mlsanityData to 7 Oct 2026

A graph convolutional autoencoder approach to model order reduction for parametrized PDEs

arXiv 2305.08573 ↗First posted 15 May 2023math.NA cs.LG cs.NAcited by 44

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Abstract (arXiv)

The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM). In the reduced order modeling (ROM) context, one is interested in obtaining real-time and many-query evaluations of parametric Partial Differential Equations (PDEs). Linear techniques such as Proper Orthogonal Decomposition (POD) and Greedy algorithms have been analyzed thoroughly, but they are more suitable when dealing with linear and affine models showing a fast decay of the Kolmogorov n-width. On one hand, the autoencoder architecture represents a nonlinear generalization of the POD compression procedure, allowing one to encode the main information in a latent set of variables while extracting their main features. On the other hand, Graph Neural Networks (GNNs) constitute a natural framework for studying PDE solutions defined on unstructured meshes. Here, we develop a non-intrusive and data-driven nonlinear reduction approach, exploiting GNNs to encode the reduced manifold and enable fast evaluations of parametrized PDEs. We show the capabilities of the methodology for several models: linear/nonlinear and scalar/vector problems with fast/slow decay in the physically and geometrically parametrized setting. The main properties of our approach consist of (i) high generalizability in the low-data regime even for complex regimes, (ii) physical compliance with general unstructured grids, and (iii) exploitation of pooling and un-pooling operations to learn from scattered data.

Key facts. Cited by 44 papers in the map (data through 7 Oct 2026). Builds on 44 of its 92 references that are in the map. First posted on arXiv 15 May 2023.

Builds on

44 of 92 references.

References · 44

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
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32 more
  1. 2019Graph U-Netscited by 489
  2. 2016Geometric deep learning on graphs and manifolds using mixture model CNNscited by 468
  3. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  4. 2018Generating 3D faces using Convolutional Mesh Autoencoderscited by 267
  5. 2021Enhancing Computational Fluid Dynamics with Machine Learningcited by 137
  6. 2020A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEscited by 105
  7. 2018Towards Sparse Hierarchical Graph Classifierscited by 100
  8. 2021Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problemscited by 100
  9. 2022MultiScale MeshGraphNetscited by 94
  10. 2022Predicting Physics in Mesh-reduced Space with Temporal Attentioncited by 82
  11. 2020A fast and accurate physics-informed neural network reduced order model with shallow masked autoencodercited by 80
  12. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
  13. 2021Understanding Pooling in Graph Neural Networkscited by 69
  14. 2020Fully Convolutional Mesh Autoencoder using Efficient Spatially Varying Kernelscited by 53
  15. 2022Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunitiescited by 52
  16. 2020A Point-Cloud Deep Learning Framework for Prediction of Fluid Flow Fields on Irregular Geometriescited by 44
  17. 2020Deep learning of thermodynamics-aware reduced-order models from datacited by 38
  18. 2022Thermodynamics-informed graph neural networkscited by 31
  19. 2022Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wallcited by 29
  20. 2021A Comparison of Neural Network Architectures for Data-Driven Reduced-Order Modelingcited by 22
  21. 2023Learning Reduced-Order Models for Cardiovascular Simulations with Graph Neural Networkscited by 20
  22. 2023Multiscale Graph Neural Network Autoencoders for Interpretable Scientific Machine Learningcited by 18
  23. 2023A DeepONet multi-fidelity approach for residual learning in reduced order modelingcited by 14
  24. 2021An artificial neural network approach to bifurcating phenomena in computational fluid dynamicscited by 11
  25. 2021Physics perception in sloshing scenes with guaranteed thermodynamic consistencycited by 10
  26. 2023GNN-based physics solver for time-independent PDEscited by 10
  27. 2022Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEscited by 7
  28. 2022QuadConv: Quadrature-Based Convolutions with Applications to Non-Uniform PDE Data Compressioncited by 6
  29. 2023Low-dimensional Data-based Surrogate Model of a Continuum-mechanical Musculoskeletal System Based on Non-intrusive Model Order Reductioncited by 6
  30. 2022A Continuous Convolutional Trainable Filter for Modelling Unstructured Datacited by 5
  31. 2022Neural-Network-Augmented Projection-Based Model Order Reduction for Mitigating the Kolmogorov Barrier to Reducibility of CFD Modelscited by 5
  32. 2022Agglomeration of Polygonal Grids using Graph Neural Networks with applications to Multigrid solverscited by 3

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Citing papers per year: 2023: 5; 2024: 18; 2025: 11; 2026: 10.

