Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs

Haan, Pim de; Weiler, Maurice; Cohen, Taco; Welling, Max

doi:10.48550/arxiv.2003.05425

Cited by 17 publications

(28 citation statements)

References 19 publications

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“…More complex architectures, with transformations like Eq. ( 1) after every convolution layer can be used too (de Haan et al, 2020;Wang et al, 2020). Our preliminary exploration shows that for this work, the one spatial transformer module applied on the latent space of the U-NET yields sufficiently superior performance (over the baseline, U-NET), but further exhaustive explorations should be conducted in future studies to find the best performing architecture for each application.…”

Section: Decoding Blockmentioning

confidence: 90%

Towards physically consistent data-driven weather forecasting: Integrating data assimilation with equivariance-preserving spatial transformers in a case study with ERA5

Chattopadhyay

Mustafa

Hassanzadeh

et al. 2021

Preprint

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Abstract. There is growing interest in data-driven weather prediction (DDWP), for example using convolutional neural networks such as U-NETs that are trained on data from models or reanalysis. Here, we propose 3 components to integrate with commonly used DDWP models in order to improve their physical consistency and forecast accuracy. These components are 1) a deep spatial transformer added to the latent space of the U-NETs to preserve a property called equivariance, which is related to correctly capturing rotations and scalings of features in spatio-temporal data, 2) a data-assimilation (DA) algorithm to ingest noisy observations and improve the initial conditions for next forecasts, and 3) a multi-time-step algorithm, which combines forecasts from DDWP models with different time steps through DA, improving the accuracy of forecasts at short intervals. To show the benefit/feasibility of each component, we use geopotential height at 500 hPa (Z500) from ERA5 reanalysis and examine the short-term forecast accuracy of specific setups of the DDWP framework. Results show that the equivariance-preserving networks (U-STNs) clearly outperform the U-NETs, for example improving the forecast skill by 45 %. Using a sigma-point ensemble Kalman (SPEnKF) algorithm for DA and U-STN as the forward model, we show that stable, accurate DA cycles are achieved even with high observation noise. The DDWP+DA framework substantially benefits from large (O(1000)) ensembles that are inexpensively generated with the data-driven forward model in each DA cycle. The multi-time-step DDWP+DA framework also shows promises, e.g., it reduces the average error by factors of 2–3. These results show the benefits/feasibilities of these 3 components, which are flexible and can be used in a variety of DDWP setups. Furthermore, while here we focus on weather forecasting, the 3 components can be readily adopted for other parts of the Earth system, such as ocean and land, for which there is a rapid growth of data and need for forecast/assimilation.

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Section: Decoding Blockmentioning

confidence: 90%

Towards physically consistent data-driven weather forecasting: Integrating data assimilation with equivariance-preserving spatial transformers in a case study with ERA5

Chattopadhyay

Mustafa

Hassanzadeh

et al. 2021

Preprint

View full text Add to dashboard Cite

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“…This will be done in the context of graph networks where gauge equivariance corresponds to changes of coordinate system in the tangent space at each point in the graph. The main references for this section are [43,33,34]. We begin by introducing some relevant background on harmonic networks.…”

Section: Explicit Gauge Equivariant Convolutionmentioning

confidence: 99%

“…Gauge-equivariant mesh CNNs. A different approach to gauge equivariant networks on meshes was given in [33]. A mesh can be viewed as a discretization of a two-dimensional manifold.…”

Section: Explicit Gauge Equivariant Convolutionmentioning

confidence: 99%

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Geometric Deep Learning and Equivariant Neural Networks

Gerken¹,

Aronsson²,

Carlsson³

et al. 2021

Preprint

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We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M = G/K, which are instead equivariant with respect to the global symmetry G on M. Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M = S 2 = SO(3)/SO(2). Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch-Gordan coefficients for G = SO(3), illustrating the power of representation theory for deep learning.

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“…A general manifold, and meshes in particular, lacks global symmetries that can be exploited, but one can define some locally equivariant operations. Cohen et al in [11] and [12] introduce a specific type of features on node tangent space and derive gauge equivariant convolution.…”

Section: Meshed Representationmentioning

confidence: 99%

Surrogate Modelling for Injection Molding Processes using Machine Learning

Uglov¹,

Nikolaev²,

Белов³

et al. 2021

Preprint

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Injection molding is one of the most popular manufacturing methods for the modeling of complex plastic objects. Faster numerical simulation of the technological process would allow for faster and cheaper design cycles of new products. In this work, we propose a baseline for a data processing pipeline that includes the extraction of data from Moldflow simulation projects and the prediction of the fill time and deflection distributions over 3-dimensional surfaces using machine learning models. We propose algorithms for engineering of features, including information of injector gates parameters that will mostly affect the time for plastic to reach the particular point of the form for fill time prediction, and geometrical features for deflection prediction. We propose and evaluate baseline machine learning models for fill time and deflection distribution prediction and provide baseline values of MSE and RMSE metrics. Finally, we measure the execution time of our solution and show that it significantly exceeds the time of simulation with Moldflow software: approximately 17 times and 14 times faster for mean and median total times respectively, comparing the times of all analysis stages for deflection prediction. Our solution has been implemented in a prototype web application that was approved by the management board of Fiat Chrysler Automobiles and Illogic SRL. As one of the promising applications of this surrogate modelling approach, we envision the use of trained models as a fast objective function in the task of optimization of technological parameters of the injection molding process (meaning optimal placement of gates), which could significantly aid engineers in this task, or even automate it.

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Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs

Cited by 17 publications

References 19 publications

Towards physically consistent data-driven weather forecasting: Integrating data assimilation with equivariance-preserving spatial transformers in a case study with ERA5

Towards physically consistent data-driven weather forecasting: Integrating data assimilation with equivariance-preserving spatial transformers in a case study with ERA5

Geometric Deep Learning and Equivariant Neural Networks

Surrogate Modelling for Injection Molding Processes using Machine Learning

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