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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. This article is part of the theme issue ‘Partial differential equations in data science’. 

Abstract Ground-based high-resolution cross-correlation spectroscopy (HRCCS;R ≳ 15,000) is a powerful complement to space-based studies of exoplanet atmospheres. By resolving individual spectral lines, HRCCS can precisely measure chemical abundance ratios, directly constrain atmospheric dynamics, and robustly probe multidimensional physics. But the subtleties of HRCCS data sets—e.g., the lack of exoplanetary spectra visible by eye and the statistically complex process of telluric removal—can make interpreting them difficult. In this work, we seek to clarify the uncertainty budget of HRCCS with a forward-modeling approach. We present an HRCCS observation simulator,scope,⁵⁵https://github.com/arjunsavel/scopethat incorporates spectral contributions from the exoplanet, star, tellurics, and instrument. This tool allows us to control the underlying data set, enabling controlled experimentation with complex HRCCS methods. Simulating a fiducial hot Jupiter data set (WASP-77Ab emission with IGRINS), we first confirm via multiple tests that the commonly used principal component analysis does not bias the planetary signal when few components are used. Furthermore, we demonstrate that mildly varying tellurics and moderate wavelength solution errors induce only mild decreases in HRCCS detection significance. However, limiting-case, strongly varying tellurics can bias the retrieved velocities and gas abundances. Additionally, in the low signal-to-noise ratio limit, constraints on gas abundances become highly non-Gaussian. Our investigation of the uncertainties and potential biases inherent in HRCCS data analysis enables greater confidence in scientific results from this maturing method.