General circulation models (GCMs) have been widely used to study climate variability, prediction, and projections. Despite decades of GCM development, most GCMs continue to suffer from many systematic biases, especially in low-latitude regions. The prominent tropical bias of most current GCMs is referred to as the double Intertropical Convergence Zone (ITCZ) syndrome, which is characterized by two parallel zonal bands of annual precipitation straddling the Equator over the central and eastern Pacific (Lin, 2007; Zhang et al., 2019). Convectively coupled equatorial waves and the Madden–Julian oscillation (MJO), which are characterized by eastward-propagating convective cloud clusters, are also not well simulated by GCMs (Ling et al., 2017; Cao and Zhang, 2017).

Many studies have attributed most of these biases to deficiencies in the parameterization schemes for atmospheric moist convection and cloud processes in the current GCMs (Zhang and Song, 2010; Cao and Zhang, 2017; Song and Zhang, 2018; Zhang et al., 2019). Cloud-related processes span a large range of spatial scales, from micrometer-scale cloud nucleation to meter-scale turbulence and from individual convective cells and organized convective systems, which are a few kilometers to hundreds of kilometers in size, to tropical disturbances, which have a spatial scale of thousands of kilometers. They directly influence the radiation balance and hydrological cycle of the Earth system and interact with the atmospheric circulation, affecting the transport and distribution of energy (Emanuel et al., 1994). Therefore, it is very important to simulate the cloud and convection processes in GCMs correctly. However, the GCMs that are currently used for climate simulations have a horizontal resolution of ∼100 km and a vertical hydrostatic coordinate. Thus, in most GCMs, in addition to parameterized cloud microphysics, convection and its influence on atmospheric circulation are represented by convective parameterization schemes, which are usually based on simplified theories, limited observations, and empirical relationships (Tiedtke, 1989; Zhang and McFarlane, 1995; Lopez-Gomez et al., 2020). These schemes regard convective heat and moisture transport as the collective effects of idealized individual kilometer-scale convective cells. They cannot represent the effects of many complicated convective structures, including organized convective systems, which leads to large uncertainties and biases in climate simulations (Bony et al., 2015).

In contrast, cloud-resolving models (CRMs) have long been used to simulate convection. Because CRMs have higher horizontal and vertical resolutions and can explicitly resolve the thermodynamic processes involved in convection, they simulate convection more accurately, including convective organization (Feng et al., 2018). In recent years, CRMs have been used for super-parameterization (SP) in low-resolution GCMs and have replaced conventional cumulus convection and cloud parameterization schemes. The most commonly used SP model is the super-parameterized Community Atmosphere Model (SPCAM) developed by the National Center for Atmospheric Research (NCAR) (Grabowski and Smolarkiewicz, 1999; Grabowski, 2001, 2004; Khairoutdinov and Randall, 2001; Randall et al., 2003; Khairoutdinov et al., 2005). Compared with conventional cumulus convection and cloud parameterization schemes, SPCAM performs better in simulating mesoscale convective systems, diurnal precipitation cycles, monsoons, the precipitation frequency distribution, and the MJO (Khairoutdinov et al., 2005; Bretherton et al., 2014; Jiang et al., 2015; Jin and Stan, 2016; Kooperman et al., 2016). However, when using a 2-D CRM for SP, the improvement of the climate mean states is not obvious (Khairoutdinov et al., 2005). In addition, SPCAM requires far more computing resources (i.e., an order of magnitude or more) than a Community Atmosphere Model (CAM) with the same resolution. Thus, the use of SPCAM in long-term climate simulations and ensemble predictions is restricted by the current computing resources. Developing novel and computationally efficient schemes for convection and cloud processes is highly desired in GCM development.

