Impact Statement
This new cloud microphysics parameterization has the potential to enhance climate model accuracy, contributing to a better understanding of climate-relevant processes, in particular taking into account the influence of highly resolved dynamics on cloud microphysical processes. More accurate and robust climate projections are essential for informed decision-making and risk assessment in addressing the global warming crisis.
1. Introduction
The parameterization of cloud microphysics is a central component in Earth system models (ESMs), working closely coupled with the convection scheme to model the behavior of clouds in coarse-scale ESMs. It calculates the formation, growth, and removal of liquid and solid water particles and captures phase changes. The accurate representation of these processes is crucial for reliable climate projections, as they significantly influence the lifetime and brightness of clouds and, in general, the transport of water and energy through the atmosphere. As uncertainties in microphysical processes, particularly those involving ice particles, can greatly affect climate models, addressing these challenges is essential for improving climate projections (Morrison et al., Reference Morrison, van Lier-Walqui, Fridlind, Grabowski, Harrington, Hoose, Korolev, Kumjian, Milbrandt, Pawlowska, Posselt, Prat, Reimel, Shima, van Diedenhoven and Xue2020).
The focus of this study is to enhance the representation of subgrid-scale cloud microphysics within the coarse-scale icosahedral nonhydrostatic modeling framework (ICON) model (Zängl et al., Reference Zängl, Reinert, Rípodas and Baldauf2014; Giorgetta et al., Reference Giorgetta, Sawyer, Lapillonne, Adamidis, Alexeev, Clément, Dietlicher, Engels, Esch, Franke, Frauen, Hannah, Hillman, Kornblueh, Marti, Norman, Pincus, Rast, Reinert, Schnur, Schulzweida and Stevens2018), which traditionally employs the single-moment microphysical scheme of Lohmann and Roeckner (Reference Lohmann and Roeckner1996). This scheme solves prognostic equations for the mass mixing ratios (mmrs) of water vapor, cloud water, cloud ice, and rain. To improve upon this baseline, we develop a machine learning (ML)-based parameterization trained on high-resolution ICON simulation data that use a more complex single-moment scheme, that is, the graupel scheme of Baldauf et al. (Reference Baldauf, Seifert, Förstner, Majewski, Raschendorfer and Reinhardt2011), Rutledge and Hobbs (Reference Rutledge and Hobbs1984). Another option would have been to use a two-moment cloud microphysics scheme (Seifert and Beheng, Reference Seifert and Beheng2006). These schemes provide a more detailed representation of cloud processes but are computationally significantly more expensive, especially for convection-permitting simulations at high resolution. Due to the computational constraints of our study, we chose a single-moment scheme. This scheme has been used in convection-permitting high-resolution simulations (simulation data for this study), as well as in coarse simulations together with an additional convection parameterization. In addition to the scheme of Lohmann and Roeckner (Reference Lohmann and Roeckner1996), it calculates the mmr of snow and graupel and provides precipitation rates for rain, snow, and graupel. All microphysical schemes considered here also solve the prognostic equations for temperature resulting from microphysical processes, such as latent heat release from phase changes.
ML made great strides in enhancing ESMs by replacing traditional parameterizations that are based on empirical and physical understanding and representing the statistical effect of a given process at the grid scale of the climate model (Rasp et al., Reference Rasp, Pritchard and Gentine2018; Reichstein et al., Reference Reichstein, Camps-Valls, Stevens, Jung, Denzler, Carvalhais and Prabhat2019; Grundner et al., Reference Grundner, Beucler, Gentine, Iglesias-Suarez, Giorgetta and Eyring2022; Fuchs et al., Reference Fuchs, Sherwood, Prasad, Trapeznikov and Gimlett2024). For this, ML models are trained on short high-resolution climate simulations or observations and then coupled to the coarse climate model, potentially eliminating long-standing biases in ESMs (Gentine et al., Reference Gentine, Eyring and Beucler2021; Eyring et al., Reference Eyring, Collins, Gentine, Barnes, Barreiro, Beucler, Bocquet, Bretherton, Christensen, Dagon, Gagne, Hall, Hammerling, Hoyer, Iglesias-Suarez, Lopez-Gomez, McGraw, Meehl, Molina and Zanna2024). Over the last years, learning the cloud microphysics parameterization with ML models has been an active field of research (Han et al., Reference Han, Zhang, Huang and Wang2020; Gettelman et al., Reference Gettelman, Gagne, Chen, Christensen, Lebo, Morrison and Gantos2021; Perkins et al., Reference Perkins, Brenowitz, Bretherton and Nugent2023; Seifert and Siewert, Reference Seifert and Siewert2024; Sharma and Greenberg, Reference Sharma and Greenberg2024). Existing ML approaches often struggle with the balance between computational efficiency and physical accuracy, particularly when upscaling from high-resolution data to coarser model grids.
So far, studies have emulated cloud microphysics parameterizations using the same model and resolution as the simulation data (Han et al., Reference Han, Zhang, Huang and Wang2020; Gettelman et al., Reference Gettelman, Gagne, Chen, Christensen, Lebo, Morrison and Gantos2021; Perkins et al., Reference Perkins, Brenowitz, Bretherton and Nugent2023). These approaches aim to improve computational efficiency by accelerating ESMs while precisely replicating existing parameterizations, without expecting to enhance the physical representation of microphysics. Alternatively, some approaches learn from more detailed microphysics parameterizations using superdroplet box models (Seifert and Siewert, Reference Seifert and Siewert2024; Sharma and Greenberg, Reference Sharma and Greenberg2024). These methods aim to enhance the representation of cloud microphysics and reduce biases in coarse-resolution ESMs through detailed high-resolution simulations. However, switching from superdroplet box models to global climate simulations can introduce issues. The differences in scale, boundary conditions, and dynamics between box models and ESMs can complicate the integration and accuracy of ML parameterizations. Furthermore, the computational requirements of ML-based superdroplet parameterizations differ significantly from classical parameterizations typically used in an ESM. This adds an extra layer of technical complexity when implementing such models in highly nonlinear systems like ICON.
