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Machine learning methods are increasingly being employed as surrogate models in place of computationally expensive and slow numerical integrators for a bevy of applications in the natural sciences. However, while the laws of physics are relationships between scalars, vectors and tensors that hold regardless of the frame of reference or chosen coordinate system, surrogate machine learning models are not coordinate-free by default. We enforce coordinate freedom by using geometric convolutions in three model architectures: a ResNet, a Dilated ResNet and a UNet. In numerical experiments emulating two-dimensional compressible Navier–Stokes, we see better accuracy and improved stability compared with baseline surrogate models in almost all cases. The ease of enforcing coordinate freedom without making major changes to the model architecture provides an exciting recipe for any convolutional neural network-based method applied to an appropriate class of problems. 

Recent methods to simulate complex fluid dynamics problems have replaced computationally expensive and slow numerical integrators with surrogate models learned from data. However, while the laws of physics are relationships between scalars, vectors, and tensors that hold regardless of the frame of reference or chosen coordinate system, surrogate machine learning models are not coordinate-free by default. We enforce coordinate freedom by using geometric convolutions in three model architectures: a ResNet, a Dilated ResNet, and a UNet. In numerical experiments emulating 2D compressible Navier-Stokes, we see better accuracy and improved stability compared to baseline surrogate models in almost all cases. The ease of enforcing coordinate freedom without making major changes to the model architecture provides an exciting recipe for any CNN-based method applied on an appropriate class of problems. 

Machine learning methods are increasingly being employed as surrogate models in place of computationally expensive and slow numerical integrators for a bevy of applications in the natural sciences. However, while the laws of physics are relationships between scalars, vectors and tensors that hold regardless of the frame of reference or chosen coordinate system, surrogate machine learning models are not coordinate-free by default. We enforce coordinate freedom by using geometric convolutions in three model architectures: a ResNet, a Dilated ResNet and a UNet. In numerical experiments emulating two-dimensional compressible Navier–Stokes, we see better accuracy and improved stability compared with baseline surrogate models in almost all cases. The ease of enforcing coordinate freedom without making major changes to the model architecture provides an exciting recipe for any convolutional neural network-based method applied to an appropriate class of problems. This article is part of the theme issue ‘Partial differential equations in data science’. 

Abstract The changes in colors across a galaxy are intimately connected to the galaxy’s formation, growth, quenching history, and dust content. A particularly important epoch in the growth of galaxies is nearz∼ 2, often referred to as “cosmic noon,” where galaxies on average reach the peak of their star formation. We study a population of 125 cluster galaxies atz∼ 1.6 in three Hubble Space Telescope filters, F475W, F625W, and F160W, roughly corresponding to the rest-frame far-ultraviolet, near-ultraviolet, andrband, respectively. By comparing to a control sample of 200 field galaxies at similar redshift, we reveal clear, statistically significant differences in the overall spatially resolved colors and color gradients in galaxies across these two different environments. On average, cluster galaxies have redder ultraviolet colors in both the inner and outer regions bounded byr₅₀, as well as an overall wider dispersion of outside-in color gradients. The presence of these observed differences, along with evidence from ancillary data from previous studies, strongly suggests that the environment drives these population-level color differences, by affecting the stellar populations and/or dust content.