order FV scheme. We then extend this approach to an unsupervised context: the GNN recovers the first-order FV scheme using only a residual loss similar to Physics-Informed Neural Networks (PINNs) with no access to ground-truth data. Finally, we push the methodology further by considering higher-order schemes: we train (i) a 2-hop and (ii) a 2-layers GNN using the same PINN loss, that autonomously discover (i) a second-order correction term to the initial scheme using a 2-hop stencil, and (ii) the classic second-order midpoint scheme. These findings follows a recent paradigm in scientific computing: GNNs are not only strong approximators, but can be active contributors to the development of novel numerical methods.