In this study, we aimed to leverage the strengths of the previously mentioned approaches. We expand upon our previous work presented in Sarauer et al. (Reference Sarauer, Schwabe, Lauer, Stier, Weiss and Eyring2024) and now provide a more comprehensive analysis. We use simulations from our target model, ICON, to generate training data for the ML parameterization. However, instead of merely emulating the model, we enhance our approach by learning from simulations with a better resolution at a convection-permitting horizontal scale of 5 km and use a more detailed single-moment scheme, that is, the graupel scheme. As a consequence, at the target resolution of 80 km, the ML model is able to preserve the influence of crucial subgrid-scale dynamics and their nonlinear interactions with the larger-scale flow on cloud microphysics. Although this approach complicates the distinction between contributions from subgrid-scale dynamics and microphysical processes, it ensures a more accurate representation of their combined influence on cloud properties and their temporal evolution at coarser resolutions. This will be especially beneficial when parameterizing cloud convection in a combined parameterization with microphysics in future studies, where other dynamical parameters such as vertical updraft velocity or aerosol activation will be taken into account as additional input parameters. Even though the ML parameterization has not yet been coupled to ICON, this approach is believed to be able to enhance the representation of cloud microphysical processes in climate models, with the potential for more accurate climate projections by the resulting hybrid ESM (Eyring et al., Reference Eyring, Collins, Gentine, Barnes, Barreiro, Beucler, Bocquet, Bretherton, Christensen, Dagon, Gagne, Hall, Hammerling, Hoyer, Iglesias-Suarez, Lopez-Gomez, McGraw, Meehl, Molina and Zanna2024).
2. Methodology
2.1. Underlying dataset
We generate high-resolution data through a simulation setup utilizing the atmospheric component of the ESM ICON Sapphire (Hohenegger et al., Reference Hohenegger, Korn, Linardakis, Redler, Schnur, Adamidis, Bao, Bastin, Behravesh, Bergemann, Biercamp, Bockelmann, Brokopf, Brüggemann, Casaroli, Chegini, Datseris, Esch, George, Giorgetta, Gutjahr, Haak, Hanke, Ilyina, Jahns, Jungclaus, Kern, Klocke, Kluft, Kölling, Kornblueh, Kosukhin, Kroll, Lee, Mauritsen, Mehlmann, Mieslinger, Naumann, Paccini, Peinado, Praturi, Putrasahan, Rast, Riddick, Roeber, Schmidt, Schulzweida, Schütte, Segura, Shevchenko, Singh, Specht, Stephan, von Storch, Vogel, Wengel, Winkler, Ziemen, Marotzke and Stevens2023; Segura et al., Reference Segura, Pedruzo-Bagazgoitia, Weiss, Muller, Rackow, Lee, Dolores-Tesillos, Benedict, Aengenheyster, Aguridan, Arduini, Baker, Bao, Bastin, Baulenas, Becker, Beyer, Bockelmann, Bruggemann, Brunner, Cheedela, Das, Denissen, Dragaud, Dziekan, Ekblom, Engels, Esch, Forbes, Frauen, Freischem, Garcia-Maroto, Geier, Gierz, Gonzalez-Cervera, Grayson, Griffith, Gutjahr, Haak, Hadade, Haslehner, ul Hasson, Hegewald, Kluft, Koldunov, Koldunov, Kolling, Koseki, Kosukhin, Kousal, Kuma, Kumar, Li, Maury, Meindl, Milinski, Mogensen, Niraula, Nowak, Praturi, Proske, Putrasahan, Redler, Santuy, Sarmany, Schnur, Scholz, Sidorenko, Spat, Sutzl, Takasuka, Tompkins, Uribe, Valentini, Veerman, Voigt, Warnau, Wachsmann, Waclawczyk, Wedi, Wieners, Wille, Winkler, Wu, Ziemen, Zimmermann, Bender, Bojovic, Bony, Bordoni, Brehmer, Dengler, Dutra, Faye, Fischer, van Heerwaarden, Hohenegger, Jarvinen, Jochum, Jung, Jungclaus, Keenlyside, Klocke, Konow, Klose, Malinowski, Martius, Mauritsen, Mellado, Mieslinger, Mohino, Pawlowska, Peters-von Gehlen, Sarre, Sobhani, Stier, Tuppi, Vidale, Sandu and Stevens2025), in a similar setup to Weiss and Stier (Reference Weiss and Stier2024). In contrast to the simulation of Hohenegger et al. (Reference Hohenegger, Korn, Linardakis, Redler, Schnur, Adamidis, Bao, Bastin, Behravesh, Bergemann, Biercamp, Bockelmann, Brokopf, Brüggemann, Casaroli, Chegini, Datseris, Esch, George, Giorgetta, Gutjahr, Haak, Hanke, Ilyina, Jahns, Jungclaus, Kern, Klocke, Kluft, Kölling, Kornblueh, Kosukhin, Kroll, Lee, Mauritsen, Mehlmann, Mieslinger, Naumann, Paccini, Peinado, Praturi, Putrasahan, Rast, Riddick, Roeber, Schmidt, Schulzweida, Schütte, Segura, Shevchenko, Singh, Specht, Stephan, von Storch, Vogel, Wengel, Winkler, Ziemen, Marotzke and Stevens2023), the sea surface temperature and sea ice concentrations are prescribed. The horizontal resolution is about 5 km. The atmosphere is discretized with 90 vertical levels, and the land is discretized with 5 soil layers. The time step
$ \Delta t $
