# Interpretable A-posteriori Error Indication for Graph Neural Network Surrogate Models

arXiv: https://arxiv.org/abs/2311.07548 · HTML: https://mlsanity.com/paper/2311.07548/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 56 references that are in the map.
- First posted on arXiv 13 Nov 2023.

## Builds on (most cited first, 25 of 25)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Evaluating the visualization of what a Deep Neural Network has learned](https://mlsanity.com/paper/1509.06321/) (2015)
- [A Survey on Neural Network Interpretability](https://mlsanity.com/paper/2012.14261/) (2020)
- [GraphCast: Learning skillful medium-range global weather forecasting](https://mlsanity.com/paper/2212.12794/) (2022)
- [Interpretable Convolutional Neural Networks](https://mlsanity.com/paper/1710.00935/) (2017)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [MultiScale MeshGraphNets](https://mlsanity.com/paper/2210.00612/) (2022)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations](https://mlsanity.com/paper/2211.00713/) (2022)
- [PhysGNN: A Physics-Driven Graph Neural Network Based Model for Predicting Soft Tissue Deformation in Image-Guided Neurosurgery](https://mlsanity.com/paper/2109.04352/) (2021)
- [Multiscale Graph Neural Network Autoencoders for Interpretable Scientific Machine Learning](https://mlsanity.com/paper/2302.06186/) (2023)
- [Conditionally Parameterized, Discretization-Aware Neural Networks for Mesh-Based Modeling of Physical Systems](https://mlsanity.com/paper/2109.09510/) (2021)
- [Assessments of epistemic uncertainty using Gaussian stochastic weight averaging for fluid-flow regression](https://mlsanity.com/paper/2109.08248/) (2021)
- [Jacobian-Scaled K-means Clustering for Physics-Informed Segmentation of Reacting Flows](https://mlsanity.com/paper/2305.01539/) (2023)

## Built on by (most cited first, 6 of 6)

- [Mesh-based Super-Resolution of Fluid Flows with Multiscale Graph Neural Networks](https://mlsanity.com/paper/2409.07769/) (2024)
- [Learning Effective Dynamics across Spatio-Temporal Scales of Complex Flows](https://mlsanity.com/paper/2502.07990/) (2025)
- [FIGNN: Feature-Specific Interpretability for Graph Neural Network Surrogate Models](https://mlsanity.com/paper/2506.11398/) (2025)
- [Interpretable Diagnostics and Adaptive Data Assimilation for Neural ODEs via Discrete Empirical Interpolation](https://mlsanity.com/paper/2510.21852/) (2025)
- [Physics-integrated neural differentiable modeling for immersed boundary systems](https://mlsanity.com/paper/2603.16277/) (2026)
- [Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decomposition](https://mlsanity.com/paper/2605.25115/) (2026)

## Abstract (arXiv)

Data-driven surrogate modeling has surged in capability in recent years with the emergence of graph neural networks (GNNs), which can operate directly on mesh-based representations of data. The goal of this work is to introduce an interpretability enhancement procedure for GNNs, with application to unstructured mesh-based fluid dynamics modeling. Given a black-box baseline GNN model, the end result is an interpretable GNN model that isolates regions in physical space, corresponding to sub-graphs, that are intrinsically linked to the forecasting task while retaining the predictive capability of the baseline. These structures identified by the interpretable GNNs are adaptively produced in the forward pass and serve as explainable links between the baseline model architecture, the optimization goal, and known problem-specific physics. Additionally, through a regularization procedure, the interpretable GNNs can also be used to identify, during inference, graph nodes that correspond to a majority of the anticipated forecasting error, adding a novel interpretable error-tagging capability to baseline models. Demonstrations are performed using unstructured flow field data sourced from flow over a backward-facing step at high Reynolds numbers, with geometry extrapolations demonstrated for ramp and wall-mounted cube configurations.