# Machine Learning Methods in Weather and Climate Applications: A Survey
## 6 Medium-to-Long-Term Climate Prediction
### Model Design
Leveraging 52,201 years of climate simulation data from CMIP5/ CMIP6, which serves to increase the sample size, the method for medium-term forecasting employs CNNs and Temporal Convolutional Neural Networks (TCNNs) to extract essential features from high-dimensional geospatial data. This feature extraction lays the foundation for probabilistic deep learning, which determines an approximate distribution of the target variables, capturing the data's structure and uncertainty [95]. The model's parameters are optimized by maximizing the Evidence Lower Bound (ELBO) within the variational inference framework. The structure is shown in Figure 6. The integration of deep learning techniques with probabilistic modeling ensures accuracy, robustness to sparse data, and flexibility in assumptions, enhancing the precision of forecasts and offering valuable insights into confidence levels and expert knowledge integration. Leveraging advanced techniques in variational inference and neural networks, the method described seeks to approximate the complex distribution \(p(Y\mid X,M)\), where \(Y\) is the target variable and \(X\) and \(M\) are predictor and GCM index information, respectively. The process is outlined as follows:  
1. _Problem Definition:_ The goal is to approximate \(p(Y\mid X,M)\), a task challenged by high-dimensional geospatial data, data inhomogeneity, and a large dataset. 2. _Model Specification:_ * Random Variable \(z\): A latent variable with a fixed standard Gaussian distribution. * Parametric Functions \(p_{\theta},q_{\phi},p_{\phi}\): Neural networks for transforming \(z\) and approximating target and posterior distributions. * Objective Function: Maximization of the Evidence Lower Bound (ELBO). Figure 6: Conditional Generative Forecasting (CGF) model. 3. _Training Procedure:_ * Initialize: Define random variable \(z\sim N(0,1)\)[96; 97] * parametric functions \(p_{\theta}(z,X,M),q_{\phi}(z\mid X,Y,M),p_{\psi}(Y\mid X,M,z)\). * Training Objective (Maximize ELBO) [98]: The ELBO is defined as: \[\text{ELBO}=\mathbb{E}_{z\sim q_{\phi}}\big{(}\log p_{\psi}(Y\mid X,M,z)\big{)} -\mathcal{D}_{\text{KL}}(q_{\phi}\|p(z\mid X,M))-\mathcal{D}_{\text{KL}}(q_{ \phi}\|p(z\mid X,Y,M))\] (8) with terms for reconstruction, regularization, and residual error. * Optimization: Utilize variational inference, Monte Carlo reparameterization, and Gaussian assumptions. 4. _Forecasting:_ Generate forecasts by sampling \(p(z\mid X,M)\), the likelihood of \(p_{\psi}\), and using the mean of \(p_{\psi}\) for an average estimate. This method embodies a rigorous approach to approximating complex distributions, bridging deep learning and probabilistic modeling to enhance forecasting accuracy and insights. \[\text{ELBO}(\lambda)=\mathbb{E}_{q(\mathbf{z}|\mathbf{x})}[\log p(\mathbf{x}, \mathbf{z})-\log q(\mathbf{z}|\mathbf{x})]\ (\text{Evidence Lower Bound})\] (9)  
In summary, the combination of deep learning and probabilistic insights presents a unique and potent method for spatial predictive analytics. The approach is marked by scalability, flexibility, and an ability to learn complex spatial features, even though challenges persist, such as intrinsic complexity in computational modeling and the requirement for a profound statistical and computer science background. Its potential in handling large data sets and adapting to varying scenarios highlights its promising applicability in modern spatial predictive analytics, representing an advanced tool in the arena of seasonal climate prediction. **Cycle-Consistent Generative Adversarial Networks. ** Cycle-Consistent Generative Adversarial Networks (CycleGANs) have been ingeniously applied to the bias correction of high-resolution Earth System Model (ESM) precipitation fields, such as GFDL-ESM4 [99]. This model includes two generators responsible for translating between simulated and real domains, and two discriminators to differentiate between generated and real observations.