is 40 s. We use ERA5 boundary conditions by Hersbach et al. (Reference Hersbach, Bell, Berrisford, Hirahara, Horányi, Muñoz-Sabater, Nicolas, Peubey, Radu, Schepers, Simmons, Soci, Abdalla, Abellan, Balsamo, Bechtold, Biavati, Bidlot, Bonavita, De Chiara, Dahlgren, Dee, Diamantakis, Dragani, Flemming, Forbes, Fuentes, Geer, Haimberger, Healy, Hogan, Hólm, Janisková, Keeley, Laloyaux, Lopez, Lupu, Radnoti, de Rosnay, Rozum, Vamborg, Villaume and Thépaut2020) and initialize the model with historical weather data, ozone concentrations, aerosol concentrations, ocean properties such as the sea surface temperature and sea ice concentration, and land properties. To keep the data volume manageable, we save the data every three simulated hours. Specifically, we store the time-averaged values of all input and output parameters used in the ML parameterization (see Appendix, Table A1) over 3-hour periods, rather than instantaneous tendencies. This allows the ML model to learn from smoothed data and reduces the likelihood of predicting extreme outliers that could destabilize online-coupled simulations (Yu et al., Reference Yu, Hu, Subramaniam, Hannah, Peng, Lin, Bhouri, Gupta, Lütjens, Will, Behrens, Busecke, Loose, Stern, Beucler, Harrop, Heuer, Hillman, Jenney, Liu, White, Zheng, Kuang, Ahmed, Barnes, Brenowitz, Bretherton, Eyring, Ferretti, Lutsko, Gentine, Mandt, Neelin, Yu, Zanna, Urban, Yuval, Abernathey, Baldi, Chuang, Huang, Iglesias-Suarez, Jantre, Ma, Shamekh, Zhang and Pritchard2024). We run the simulation for 22 days, from 20 January to 9 February 2020, and discard the first 10 days as a spin-up period, which leaves us with 12 days of data. An illustration of the microphysical hydrometeors, used as inputs for our ML framework, derived from the simulation is given in Figure 1.
Map of mmrs. Obtained from the simulation on model level 70, corresponding to a height of about 3 km. The upper figure shows water vapor (red), cloud water (blue), and cloud ice (green). The lower figure shows rain (red), snow (blue), and graupel (green). Earth illustration from NASA Visible Earth (2024).
Subsequently, we apply horizontal coarse-graining mapping the data to a lower-resolution ICON grid (80 km) using the method from Grundner et al. (Reference Grundner, Beucler, Gentine, Iglesias-Suarez, Giorgetta and Eyring2022). The coarse-graining is performed on the entire dataset including both inputs (temperature, pressure, mmrs) and outputs (tendency temperature, tendencies mmrs). Coarse-graining is done by estimating grid-scale mean values through a weighted sum of high-resolution grid cells, where weights represent the fraction of the grid cell volume that each high-resolution cell fills. This allows us to approximate the large-scale state variables that are inputs for our cloud microphysics parameterization and ensures that the parameterization performs consistently across various spatial resolutions. After the coarse-graining, we select a random subset within the first 9 days after spin-up of our simulation for training and validation. We test our method on the 3 remaining days of the simulation. We scale the data with respect to the standard deviation and the mean of the training set.
2.2. Overview of ML model
Since the ultimate goal is to develop an ML-based cloud microphysics parameterization that runs online in an ICON simulation, we prioritize a multilayer perceptron (MLP)-based architecture developed in PyTorch (Paszke et al., Reference Paszke, Gross, Massa, Lerer, Bradbury, Chanan, Killeen, Lin, Gimelshein, Antiga, Desmaison, Köpf, Yang, DeVito, Raison, Tejani, Chilamkurthy, Steiner, Fang, Bai and Chintala2019), which can be applied to each grid cell of the ICON model individually. Figure 2 shows an overview of the ML-based parameterization developed in this work. The considered input and output parameters are the same as in the traditional graupel scheme. The tendencies are solely based on microphysical processes, for example, tendency temperature considered here is only a result of latent heat release or cooling by phase changes. For the used set of input and output parameters and their description, the reader is referred to Appendix, Table A1. The ML-based cloud microphysics parameterization is split into a classifier and a regression task: if the classifier determines that in a specific grid cell, the microphysics scheme is triggered and the workflow continues by calling the regression model. Otherwise, we assume that all tendencies are zero for this grid cell and pass a zero vector. While end-to-end training of the classifier and regression tasks showed a similar performance as the modular approach, we opted for separate training to maintain modularity, which is crucial for the explainability of individual components. Nevertheless, for future work involving online coupling, an end-to-end trained model may be considered.
Overview of the presented ML parameterization pipeline. Left, the input variables. Right, the output variables are listed, and for more details, see Table A1. In the center, step 1 illustrates the classifier MLP, and step 2 illustrates the regression MLP.
