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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. 

Abstract By using surface brightness maps of Tycho’s supernova remnant (SNR) in radio and X-rays, along with the properties of thermal and synchrotron emission, we have derived the postshock density and magnetic field (MF) strength distributions over the projection of this remnant. Our analysis reveals a density gradient oriented toward the northwest, while the MF strength gradient aligns with the Galactic plane, pointing eastward. Additionally, utilizing this MF map, we have derived the spatial distributions of the cutoff frequency and maximum energy of electrons in Tycho’s SNR. We further comment on the implications of these findings for interpreting the gamma-ray emission from Tycho’s SNR.