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  1. 2023Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networkscited by 17
  2. 2024Graph Neural PDE Solvers with Conservation and Similarity-Equivariancecited by 15
  3. 2025Machine learning for modelling unstructured grid data in computational physics: a reviewcited by 12
  4. 2024Generating synthetic data for neural operatorscited by 11
  5. 2024GFN: A graph feedforward network for resolution-invariant reduced operator learning in multifidelity applicationscited by 10
  6. 2024Mesh-based Super-Resolution of Fluid Flows with Multiscale Graph Neural Networkscited by 10
  7. 2024A practical existence theorem for reduced order models based on convolutional autoencoderscited by 9
  8. 2026MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanicscited by 8
  9. 2025BARNN: A Bayesian Autoregressive and Recurrent Neural Networkcited by 5
  10. 2024A comparison of Single- and Double-generator formalisms for Thermodynamics-Informed Neural Networkscited by 4
  11. 2024PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEscited by 4
  12. 2024Neural empirical interpolation method for nonlinear model reductioncited by 4
32 more
  1. 2024Enhancing Graph U-Nets for Mesh-Agnostic Spatio-Temporal Flow Predictioncited by 4
  2. 2023Branched Latent Neural Mapscited by 3
  3. 2023On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fieldscited by 3
  4. 2024Data-driven identification of latent port-Hamiltonian systemscited by 3
  5. 2024TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problemscited by 3
  6. 2025A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEscited by 3
  7. 2023Optimal Transport-inspired Deep Learning Framework for Slow-Decaying Kolmogorov n-width Problems: Exploiting Sinkhorn Loss and Wasserstein Kernelcited by 2
  8. 2024Mitigating distribution shift in machine learning-augmented hybrid simulationcited by 2
  9. 2024Polytopic Autoencoders with Smooth Clustering for Reduced-order Modelling of Flowscited by 2
  10. 2024Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reductioncited by 2
  11. 2025Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solverscited by 2
  12. 2025Deep Symmetric Autoencoders from the Eckart-Young-Schmidt Perspectivecited by 2
  13. 2025Autoencoder-based non-intrusive model order reduction in continuum mechanicscited by 2
  14. 2025Time Extrapolation with Graph Convolutional Autoencoder and Tensor Train Decompositioncited by 2
  15. 2026Surrogates for Physics-based and Data-driven Modelling of Parametric Systems: Review and New Perspectivescited by 2
  16. 2026PODiff: Latent Diffusion in Proper Orthogonal Decomposition Space for Scientific Super-Resolutioncited by 2
  17. 2024Multi-Hierarchical Surrogate Learning for Structural Dynamical Crash Simulations Using Graph Convolutional Neural Networkscited by 1
  18. 2024Sparsifying dimensionality reduction of PDE solution data with Bregman learningcited by 1
  19. 2024Edge-Wise Graph-Instructed Neural Networkscited by 1
  20. 2025A discrete physics-informed training for projection-based reduced order models with neural networkscited by 1
  21. 2025Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometrycited by 1
  22. 2025Variational Rank Reduction Autoencoders for Generative Thermal Designcited by 1
  23. 2023MP-FVM: Enhancing Finite Volume Method for Water Infiltration Modeling in Unsaturated Soils via Message-passing Encoder-decoder Networkcited by 0
  24. 2024Parametric Taylor series based latent dynamics identification neural networkscited by 0
  25. 2025Derivative-informed Graph Convolutional Autoencoder with Phase Classification for the Lifshitz-Petrich Modelcited by 0
  26. 2026Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEscited by 0
  27. 2026Tackling multiphysics problems via finite element-guided physics-informed operator learningcited by 0
  28. 2026Graph-Instructed Neural Networks for parametric problems with varying boundary conditionscited by 0
  29. 2026Variational Graph Neural Networks for Uncertainty Quantification in Inverse Problemscited by 0
  30. 2026Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometrycited by 0
  31. 2026Conformal risk control for model-form uncertainty in parametric non-intrusive reduced-order modelscited by 0
  32. 2026Variational Parameter Calibration with Physics-Aware Latent-Space Surrogatescited by 0