2.3. Microphysics trigger classifier
Similarly to Gettelman et al. (Reference Gettelman, Gagne, Chen, Christensen, Lebo, Morrison and Gantos2021) and Perkins et al. (Reference Perkins, Brenowitz, Bretherton and Nugent2023), we choose an approach to first classify the grid cells into those with and without active clouds, that is, according to whether we expect any change in our grid cell due to the cloud microphysical parameterization. Similar to Perkins et al. (Reference Perkins, Brenowitz, Bretherton and Nugent2023), we need to apply proper preselection criteria to our dataset before the model training. Additionally, for the training and testing datasets, we only consider a grid cell as triggered when the sum of all microphysical hydrometeor tendencies
$ \Delta {m}_q $
fulfills the criterion
$$ \sum \limits_q\mid \Delta {m}_q\mid \ge \unicode{x025B}, $$
with the computational threshold
$ \unicode{x025B} ={10}^{-10} $
. The tendency
$ \Delta {m}_q $
represents the rate of change of the mmr per internal model time step
$ \Delta t $
in the ICON simulation, which is 40 s in this study. By examining the underlying dataset of this study, we find that 81% of the grid cells are not changed by the microphysics scheme, which makes it especially important to sort out the inactive cells before performing the regression task. This happens in step 1 of our ML-based cloud microphysics parameterization. We train a simple MLP with 8 input nodes for the microphysical input parameters, as shown in Figure 2 with 2 hidden layers with 256 and 512 nodes and 1 output node for the classification task. We use dropout layers in between to avoid overfitting to the training dataset. The network uses rectified linear unit (ReLU) (Agarap, Reference Agarap2019) activation functions after the first and second layers to introduce nonlinearity and a sigmoid activation function at the end to produce a probability between 0 and 1. If the output value is larger than 0.5, we consider the microphysics scheme to be triggered and continue by calling the regression model. Otherwise, we assume that all tendencies are zero for this grid cell and pass a zero vector. We train the microphysics trigger classifier for 20 epochs on a random subset of 25 million samples selected from the 9 training days and evaluate on the 3 validation days.
2.4. Physics-constrained microphysics regression model
After training the trigger classifier, we predict the tendencies of triggered grid cells with the MLP-based regression model. It is structured as follows: 8 input nodes for the microphysical input parameters, 3 hidden layers with 512 nodes each using the ReLU activation (Agarap, Reference Agarap2019), and 7 output nodes using linear activation. Additionally, we include a residual connection between the input and output layers to preserve mass throughout the prediction for physical consistency. We use dropout layers and batch normalization to avoid overfitting to the training dataset. We train the microphysics regression model for 30 epochs on a random subset of 10 million samples selected from the 9 training days and evaluate on the 3 validation days.
During the preselection procedure of our simulated dataset, we find that the classical parameterization produces negative masses, or tendencies that lead to negative masses after the application of the classical parameterization in 4
$ \% $
of the samples in the preselected dataset. When these effects occur in the classical parameterization, the ICON model sets the negative masses to zero and recalculates the tendencies. To incorporate this correction into our model, we apply the method of physics constraining for negative masses following the approach of Harder et al. (Reference Harder, Watson-Parris, Stier, Strassel, Gauger and Keuper2022). We modify the weights of the output layer during the optimization of the MLP, constraining all tendencies
$ \Delta q $
not fulfilling the condition
$$ {m}_{q,{t}_1}={m}_{q,{t}_0}+\Delta t\cdot \Delta {m}_q\ge 0. $$
where
$ {m}_{q,{t}_1} $
is the mmr after the application of the parameterization, and
$ {m}_{q,{t}_0} $
is the mmr before the application of the parameterization.
Additionally, we incorporate a constraint that ensures water mass conservation for the microphysical hydrometeors
$$ \sum \limits_q{m}_{q,{t}_1}=\sum \limits_q\left({m}_{q,{t}_0}+\Delta t\cdot \Delta {m}_q\right). $$
The total mmr of hydrometeors after applying the microphysics scheme,
$ {m}_{q,{t}_1} $
, must equal the sum of mmrs of the initial hydrometeors
$ {m}_{q,{t}_0} $
plus the sum of all tendencies
$ \Delta {m}_q $
.
2.5. Explainability
As discussed by de Burgh-Day and Leeuwenburg (Reference de Burgh-Day and Leeuwenburg2023), incorporating domain knowledge into ML parameterizations increases the performance. Therefore, the strategy of this work is to stay as physically consistent as possible. Therefore, we introduce constraints to avoid unphysical negative mmrs. To further ensure consistency, we examine the model’s behavior with explainability methods in each substep. For the purpose of explainability, we use the SHAP (SHapley Additive exPlanations package [Lundberg and Lee, Reference Lundberg, Lee, Guyon, Luxburg, Bengio, Wallach, Fergus, Vishwanathan and Garnett2017]). This package follows the game theory approach of the calculation of Shapley values. The Shapley values quantify the contribution of each input feature to the prediction of a model by comparing it to all possible combinations of features. The effect of this feature value on the output is determined by how much it increases or decreases the prediction relative to the model’s baseline prediction, with the Shapley value reflecting this impact. We perform this analysis for the classifier and regression model individually. A high feature value in combination with a positive Shapley value means that with an increasing value of this input variable, we expect an increase in the respective model output. A low feature value and positive Shapley value means that by decreasing this feature value, we expect an increase in the output variable. The closer the Shapley value is to zero, the harder it is to explain the model’s behavior. With these analyses, we can find the most relevant features for the two parts of our ML framework. Additionally, for each input feature, we can examine whether it has a negative or positive correlation with the model output. This helps us to see if the model is learning processes and relationships that we would actually expect from our physical understanding.
3. Results
Our study evaluates the performance of a physics-informed, ML-based cloud microphysics parameterization using coarse-grained data from a global, high-resolution ICON simulation for training, testing, and validation. We focus on identifying the MLP algorithms that best capture the data’s characteristics while being as simple as possible for computational efficiency. This includes both the microphysics trigger classifier and the regression model.
