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In this chapter we review stochastic modelling methods in climate science. First we provide a conceptual framework for stochastic modelling of deterministic dynamical systems based on the Mori-Zwanzig formalism. The Mori-Zwanzig equations…

Simulating abundances of stable water isotopologues, i.e. molecules differing in their isotopic composition, within climate models allows for comparisons with proxy data and, thus, for testing hypotheses about past climate and validating…

Parameters in climate models are usually calibrated manually, exploiting only small subsets of the available data. This precludes both optimal calibration and quantification of uncertainties. Traditional Bayesian calibration methods that…

We propose a method to efficiently determine the optimal coarse-grained force field in mesoscopic stochastic simulations of Newtonian fluid and polymer melt systems modeled by dissipative particle dynamics (DPD) and energy conserving…

Recent years have seen a surge in interest in building deep learning-based fully data-driven models for weather prediction. Such deep learning models if trained on observations can mitigate certain biases in current state-of-the-art weather…

The stochastic variational approach for geophysical fluid dynamics was introduced by Holm (Proc Roy Soc A, 2015) as a framework for deriving stochastic parameterisations for unresolved scales. This paper applies the variational stochastic…

Due to computational constraints, running global climate models (GCMs) for many years requires a lower spatial grid resolution (${\gtrsim}50$ km) than is optimal for accurately resolving important physical processes. Such processes are…

Real-time simulation of elastic structures is essential in many applications, from computer-guided surgical interventions to interactive design in mechanical engineering. The Finite Element Method is often used as the numerical method of…

Errors in the representation of clouds in convection-permitting numerical weather prediction models can be introduced by different sources. These can be the forcing and boundary conditions, the representation of orography, the accuracy of…

Sub-seasonal climate forecasting (SSF) focuses on predicting key climate variables such as temperature and precipitation in the 2-week to 2-month time scales. Skillful SSF would have immense societal value, in areas such as agricultural…

This paper presents a novel generative model to synthesize fluid simulations from a set of reduced parameters. A convolutional neural network is trained on a collection of discrete, parameterizable fluid simulation velocity fields. Due to…

AI weather prediction ensembles with latent noise injection and optimized with the continuous ranked probability score (CRPS) have produced both accurate and well-calibrated predictions with far less computational cost compared with…

The under-representation of cloud formation is a long-standing bias associated with climate simulations. Parameterisation schemes are required to capture cloud processes within current climate models but have known biases. We overcome these…

Neural networks (NNs) are increasingly used for data-driven subgrid-scale parameterization in weather and climate models. While NNs are powerful tools for learning complex nonlinear relationships from data, there are several challenges in…

Machine-learning (ML) parameterizations of subgrid processes (here of turbulence, convection, and radiation) may one day replace conventional parameterizations by emulating high-resolution physics without the cost of explicit simulation.…

Machine learning (ML)-based models have demonstrated high skill and computational efficiency, often outperforming conventional physics-based models in weather and subseasonal predictions. While prior studies have assessed their fidelity in…

Machine learning (ML) techniques, especially neural networks (NNs), have shown promise in learning subgrid-scale parameterizations for climate models. However, a major problem with data-driven parameterizations, particularly those learned…

State-of-the-art training algorithms for deep learning models are based on stochastic gradient descent (SGD). Recently, many variations have been explored: perturbing parameters for better accuracy (such as in Extragradient), limiting SGD…

Global Climate Models (GCMs) are numerical models that simulate complex physical processes within the Earth's climate system and are essential for understanding and predicting climate change. However, GCMs suffer from systemic biases due to…

Radiation is typically the most time-consuming physical process in numerical models. One solution is to use machine learning methods to simulate the radiation process to improve computational efficiency. From an operational standpoint, this…