In the last 5 years, the rapid development of machine learning (ML) techniques, especially deep-learning techniques such as neural networks (NNs), has provided novel approaches to constructing parameterization schemes. Machine learning can identify, discover, and model complex nonlinear relationships that exist in large datasets. Several studies have used ML methods to develop convection and cloud parameterization schemes (e.g., Gentine et al., 2018; Rasp et al., 2018). These studies followed a similar approach. The first step is to derive a target dataset from a reference simulation, which is later used to train the ML models. Following this, the trained ML models are often evaluated offline against other independent reference simulations, and they are finally implemented in a GCM to replace the conventional parameterization schemes.

Krasnopolsky et al. (2013) first proposed a proof of concept for developing convection parameterization based on the NN technique. Specifically, an ensemble of shallow NNs was applied to learn the convective temperature and moisture tendencies, and the training data for the CRM simulations was forced using observations in the tropical western Pacific. The resulting convective parameterization scheme was able to simulate the main features of the clouds and precipitation in the NCAR CAM4 diagnostically. However, the key issue of prognostic validation in 3-D GCMs has not been addressed. Recent studies have investigated ML parameterizations in prognostic mode in simplified aqua-planet GCMs. For example, Rasp et al. (2018) developed a fully connected deep NN (DNN) to predict convection and clouds, which was trained using data from an aqua-planet SPCAM. The DNN-based parameterization was then implemented in the corresponding aqua-planet CAM and produced multi-year prognostic results that were close to the SPCAM data. For this DNN-based parameterization, Rasp (2020) found that minor changes, either to the training dataset or to the input/output vectors, can lead to model integration instabilities. Brenowitz and Bretherton (2019) fitted a DNN for convection and clouds to the coarse-grained data from a near-global aqua-planet cloud-resolving simulation using the System for Atmospheric Modeling (SAM). The NN scheme was then tested prognostically in a coarse-grid SAM. Their results showed that non-physical correlations were learned by the network, and the information in the upper levels obtained from the input data had to be removed to produce stable long-term simulations. Rather than using NNs, Yuval and O'Gorman (2020) used the random forest algorithm to develop an ML parameterization based on training data from a high-resolution idealized 3-D model with a setup on the equatorial beta plane. They used two independent random forests to separately emulate different processes. Later, Yuval et al. (2021) ensured the physical constrains by using an NN parameterization with a special structure to predict the subgrid fluxes instead of tendencies. Both methods achieved stable simulations for coarse resolution aqua-planet GCMs. To determine why some methods can achieve stable prognostic simulations and others cannot. Brenowitz et al. (2020) also proposed methods for interpreting and stabilizing ML parameterization for convection. In their study, a wave spectra analysis tool was introduced to explain why the ML coupled GCMs blew up.

In real-world climate models with varying underlying surfaces, convection and clouds are more diverse under different climate backgrounds, which makes the task of developing ML-based parameterizations more complicated. A few earlier studies demonstrated the feasibility of using neural networks to emulate cloud processes in real-world models. Han et al. (2020) used a 1-D deep residual convolutional neural network (ResNet) to emulate moist physics in SPCAM. This ResNet-based parameterization fit the targets with a high accuracy and was successfully implemented in a single column model. Mooers et al. (2021) developed a high-skill DNN using an automated ML technique and forced an offline land model using DNN-emulated atmospheric fields. However, neither of these studies tested their NNs prognostically for long-term simulations. Similar to the idea of using several NNs for different processes proposed by Yuval and O'Gorman (2020), in this study a set of NNs was used to emulate convection and cloud processes in SPCAM with the actual global land–ocean distribution. We used the residual connections of Han et al. (2020) to acquire super deep neural networks with a great nonlinear fitting ability. Furthermore, we conducted systematic trial-and-error analysis to filter out unstable NN parameterizations and to obtain the best residual deep neural network (ResDNN) set in terms of both accuracy and long-term stability. The NN parameterization scheme was then implemented in a realistically configured CAM to obtain long-term stable simulations. NNs are commonly implemented using high-level programming languages such as Python and deep-learning libraries. However, GCMs are mainly written in Fortran, making integrating them with deep-learning algorithms inconvenient. Therefore, we introduced an NN–GCM coupling platform in which NN models and GCMs can interact through data transmission. This coupling strategy facilitates the development of ML–physical hybrid models with a high flexibility. Under real geographic boundary conditions, we achieved more than 10-year-long stable climate simulations in Atmospheric Model Intercomparison Project (AMIP)-style experiments using a hybrid ML–physical GCM. The simulation results exhibited some biases in the mean climate fields, but they successfully reproduced the variability in SPCAM. To our knowledge, this is the first time a decade-long stable real-world climate simulation has been achieved using an NN-based parameterization.