3.1. Microphysics trigger classifier
For the first part of our ML framework, the microphysics trigger classifier, we develop an MLP-based classifier to predict whether a given grid cell state will result in changes to microphysical properties. The model achieves a good performance, with an F1 score of 0.93 and a receiver operating characteristic area under curve (ROC–AUC) (Fawcett, Reference Fawcett2006) of 0.99. The F1 score measures a model’s balance between precision and recall, while ROC–AUC assesses its overall ability to distinguish between classes. The performance is also illustrated in Figure 3. The confusion matrix shows the performance of the classification model by displaying the number of correct and incorrect predictions across different classes. The ratios on the diagonal tell the proportion of correctly classified samples out of the total actual samples of that class. The off-diagonal ratios represent the proportion of misclassified samples relative to the actual total in that class. We observe a good classification performance. Additionally, the similar ratios of false positives and false negatives indicate that the model does not exhibit a significant bias toward the dominant class, confirming the robustness of our classifier. Further analysis using explainability methods (Figure 3) shows that the mmr of rain is the most important feature for the microphysics trigger classifier. Across all mmrs of microphysical hydrometeors, lower values reduce the probability of microphysics triggering, while higher values increase it. This is consistent with our physical understanding, as more microphysical processes occur in regions with higher cloud water or precipitation content. Before implementing the classifier, we used threshold-based criteria to predict grid cell activation. Our Shapley plot supports these previous threshold methods, confirming that cells with sufficient cloud mmr (cloud water
$ {m}_{\mathrm{c}} $
and cloud ice
$ {m}_{\mathrm{i}} $
) or precipitation mmr (rain
$ {m}_{\mathrm{r}} $
, snow
$ {m}_{\mathrm{s}} $
, and graupel
$ {m}_{\mathrm{g}} $
) are more likely to activate the microphysics scheme, previously (Sarauer et al., Reference Sarauer, Schwabe, Lauer, Stier, Weiss and Eyring2024) defined by the following thresholds:
$$ \left({m}_{\mathrm{c}}+{m}_{\mathrm{i}}\right)\ge {10}^{-5}\hskip0.1em \mathrm{kg}\cdot {\mathrm{kg}}^{-1},\hskip1em \left({m}_{\mathrm{r}}+{m}_{\mathrm{s}}+{m}_{\mathrm{g}}\right)\ge {10}^{-7}\hskip0.1em \mathrm{kg}\cdot {\mathrm{kg}}^{-1}. $$
Classification performance. The left image shows the confusion matrix, comparing predicted and true classes. The color intensity represents the ratio of samples in each class to the total number for that class. The right image shows the Shapley analysis. The input variables are listed in importance for the respective output variable from top to bottom. For each feature, red represents large values of a variable, and blue represents small ones. The x-axis shows the Shapley values.
As expected, a broader range of features, such as temperature, air pressure, and water vapor concentration, are correlated with microphysics triggering, as microphysical processes occur across a wide range of values for these parameters and water vapor is present throughout the troposphere, where cloud microphysics takes place. This evaluation of the microphysics trigger classifier shows that the grid cells of interest are identified in a physically consistent manner. This provides a reliable basis for further analysis and thus can be used as a strong foundation for the subsequent regression tasks.
3.2. Microphysics regression model
In the second part of our ML framework, we address the regression task for the grid cells classified as triggered in the previous step. Despite the complexity of the cloud microphysics parameterization and low first-order correlations (cf. Appendix, Figure A1), we are able to reach an
$ {R}^2 $
score of 0.72 averaged over all output features, achieving the best regression output for the tendencies of cloud ice and graupel mmr (both 0.84
$ {R}^2 $
score) and the worst regression output for the tendency of cloud water mmr (0.61
$ {R}^2 $
score). The specific goodness of fit for the individual output features is presented in Table 1.
Fit performance measures for the regression output in the ML framework
Note. The table shows for each output variable: arithmetic average (mean) and standard deviation (
$ \sigma $
), root mean squared error (RMSE), and coefficient of determination (
$ {R}^2 $
).
The low score for cloud water in the model likely stems from its complex and sensitive nature, driven by many nonlinear, small-scale processes like droplet formation and growth, which are difficult to accurately parameterize. Cloud water varies greatly at smaller scales, making it harder to represent accurately. Additionally, the multitude of interactions of cloud water with other microphysical variables can complicate an accurate prediction. On the other hand, cloud ice, graupel, snow, and rain are easier to learn for the ML model, likely because their process representation is simpler in the classical, single moment graupel scheme. Moreover, variables such as temperature and water vapor are typically less sensitive, resulting in better model performance for these variables.
As illustrated in Figure 4, we see a good alignment between the prediction and test datasets (ground truth) for cloud ice. A similar illustration for the other variables can be found in Appendix, Figures A2 and A3. For all output features, the model effectively represents values near zero but has difficulty capturing extreme outliers. Despite experimenting with various scaling methods such as the Standard Scaler, MinMax Scaler, Robust Scaler, and Log Scaler, accurately capturing these distributions remains a challenge for ML algorithms, as none of the approaches investigated fully captured the variability in the data while maintaining optimal training performance. In this work, we use the Standard Scaler, a commonly applied preprocessing technique for training such models (e.g., Heuer et al., Reference Heuer, Schwabe, Gentine, Giorgetta and Eyring2024). To gain more insight into the relationships between input features and resulting model output, that is, microphysical tendencies, we apply explainability techniques.
Regression performance. The left image shows the ML predictions of the regression model versus ground truth of the tendency of cloud ice mmr. The colors illustrate the density of the data on a logarithmic scale. The right image shows the Shapley analysis for the tendency of cloud ice mmr. The input variables are listed in importance for the respective output variable from top to bottom. For each feature, red represents large values of a variable, and blue represents small ones. The x-axis shows the Shapley values.
Figure 4 shows illustrations of our explainability method for the microphysical tendency of cloud ice. The Shapley illustrations of the other tendencies can be found in Appendix, Figures A2 and A3. Generally, across all features, we observe a strong correlation between microphysical tendencies and both air pressure and temperature. This correlation is expected, as water vapor is of great importance to cloud formation and evolution, which itself is strongly influenced by temperature and pressure. The cloud ice tendency is positively correlated with a higher amount of cloud water and cloud ice content, as indicated by positive Shapley values for positive feature values. This could be a result of the frequent coexistence of cloud water and cloud ice in mixed clouds. Conversely, high feature values for water vapor go together with mostly negative Shapley values. Additionally, we would expect with our physical understanding that low temperature values should have positive Shapley values for ice, as low temperatures are required for the formation of cloud ice. However, the ML model shows positive Shapley values for cloud ice mmr for high temperature values. This could be due to higher temperatures resulting in a higher amount of water vapor that can, for example, condensate, which can lead to higher microphysical tendencies.
