Conservation laws in a neural network architecture: Enforcing the atom balance of a Julia-based photochemical model (v0.2.0)

Models of atmospheric phenomena provide insight into climate, air quality, and meteorology, and provide a mechanism for understanding the effect of future emissions scenarios. To accurately represent atmospheric phenomena, these models consume vast quantities of computational resources. Machine learning (ML) techniques such as neural networks have the potential to emulate compute-intensive components of these models to reduce their computational burden. However, such ML surrogate models may lead to nonphysical predictions that are difficult to uncover. Here we present a neural network architecture that enforces conservation laws. Instead of simply predicting properties of interest, a physically interpretable hidden layer within the network predicts fluxes between properties which are subsequently related to the properties of interest. As an example, we design a physics-constrained neural network surrogate model of photochemistry using this approach and find that it conserves atoms as they flow between molecules to machine precision, while outperforming a naïve neural network in terms of accuracy and non-negativity of concentrations.

Precise Numerical Differentiation of Thermodynamic Functions with Multicomplex Variables

The multicomplex finite-step method for numerical differentiation is an extension of the popular Squire–Trapp method, which uses complex arithmetics to compute first-order derivatives with almost machine precision. In contrast to this, the multicomplex method can be applied to higher-order derivatives. Furthermore, it can be applied to functions of more than one variable and obtain mixed derivatives. It is possible to compute various derivatives at the same time. This work demonstrates the numerical differentiation with multicomplex variables for some thermodynamic problems. The method can be easily implemented into existing computer programs, applied to equations of state of arbitrary complexity, and achieves almost machine precision for the derivatives. Alternative methods based on complex integration are discussed, too.

A structure-preserving finite element method for compressible ideal and resistive magnetohydrodynamics

We construct a structure-preserving finite element method and time-stepping scheme for compressible barotropic magnetohydrodynamics both in the ideal and resistive cases, and in the presence of viscosity. The method is deduced from the geometric variational formulation of the equations. It preserves the balance laws governing the evolution of total energy and magnetic helicity, and preserves mass and the constraint $\text {div}B = 0$ to machine precision, both at the spatially and temporally discrete levels. In particular, conservation of energy and magnetic helicity hold at the discrete levels in the ideal case. It is observed that cross-helicity is well conserved in our simulation in the ideal case.

Grain growth for astrophysics with Discontinuous Galerkin schemes

Depending on their sizes, dust grains store more or less charges, catalyse more or less chemical reactions, intercept more or less photons and stick more or less efficiently to form embryos of planets. Hence the need for an accurate treatment of dust coagulation and fragmentation in numerical modelling. However, existing algorithms for solving the coagulation equation are over-diffusive in the conditions of 3D simulations. We address this challenge by developing a high-order solver based on the Discontinuous Galerkin method. This algorithm conserves mass to machine precision and allows to compute accurately the growth of dust grains over several orders of magnitude in size with a very limited number of dust bins.

A consistent implementation of point sources on finite-difference grids

SUMMARY We present a method to position point sources at arbitrary locations on finite-difference (FD) grids. We show that implementing point sources on single nodes can cause considerable errors when modelling with the FD method. In contrast, we propose to create a spatially distributed source (over multiple nodes) that nonetheless creates the desired point-source response. The spatial point source is formulated in the wavenumber domain and is based on the FD coefficients used for the wave propagation. Using this ‘FD-consistent source’ on 1-D and 2-D examples, we show that we can obtain superior fits to analytical solutions compared to single-node or sinc-function source implementations, and we show that sources can be offset to arbitrary locations from ‘on’ the grid to ‘off’ the grid, while resulting in solutions that are identical to within machine precision