The remainder of this paper is organized as follows. Section 2 briefly describes the model, the experiments, the NN algorithm, and the NN–GCM coupling platform. Section 3 analyzes the simulation stability of the GCM using neural network parameterizations (NNCAM). Section 4 presents the offline validation of the NN scheme, focusing on the output temperature and moisture tendencies. The results of the multi-year simulations conducted using the NN parameterization scheme are presented in Sect. 5. A summary and the conclusions are presented in Sect. 6.

In this study, we chose SPCAM as the reference model to generate the target simulations. A set of NNs was trained using the target simulation data and optimized hyperparameters. Following this, they were organized as a subgrid physics emulator and were implemented in SPCAM, replacing both the CRM-based SP and the radiation effects of the CRM. This NN-enabled GCM is hereinafter referred to as NNCAM.

Before evaluating the prognostic results, the offline performance with geographic information needs to be demonstrated for the following purposes: (1) to show how well our NN parameterization emulates the SP for a realistic configuration compared with the baseline CAM5 physics and previous studies and (2) to reveal the strengths and weaknesses of the NN emulations with the correct input and to provide clues to the analysis of the prognostic results in the following section. We performed offline testing using a realistically configured SPCAM from 1 January 1999 to 31 December 2000, in which the NN parameterization was diagnostically run parallel to the SP in addition to the CAM5 physics. The results for the entire second year of the simulation period were chosen for evaluation, which was completely independent from the training dataset. Following the conventions of Han et al. (2020) and Mooers et al. (2021), we used the mean fields and the coefficient of determination (R2) as the evaluation metrics. It should be noted that the NN parameterization was tuned to emulate the SP, and the CAM's parameterization was tuned to obtain close results to the observations. The latter is merely introduced as a baseline.

The mean diabatic heating and drying rates produced by convection, large-scale condensation, and cloud radiation effects in SPCAM and the NN parameterization are in close agreement. Figure 6 shows the latitude–height cross sections of the annual mean heating and moistening rates in SPCAM and the corresponding NN parameterization. At 5∘ N, SPCAM exhibits the maximum latent heating in the deep troposphere, corresponding to the deep convection in the ITCZ. In the subtropics, heating and moistening occur in the lower troposphere, corresponding to the stratocumulus and shallow convection in the subtropics. In the mid-latitudes, there is a secondary heating maximum below 400 hPa due to the mid-latitude storm tracks. All of these features are well reproduced by the NN parameterization. It should be noted that the peak in the drying rate in the ITCZ in the mid-troposphere is slightly weaker in the NN parameterization than in SPCAM (Fig. 6c and d).

In addition to the mean fields, the high prediction skill of the NN parameterization is also demonstrated by the spatial distribution of the R2 values. To illustrate the R2 values of the 3-D variables, such as the diabatic heating and moistening, the zonal averages were calculated in advance before the R2 calculation for each location in the pressure–latitude cross section following Mooers et al. (2021). For the diabatic heating, the R2 value is >0.7 throughout the middle and lower troposphere, and the high-skill regions with R2 values of greater than 0.9 are concentrated in the low levels but extend into the mid-troposphere in the storm tracks (Fig. 7a). For the moistening rate, the high skill zones are concentrated in the middle and upper troposphere (Fig. 7b), with low skill areas below. The regions with lower accuracies are generally located in the middle and lower troposphere in the tropics and subtropics, which correspond to the deep convection in the ITCZ and the shallow convection in the subtropics. Nonetheless, the tendencies of the diagnostic CAM5 parameterization are not similar to those simulated by the SP, except for a few locations in the middle and upper troposphere in the tropics and polar regions (Fig. 7c and d).

