Machine learning for weather and climate are worlds apart

Modern weather and climate models share a common heritage and often even components; however, they are used in different ways to answer fundamentally different questions. As such, attempts to emulate them using machine learning should reflect this. While the use of machine learning to emulate weather forecast models is a relatively new endeavour, there is a rich history of climate model emulation. This is primarily because while weather modelling is an initial condition problem, which intimately depends on the current state of the atmosphere, climate modelling is predominantly a boundary condition problem. To emulate the response of the climate to different drivers therefore, representation of the full dynamical evolution of the atmosphere is neither necessary, or in many cases, desirable. Climate scientists are typically interested in different questions also. Indeed emulating the steady-state climate response has been possible for many years and provides significant speed increases that allow solving inverse problems for e.g. parameter estimation. Nevertheless, the large datasets, non-linear relationships and limited training data make climate a domain which is rich in interesting machine learning challenges. Here, I seek to set out the current state of climate model emulation and demonstrate how, despite some challenges, recent advances in machine learning provide new opportunities for creating useful statistical models of the climate. This article is part of the theme issue ‘Machine learning for weather and climate modelling’.

Pricing Stock Loans with the CGMY Model

The empirical research shows that the log-return of stock price in finance market rejects the normal distribution and admits a subclass of the asymmetric distribution. Hence, the pricing problem of stock loan is investigated under the assumption that the log-return of stock price follows the CGMY process in this work. Under this framework, the pricing model of stock loan can be described by a free boundary condition problem of space-fractional partial differential equation (FPDE). First of all, in order to change the original model defined in a fixed domain, a penalty term is introduced, and then a first order fully implicit difference schemes is developed. Secondly, based on the numerical scheme, we prove the stock loan value generated by our method does not fall below the value obtained when the contract of stock loan is exercised early. Finally, the numerical experiments are implemented and the impacts of key parameters in the CGMY model on the value and optimal redemption price of stock loan are analyzed, and some reasonable explanation should be given.

Initial wave function of the universe is arbitrary

We consider quantization of the gravity-scalar field system in the minisuperspace approximation. It turns out that in the gauge fixed deparametrized theory where the scale factor plays the role of time, the Hamiltonian can be uniquely defined without any ordering ambiguity as the square root of a self-adjoint operator. Moreover, the Hamiltonian degenerates to zero and the Schrödinger equation becomes well behaved as the scale factor vanishes. Therefore, there is no technical or physical obstruction for the initial wave function of the universe to be an arbitrary vector in the Hilbert space, which demonstrates the severeness of the initial condition problem in quantum cosmology.

A plausible solution to the problem of initial conditions for inflation

We propose yet another solution to the initial condition problem of inflation associated with homogeneity beyond the horizon at the onset of inflation, in cases where inflation is preceded by a radiation era. One may argue that causality will allow for smoothness over the causal horizon scale [Formula: see text], but for thermal inflationary scenarios, the background inflaton field will only be correlated over the thermal correlation length [Formula: see text] which is much smaller than [Formula: see text]. We argue, with examples, that if the number of relativistic degrees of freedom in the preinflationary era is very large [Formula: see text] then the thermal correlation length can be of the order of the causal horizon size alleviating the initial conditions problem of inflation.

A bifurcation result involving Sobolev trace embedding and the duality mapping of W1,p

We consider the perturbed nonlinear boundary condition problem$$\left\{ {\matrix{ { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u + f\left( {\lambda ,x,u} \right)\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$Using the Sobolev trace embedding and the duality mapping defined on W1,p(Ω), we prove that this problem bifurcates from the principal eigenvalue λ1 of the eigenvalue problem$$\left\{ {\matrix{ { - \Delta _p u} \hfill & = \hfill & {\left| u \right|^{p - 2} u\,{\rm{in}}\,\Omega } \hfill \cr {\left| {\nabla u} \right|^{p - 2} \nabla u.\nu } \hfill & = \hfill & {\lambda \rho \left( x \right)\left| u \right|^{p - 2} u\,{\rm{on}}\,\Gamma .} \hfill \cr } } \right.$$