Constraining unphysical values during the training through a modified loss function, prohibiting negative masses, and ensuring water mass conservation as described above does not have an impact on the overall performance of the ML parameterization. Nevertheless, we expect the constrained ML parameterization to lead to an improved and stable simulation when coupled with a climate model in future work.
3.3. Combined model
To evaluate the overall performance of our ML-based cloud microphysics parameterization, we combine the results of the microphysics trigger classifier and the physics-informed regression model. In this study, the classifier achieved an F1 score of 0.93. This is particularly important because the dataset is imbalanced, with only 19% of the grid cells with triggered cloud microphysics processes. For the regression task, the model achieved an average
$ {R}^2 $
score of 0.72. This
$ {R}^2 $
score reflects the regression model’s ability to predict the continuous outputs related to cloud microphysics processes within the triggered cells. To provide a unified measure of performance and to allow for a fair comparison with column-based methods like Perkins et al. (Reference Perkins, Brenowitz, Bretherton and Nugent2023), we introduce a unified
$ {R}^2 $
metric. This approach treats the classification of non-triggered cells as regression outputs mapped to zero. Specifically, for all grid cells, we compare the predicted values, which combine the classifier’s predictions and the regression model’s outputs, against the true values, where nontriggered cells have an expected output of 0. This unified approach ensures that the metric reflects performance across both tasks, including cases where the regressor is incorrectly activated. This calculation results in an overall score of 0.78, which reflects a strong model performance, balancing high classification accuracy with solid regression precision.
To conclude the analysis of our results, we consider the complex graupel mmr tendency predicted by the ML model in comparison with the coarse-grained high-resolution data and a coarse reference simulation (Figure 5). Similar illustrations for the microphysical tendencies can be found in Appendix, Figures A4 and A5. The white spaces in the figures indicate grid cells where the microphysics scheme is not activated, which illustrates the good representation of the triggered cells by the microphysics trigger classifier. Nevertheless, when looking at the predicted values with the regression, we see an overestimation of the tendency of graupel mmr, especially in the subtropics compared to the ground truth. Additionally, we observe biases in the ML model for some extreme values of the other tendencies, for example, underestimating the extremes in the tendency of cloud ice mmr and overestimating the extremes in the tendency of cloud water mmr. This could also introduce discrepancies when coupling the model with a coarse global climate model. Appendix includes the distributions of these tendencies and a brief discussion. Despite this, the model demonstrates a reasonable distinction between areas of positive and negative tendencies. It is important to highlight that our model achieves these results using only cell-based information. This makes the model’s performance particularly interesting, as it successfully captures physical processes and spatial distinctions. We compare our results with a reference simulation performed at a coarser horizontal resolution. For this, we use the same setup as before and coarse-grain all initial inputs for the simulation and run the simulation on the 80-km grid. This comparison shows that graupel processes are much less pronounced in comparison to the high-resolution simulation, even after the application of coarse-graining. However, since a convection parameterization is typically used at these coarse resolutions, we expect to see less graupel. These findings support our hypothesis that our ML-based cloud microphysics parameterization has the potential to improve the representation of microphysical processes by implicitly better resolving relevant dynamics and variability compared to an approach purely based on coarse-resolution output when integrated into the ICON model.
ML prediction (left) of the tendency of graupel mmr averaged over 3 validation days (24 timesteps in total) compared with the ground truth (center) and a coarse reference simulation (right) for a random day within the validation period. Colors represent the magnitude of the mmr tendency, averaged over longitude. The x-axis is latitude, the y-axis is height, and white areas indicate no change due to microphysics.
4. Conclusions
In this study, we conduct an analysis of cloud microphysical processes using a 22-day high-resolution ICON simulation. Based on a simulation with 5-km resolution with 90 vertical layers and coarse-graining of the data to 80-km resolution, we develop a novel ML-based framework to parameterize cloud microphysics. Our approach combines a physics-informed MLP algorithm, which includes a microphysics trigger classifier and a regression model, to address the complex problem of cloud microphysics. The use of a high-resolution model simulation and a cell-based approach provides valuable insights into the microphysical processes by learning from better resolved atmospheric dynamics that cannot be explicitly resolved in low-resolution models.
The microphysics trigger classifier, which achieved an F1 score of 0.93 and an ROC–AUC of 0.99, demonstrates good performance in identifying grid cells where microphysical processes are likely to occur. This strong classification performance is important to select 19% of the grid cells that are influenced by cloud microphysics processes. The classifier’s ability to accurately identify these cells ensures that our regression model is applied only to the relevant cells. This will be especially important when coupling this model to an ESM online.
The regression model, trained on the identified triggered grid cells, achieved an average
$ {R}^2 $
score of 0.72 across all output features. In particular, the model performs well in predicting the tendencies of variables such as cloud ice and graupel, with the highest performance observed for both (0.84
$ {R}^2 $
). However, challenges remain with the mmr of cloud water, which could stem from its representation in the single moment graupel scheme and that cloud water is affected by a multitude of complex, small-scale processes and therefore challenging to predict.
Our analysis aims for physical consistency using explainability methods. The Shapley values indicate that features related to cloud water and cloud ice are the most important quantities in determining microphysics triggering. This aligns with our physical understanding, where the presence of higher cloud water or precipitation content increases the likelihood of phase changes due to microphysical processes. The combined performance of our ML models, with an overall performance score of 0.78, shows that the combination of classification and regression is effective. The model is able to balance classification accuracy with regression precision. The combined model’s prediction of graupel mmr tendencies, while demonstrating an overestimation in certain regions, still captures important spatial distinctions and provides a promising approach to cloud microphysics parameterization.