The global distribution of the R2 values of the precipitation predictions is shown in Fig. 8. Our NN parameterization produced excellent predictions in most of the in mid- and high-latitude regions, especially in the storm tracks. However, the prediction skill is relatively low in many of the ocean areas between 30∘ S and 30∘ N and in some mid-latitude areas over continents (Fig. 8a). In particular, the results are not ideal along the equatorial regions, in the subtropical Eastern Pacific, and in the subtropical Eastern Atlantic. These areas correspond to the low skill zones of the moistening rate in the middle and lower troposphere from the Equator to the subtropics (Fig. 7b). As a baseline, the total precipitation simulated using the CAM5 parameterizations is much less analogous to the SP than the NN parameterization and has a systematically lower accuracy globally. The CAM5 precipitation can achieve a relatively high accuracy along the mid-latitude storm tracks, but it fails in most regions in the tropics (Fig. 8b).

Generally, the NN parameterization performed far better than the CAM5 parameterization in the 1-year period in the offline testing, and it had an accuracy similar to that of the DNN used by Mooers et al. (2021). The use of real geographic data can significantly decrease the emulation skill of a deep-learning model (Mooers et al., 2021). This is because the convection backgrounds of real geographic data are much more complex with meridional and zonal asymmetric and seasonally varying circulations. In addition, the orography and various types of underlying land surfaces also add complexity. In this case, the ResDNN is a valuable NN architecture that performs well as an automated hyperparameter tuning algorithm that does not need to search for hundreds of NN candidates. Our NN parameterization still produced low-accuracy predictions along the Equator over the oceans where the convection is complex and vigorous and in subtropical ocean areas where the convection is weak and concentrated at low levels. This indicates that the NN parameterization is still inadequate in rems of its emulation skill when simulating various types of deep and shallow convection in the tropics.

The NN parameterization produced the best prognostic performance in Sect. 3.1. It was coupled in the realistically configured SPCAM to replace the SP and its cloud-scale radiation effects. This coupled model is referred to as NNCAM hereinafter and is compared with SPCAM and CAM5. The start time of all three models was 1 January 1998. They were all run for 6 years, with the first year used as spin up and the next 5 years (1 January 1999 to 31 December 2003) used for evaluation and comparison. Later, the simulation of NNCAM was extended for another 5 years (to 31 December 2008) to demonstrate its stability. Due to the excessive computing resources required, the SPCAM simulation was not extended. In the analysis of the prognostic results, the following variables were selected to demonstrate the multi-year climatology and variability:

the mean temperature and humidity fields,

In this study, the potential of deep neural network-based parameterizations in SPCAM to reproduce long-term climatology and climate variability was investigated. We developed an NN parameterization via a ResDNN set to emulate the SP with a 2-D CRM and its cloud-scale radiation for a realistically configured SPCAM with a true land–ocean distribution and orography. The input variables of the NN parameterization include the specific humidity, temperature, large-scale water vapor and temperature forcings, surface pressure, and solar insolation. The output variables of the NN parameterization include the subgrid tendencies of the moisture and dry static energy and the radiation fluxes. We propose a set of 14-layer deep residual neural networks, in which each NN is in charge of one group of output variables. With such a design, we gained a best emulation accuracy for each predictor. Through systematic trial-and-error searching, we were able to select sets of ResDNNs that support stable prognostic climate simulations, and we then chose the best set with the lowest climate errors as the formal NN parameterization. Moreover, the mechanism of the unreal perturbation amplification was identified in the GCM simulations with unstable NN parameterizations using the spectrum diagnostic tool invented by Brenowitz et al. (2020).

