Multi-resolution Graph Neural Networks for PDE Approximation

Liu, Wenzhuo; Yagoubi, Mouadh; Schoenauer, Marc

doi:10.1007/978-3-030-86365-4_13

Cited by 4 publications

(6 citation statements)

References 10 publications

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“…In the field of numerical simulations several coarsening algorithms have been developed to guarantee the stability and accuracy of the simulations. Liu et al (2021) and Belbute-Peres et al ( 2020) used coarsening techniques from numerical simulations and then interpolated the nodes attributes into the coarsened set of nodes. We found that our cell-grid coarsening and learnt MP from the high to the low-resolution graph (see section 3.2) performs significantly better than such coarseninginterpolation approach.…”

Section: Coarsening Comparisonmentioning

confidence: 99%

“…As a comparison toPfaff et al (2021), a GNN with 16 sequential MP-layers (GN-Blocks) results in a MAE of 5.852 × 10 −2 on our NSMidRe dataset; whereas MultiScaleGNN with the same number and type of MP layers, but distributed among 3 scales, results in a lower MAE of 3.081 × 10 −2 . A comparison of our coarsening/pooling algorithm toLiu et al (2021) is included in Appendix C.Table1: MAE ×10 −2 on the advection testing datasets for MultiScaleGNN models with L = 1, 2, 3, 4…”

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confidence: 99%

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Towards Fast Simulation of Environmental Fluid Mechanics with Multi-Scale Graph Neural Networks

Lino¹,

Fotiadis²,

Bharath³

et al. 2022

Preprint

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Numerical simulators are essential tools in the study of natural fluid-systems, but their performance often limits application in practice. Recent machine-learning approaches have demonstrated their ability to accelerate spatio-temporal predictions, although, with only moderate accuracy in comparison. Here we introduce MultiScaleGNN, a novel multi-scale graph neural network model for learning to infer unsteady continuum mechanics in problems encompassing a range of length scales and complex boundary geometries. We demonstrate this method on advection problems and incompressible fluid dynamics, both fundamental phenomena in oceanic and atmospheric processes. Our results show good extrapolation to new domain geometries and parameters for long-term temporal simulations. Simulations obtained with MultiScaleGNN are between two and four orders of magnitude faster than those on which it was trained.

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Section: Coarsening Comparisonmentioning

confidence: 99%

mentioning

confidence: 99%

Towards Fast Simulation of Environmental Fluid Mechanics with Multi-Scale Graph Neural Networks

Lino¹,

Fotiadis²,

Bharath³

et al. 2022

Preprint

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“…To the best of our knowledge Alet et al (2019) were the first to explore the use of GNNs to infer continuum physics by solving the Poisson PDE, and subsequently Pfaff et al (2021) proposed a mesh-based GNN to simulate a wide range of continuum dynamics. Multi-resolution graph models were later introduced by Li et al (2020); Lino et al (2021); Liu et al (2021) and Chen et al (2021).…”

Section: Related Workmentioning

confidence: 99%

“…MultiScaleGNNg is a modified version of MultiScaleGNN to follow the pooling and unpooling used by Liu et al (2021). For this model, the low-resolution sets of nodes were generated using Guillard's coarsening algorithm (Guillard, 1993) as in REMuS-GNN.…”

Section: Models Detailsmentioning

confidence: 99%

REMuS-GNN: A Rotation-Equivariant Model for Simulating Continuum Dynamics

Lino¹,

Fotiadis²,

Bharath³

et al. 2022

Preprint

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Numerical simulation is an essential tool in many areas of science and engineering, but its performance often limits application in practice or when used to explore large parameter spaces. On the other hand, surrogate deep learning models, while accelerating simulations, often exhibit poor accuracy and ability to generalise. In order to improve these two factors, we introduce REMuS-GNN, a rotation-equivariant multi-scale model for simulating continuum dynamical systems encompassing a range of length scales. REMuS-GNN is designed to predict an output vector field from an input vector field on a physical domain discretised into an unstructured set of nodes. Equivariance to rotations of the domain is a desirable inductive bias that allows the network to learn the underlying physics more efficiently, leading to improved accuracy and generalisation compared with similar architectures that lack such symmetry. We demonstrate and evaluate this method on the incompressible flow around elliptical cylinders.

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“…In this work, we use a Cartesian grid to discretize each subdomain. However, GNNs [48,49,50,51,52] as well as FEM use other discretizations such as triangle or polygonal meshes. In future, these ideas can be used to handle unstructured meshes in subdomains.Additionally, other ML methods improve the learning of ML models by compressing PDE solutions on to lower-dimensional manifolds.…”

Section: Significant Contributions Of This Workmentioning

confidence: 99%

A composable machine-learning approach for steady-state simulations on high-resolution grids

Ranade¹,

Hill²,

Ghule³

et al. 2022

Preprint

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In this paper we show that our Machine Learning (ML) approach, CoMLSim (Composable Machine Learning Simulator), can simulate PDEs on highly-resolved grids with higher accuracy and generalization to out-of-distribution source terms and geometries than traditional ML baselines. Our unique approach combines key principles of traditional PDE solvers with local-learning and low-dimensional manifold techniques to iteratively simulate PDEs on large computational domains. The proposed approach is validated on more than 5 steady-state PDEs across different PDE conditions on highly-resolved grids and comparisons are made with the commercial solver, Ansys Fluent as well as 4 other state-of-the-art ML methods. The numerical experiments show that our approach outperforms ML baselines in terms of 1) accuracy across quantitative metrics and 2) generalization to out-ofdistribution conditions as well as domain sizes. Additionally, we provide results for a large number of ablations experiments conducted to highlight components of our approach that strongly influence the results. We conclude that our local-learning and iterative-inferencing approach reduces the challenge of generalization that most ML models face.

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Multi-resolution Graph Neural Networks for PDE Approximation

Cited by 4 publications

References 10 publications

Towards Fast Simulation of Environmental Fluid Mechanics with Multi-Scale Graph Neural Networks

Towards Fast Simulation of Environmental Fluid Mechanics with Multi-Scale Graph Neural Networks

REMuS-GNN: A Rotation-Equivariant Model for Simulating Continuum Dynamics

A composable machine-learning approach for steady-state simulations on high-resolution grids

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