While our results are promising, generating a more balanced training and validation dataset through additional simulations at various times of the year could further enhance model performance. Furthermore, vertical updrafts, which are critical for accurately capturing some of the cloud microphysical key processes, are only partially resolved even at this resolution. Updrafts influence microphysical rates such as condensation, evaporation, and precipitation formation and are a significant source of biases in coarse-resolution microphysical schemes where these dynamics are entirely parameterized (Stevens et al., Reference Stevens, Acquistapace, Hansen, Heinze, Klinger, Klocke, Rybka, Schubotz, Windmiller, Adamidis, Arka, Barlakas, Biercamp, Brueck, Brune, Buehler, Burkhardt, Cioni, Costa-Surós, Crewell, Crüger, Deneke, Friederichs, Henken, Hohenegger, Jacob, Jakub, Kalthoff, Köhler, van Laar, Li, Löhnert, Macke, Madenach, Mayer, Nam, Naumann, Peters, Poll, Quaas, Röber, Rochetin, Scheck, Schemann, Schnitt, Seifert, Senf, Shapkaliijevski, Simmer, Singh, Sourdeval, Spickermann, Strandgren, Tessiot, Vercauteren, Vial, Voigt and Zängl2020). Our approach bridges this gap by indirectly learning the effects of unresolved dynamics on microphysics from the high-resolution simulations, enabling the ML parameterization to some of these even when applied to coarse-scale models. Future improvements could focus on utilizing even higher-resolution data, such as those from Large Eddy Simulations (LES), which resolve vertical motion and mixing even more accurately. Such advancements could further reduce biases and enhance the physical accuracy of ML-based parameterizations for cloud microphysics. Finally, while the ML-based microphysics parameterization has not yet been coupled to ICON, implementing it in a global climate model will be crucial for future work to fully assess potential benefits and impacts on climate projections. This step poses significant technical challenges, including instabilities in simulation runs, the need for model retuning, and the complexity of Fortran-Python bridges. Addressing these points is an ongoing effort, and the technical details will be documented in future work. Overall, this study presents a novel approach in parameterizing cloud microphysics, contributing to a better understanding of climate-relevant processes and with the potential to improve the accuracy and robustness of climate projections, once coupled with other ML-based parameterizations to advanced hybrid ESMs (Eyring et al., Reference Eyring, Collins, Gentine, Barnes, Barreiro, Beucler, Bocquet, Bretherton, Christensen, Dagon, Gagne, Hall, Hammerling, Hoyer, Iglesias-Suarez, Lopez-Gomez, McGraw, Meehl, Molina and Zanna2024).
Open peer review
To view the open peer review materials for this article, please visit http://doi.org/10.1017/eds.2025.10016.
Author contribution
Conceptualization: E.S., M.S., P.W., A.L., P.S., V.E.; Data curation: E.S., P.W.; Data visualization: E.S., P.W.; Methodology: E.S.; Writing original draft: E.S. All authors contributed to the writing of the manuscript and approved the final submitted draft.
Competing interests
The authors declare none.
Data availability statement
The Machine Learning code for this work will be published under https://github.com/EyringMLClimateGroup/sarauer24eds_ml_microphysics_parameterization. The simulation data used to train and evaluate the machine learning algorithms were generated with the ICON model. The source code is available on the GitLab of the DKRZ (https://gitlab.dkrz.de/icon/icon-mpim) under a BSD 3-clause license (https://gitlab.dkrz.de/icon/icon-mpim/-/tree/master/LICENSES). The simulations were performed with the branch feature-nextgems-aerosol-microphysics at commit 260364f1.
Ethics statement
The research meets all ethical guidelines, including adherence to the legal requirements of the study country.
Funding statement
This project was made possible by the DLR Quantum Computing Initiative and the Federal Ministry for Economic Affairs and Climate Action (qci.dlr.de/projects/klim-qml). We acknowledge funding by the European Research Council (ERC) Synergy Grant “Understanding and Modelling the Earth System with Machine Learning (USMILE)” under the Horizon 2020 research and innovation program (grant agreement no. 855187). V.E. was additionally supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) through the Gottfried Wilhelm Leibniz Prize awarded to V.E. (reference no. EY 22/2-1). P.S and P.W. acknowledge funding from the Horizon 2020 projects nextGEMS under grant agreement number 101003470 and FORCeS under grant agreement number 821205. This work used resources of the Deutsches Klimarechenzentrum granted by its Scientific Steering Committee under project ID bd1179. The authors also gratefully acknowledge the Earth System Modelling Project (ESM) for funding this work by providing computing time on the ESM partition of the supercomputer JUWELS Jülich Supercomputing Centre 2021 at the Jülich Supercomputing Centre.
A. Appendix
Figure A1 shows the correlation between the considered input and output variables. The figure shows that the input and output features show low correlation.
Figure A2 illustrates the regression results individually for the output variables tendency of temperature, water vapor mmr, and cloud water mmr and their corresponding Shapley plots.
Figure A3 illustrates the regression results individually for the output variables tendency of rain mmr, snow mmr, and graupel mmr and their corresponding Shapley plots.
Figure A4 and A5 present an illustration of the performance of the combined ML model.
Figure A6 shows the corresponding frequency distributions of the microphysical tendencies (see Section 3.3).
Table A1 lists the variables considered in this work.
A.1. Computing resources
We are only able to run the simulation for this short amount of time because of the substantial computing resources necessary for running such a simulation (12 k node hours on the Levante cluster of the DKRZ for 30 days simulation) and for storing such a large volume of data (25 TB for the 20 days of high-resolution data when reduced to 3-hourly output).