The offline tests demonstrated the good skills of the NN parameterization in emulating the SP outputs and the cloud scale radiation effects of SPCAM. The overall diabatic heating and drying rates in the NN parameterization and SPCAM are in close agreement. When implemented in the host SPCAM to replace its time-consuming SP and its radiation effects, the NN parameterization successfully produced an extensive stable long-term prognostic simulation and predicted reasonable mean vertical temperature and humidity structures and precipitation distributions. Compared with the SPCAM target simulation, NNCAM still produces some biases in the mean fields, such as a warmer troposphere over the polar regions and in the tropopause and underestimation of strong precipitation in the equatorial regions. In addition, the better climate variability of SPCAM compared to CAM5 was learned well by our NN parameterization and was reproduced by NNCAM, with better frequency for extreme rainfall and a similar MJO spectrum, propagation direction, and speed. Despite the current biases in the climate states, NNCAM can still be regarded as a first attempt to couple a NN-based parameterization and a realistically configured 3-D GCM.

Many previous studies have investigated ML parameterizations implemented in aqua-planet configured 3-D GCMs. Some faced instability problems in coupled simulations (Brenowitz and Bretherton, 2019), while others succeeded in producing stable long-term prognostic simulations with deep fully connected neural networks (Rasp et al., 2018; Yuval et al., 2021), as well as random forest algorithms (Yuval and O'Gorman, 2020). In contrast to aqua-planet simulations, the spatial heterogeneity is prominent over the land in GCMs, which are configured using real geographic boundary conditions. The convection, clouds, and interactions with the radiation in the CRM and the real geographic boundary conditions are without a doubt far more complicated than in idealized models. To meet the new demand for realistic configurations, we designed a ResDNN with sufficient depth to further improve the nonlinear fitting ability of the NN parameterization. With the skip connections, the 7-layer DNN models can be extended to 14 layers, thereby significantly improving the offline accuracy. In the prognostic tests, a dozen ResDNN parameterizations supported a stable long-term run, while all of the DNN parameterizations tested were found to be unstable.

Trial and error is still our only way to find stable NN-based parameterizations. Thus far, we have not developed an a priori method that guarantees stability. However, we did find some clues in the sensitivity tests. We believe sufficient offline accuracy is essential for online stability and can be achieved by confirming all of the inaccurate NN parameterizations as unstable. In addition, some of the highly accurate ones still crash the prognostic simulation. In this case, the total energy was found to increase rapidly. This mechanism is that unstable NNs cannot damp the neural network emulation errors, and they amplify and propagate them to the entire system through gravity waves.

The prognostic biases of the mean fields are speculated to be a result of the combined effect of the emulation errors of all of the NN parameterization prediction fields. Further study is required. Still, it may be related to the spatially non-uniform accuracy of the NN parameterization, such as the relatively low fitting accuracy in the tropical deep-convective regions and the shallow subtropical convection and stratiform cloud regions. Such problems have also been reported in previous studies (Gentine et al., 2018; Mooers et al., 2021). We believe that an NN parameterization with heterogeneous characteristics across different regions, rather than a globally uniform scheme, can further improve the fitting accuracy in these tropical and subtropical regions.

Embedding deep neural networks into Fortran-based atmospheric models is still a handicap. Before this study, researchers mainly used hard coding to build neural networks (Rasp et al., 2018; Brenowitz and Bretherton, 2019). An easier method is to use Fortran-based neural network libraries that can flexibly import network parameters (Ott et al., 2020). These methods have been used to successfully implement NNs in GCMs, but they can only support dense, layer-based NNs. As a result, developers cannot take advantage of the most advanced neural network structures, such as convolution, shortcut, self-attention, and variational autoencoder structures, to build powerful ML-based parameterizations. In this study, using an NN–GCM coupler, the NN parameterization could support the mainstream GPU-enabled ML frameworks. Thanks to the simple and effective implementation of the NN–GCM coupler, our NNCAM achieved an SYPD 30 times that of SPCAM by using a ResDNN set and NN parameterization, even though these DNNs are much deeper than the previous state-of-the-art fully connected NNs in this field.