A.2. Frequency distributions of microphysical tendencies
Histograms were generated to visualize the frequency distributions of the microphysical tendencies for the coarse-resolution simulation, the high-resolution ground truth, and the ML predictions (Figure A6). All distributions of the microphysical tendencies show a large peak of near-zero values. For this reason, a logarithmic scaling of the y-axis has been chosen to better visualize all relevant tendencies. The ML model (red) fails reproduce the tails of the tendency distributions of rain, snow, graupel, and cloud ice. The tendency distribution of temperature is well captured by the ML model, and the tails of the water vapor and cloud liquid water distributions are overestimated compared to the ground truth (gray). Even though the ML prediction does not capture the tails of these distributions very well, the model performs well for values near zero, which explains its high overall performance for these variables. In contrast, the coarse-resolution simulation (green) overestimates the frequencies of near-zero tendencies compared to the ground truth for most variables, except for cloud water. The ML predictions generally underestimate near-zero values, except in the case of cloud water, where an overestimation occurs. The histograms demonstrate reasonable similarities between the ML predictions and the ground truth in terms of the general shape of the distributions, confirming the good agreement of the zonal mean tendencies, as shown in Figures 5, A4 and A5.
Pearson autocorrelation matrix for all raw input and output parameters. For an explanation of the variable short names, the reader is referred to Table A1. The color bar ranges from negative correlation (blue) over no correlation (white) to positive correlation (red).
Regression performance. The left images show the ML predictions of the regression model versus ground truth of the tendency of temperature, water vapor mmr, and cloud water mmr. The colors illustrate the density of the data on a logarithmic scale. The right images show the Shapley analysis for the tendency of temperature, water vapor mmr, and cloud water mmr. The input variables are listed in importance for the respective output variable from top to bottom. For each feature, red represents large values of a variable, and blue represents small ones. The x-axis shows the Shapley values.
Regression performance. The left images show the ML predictions of the regression model versus ground truth of the tendency of rain mmr, snow mmr, and graupel mmr. The colors illustrate the density of the data on a logarithmic scale. The right images show the Shapley analysis for the tendency of rain mmr, snow mmr, and graupel mmr. The input variables are listed in importance for the respective output variable from top to bottom. For each feature, red represents large values of a variable, and blue represents small ones. The x-axis shows the Shapley values.
ML prediction (left) of the tendency of temperature, water vapor, and cloud water mmr averaged over 3 validation days (24 timesteps in total) compared with the ground truth (center) and a coarse reference simulation (right) for a random day within the validation period. Colors represent the magnitude of the mmr tendency, averaged over longitude. The x-axis is latitude, the y-axis is height, and white areas indicate no change due to microphysics.
ML prediction (left) of the tendency of cloud ice, rain, and snow mmr averaged over 3 validation days (24 timesteps in total) compared with the ground truth (center) and a coarse reference simulation (right) for a random day within the validation period. Colors represent the magnitude of the mmr tendency, averaged over longitude. The x-axis is latitude, the y-axis is height, and white areas indicate no change due to microphysics.
Histograms of the microphysical tendencies averaged over 3 validation days (24 timesteps in total) for the high-resolution ground truth (gray), coarse-resolution simulation (green), and ML predictions (red). The histograms are standardized, and the number of entries is shown on a logarithmic scale to improve readability.
Overview of all considered input and output features of the MLP model
Open access
Comments
From:
Ellen Sarauer (German Aerospace Center)
To:
Joachim Denzler (Friedrich Schiller University Jena)
Patrick Emami (National Renewable Energy Laboratory)
Emily Gordon (Stanford University)
Panayiotis Moutis (City College New York)
Zoltan Nagy (University of Texas Austin)
Tejasri Nampally (Climate Change AI)
Mark Roth (Climate Change AI)
Environmental Data Science Journal
Cambridge University Press
02.09.2024
Dear Environmental Data Science Editorial Team,
we are pleased to submit our paper: “A physics-informed machine learning parameterization for cloud microphysics in ICON” to be considered for publication in the Environmental Data Science journal as part of the special collection “Tackling Climate Change with Machine Learning”.
We propose an application paper within the scope of improving climate modelling focusing on the atmosphere with machine learning. With our work, we aim to enhance the accuracy of climate models and deepen our understanding of subgrid-scale cloud processes. Therefore, our paper presents a new, data-driven cloud microphysics parameterization. The research presented in this paper builds upon our previous work, which we initially presented at the ICLR 2024 CCAI workshop, and has been significantly extended in the last months.
We employ advanced machine learning methods and achieve promising results by training and evaluating on high-resolution Icosahedral Non-hydrostatic modeling framework (ICON) simulations. Therefore, we improve cloud microphysical process representations by particularly taking into account the influence of highly resolved dynamics on cloud microphysical processes. We use explainability techniques to explore the correlations between input and output features in each step of our architecture, finding a strong alignment with physical understanding of cloud microphysical processes. Additionally, we use physics-constrained loss functions to ensure for water mass conservation and mass positivity. These methods will contribute to a better understanding of climate-relevant processes, once coupled with other ML-based parameteriztions to advanced hybrid Earth system models. Our work is relevant in the scope of climate change, because more accurate and robust climate projections are essential for informed decision-making and risk assessment in addressing the global warming crisis.
The article “A physics-informed machine learning parameterization for cloud microphysics in ICON” has not been published in another journal and reflects original research conducted by the authors. None of the authors have any competing interest concerning this paper.
We believe our paper would be a strong fit for your journal and would contribute valuable insights to the special collection.
Thank you for considering our submission. We look forward to your feedback.
Sincerely,
Ellen Sarauer
Institut für Physik der Atmosphäre
Deutsches Zentrum für Luft- und Raumfahrt (German Aerospace Center)
E-Mail: ellen.sarauer@dlr.de