The modRSW model used in this research represents an extended version of the 1D shallow water equations, i.e., 1D fluid flow over orography. Its prognostic variables are fluid height h, wind speed u, and a rain mass fraction r. Based on the model by , it implements two threshold heights, Hc<Hr, which initiate convection and rain production, respectively. Convection is stimulated by modifying the pressure term to remain constant where h rises above Hc. In contrast to , the modRSW model does not apply diffusion or stochastic forcing. The model is mass conserving, meaning that the domain mean of h is constant over time. In this study, a small but significant model-intrinsic drift in the domain mean of u is removed by adding a relaxation term. This term is defined by using a corresponding timescale trelax, as (u‾0-u‾t)⋅trelax, where the overbar denotes the domain mean. Depending on the orography used, this model yields a range of dynamical organization between regular and chaotic behaviors. Orography is defined as a superposition of cosines with the wavenumbers k=1/L,…,kmax/L (L domain length). Amplitudes are given as A(k)=1/k, while phase shifts for each term are randomly chosen from [0,L]. In this work, two realizations of the orography are selected to represent the regular and more chaotic dynamical behaviors. Figure  displays a 24 h segment of the simulation corresponding to each orography.

Conceptually, the ANN's task is to correct a low-resolution (LR) model forecast towards the model truth, which is a coarse-grained high-resolution (HR) model simulation. The coarse-graining factor in this study is set to 4, which is analogous to the range of scales found in the gray zone where deep cumulus convection is partially resolved (e.g., 2.5–10 km). show that the choice of coarse-graining factor can substantially affect the performance of ML methods. In our case, however, choosing a larger factor would correspond to a coarse model grid length that is larger than the typical cloud size, changing the nature of the problem from learning to improve poorly resolved existing features in the coarse simulation to parameterizing features that might not be seen at all. The dynamical time step of the model is determined at each iteration, based on the Courant–Friedrichs–Lewy (CFL) criterion. To achieve temporally equidistant output states for both resolutions, the time step is truncated accordingly when necessary.

A training sample (input target pair) is defined by the model truth at some time ti and the difference between the model truth and the corresponding LR forecast at ti+1=ti+dt, respectively (see Fig. ). To generate the model truth, HR data are obtained by integrating the modRSW model forward using the parameters shown in Table . All states and the orography are subsequently coarse grained to LR, resulting in model truth (LR1). Each LR1 state is integrated forward for a single time step, using the modRSW model on LR with the coarse grained orography, resulting in a single step prediction (LR2). The synchronized differences LR1(ti)-LR2(ti) then define the training targets corresponding to the input LR1(ti-1), which includes the orography. A time series of T=200000 time steps, which is equivalent to approximately 57 d in real time, is generated for both orographies. The first day of the simulation is discarded as spin up, the subsequent 30 d are used for training, and the remaining 26 d are used for validation purposes. The decorrelation length scale of the model is roughly 4 h.

A characteristic property of convolutional ANNs is that they reflect spatial invariance and localization. These two properties also apply to the dynamics of many physical systems, such as the one investigated here. They differ from, e.g., dense networks by the use of a so-called kernel. This vector is shifted across the domain grid, covering k grid points at each position. At each position, the dot product of the kernel and current grid values is computed, determining (along with an activation function) the corresponding output value. For more details on convolutional ANNs, we refer to .

The ANN structure used in this research is described in the following. There are five hidden layers applied, each using the ReLU activation function. The input layer uses ReLU as well, while the output layer uses a linear activation function. All hidden layers have 32 filters. The input and output layer shapes are defined by the input and target data. The kernel size is set uniformly to three grid points.

The loss is determined during training by comparing the ANN output to the corresponding target. A standard measure for loss is the mean squared error (MSE). However, any loss function can be used to tailor the application. For example, additional terms can be added to impose weak constraints on the training process as, for example, done in . This possibility is exploited here to impose mass conservation in a weak sense. The constraint is implemented by penalizing the deviation of the square of the domain mean corrections of h from zero, resulting in the following loss function: 1MSE(yout,ytarget)+wmass⋅youth‾2, where the second term represents a weighted mass conservation constraint. In this expression, yout and ytarget are the output and corresponding target of the ANN, respectively, MSE denotes the mean squared error, the tunable scalar wmass is the mass conservation constraint weighting, youth is the ANN output for h, and the overbar denotes the domain mean.

The Adam algorithm, with a learning rate of 10-3, is used to minimize the loss function over the ANN weights in batches of 256 samples. Since the loss function is typically not convex, the ANN likely converges to a local rather than the global minimum. To sample this error, we repeat the training of each ANN presented in this paper, with randomly chosen initial weights, five times. For all ANNs, a total of 1000 epochs is performed. The ANN architecture and hyperparameters were selected based on a loose tuning procedure where no strong sensitivities were detected. The training is done using Python library Keras .

Figure  shows the results for the single time step predictions. The improvements achieved by the ANN with respect to LR in terms of RMSE are substantial for h, u, and r, amounting to 97 %, 89 %, and 90 % for the regular case and 96 %, 84 %, and 92 % for the chaotic case. Also, the negative biases present in u and r for LR virtually disappear when the ANN is applied. However, as the ANN is not explicitly instructed to conserve mass, a small SME is introduced in variable h. Its significance will become apparent when analyzing the 48 h forecasts. It is interesting to note that the ANN's architecture and learning process is unbiased (small bias‾), although single ANN realizations may be biased (large SDANN). Also, in contrast to the SME and bias, for the RMSE, the initial conditions are the main source of variability (SDtime≫SDANN). This is better visible in Figs.  and , where the results for wmass=0 are plotted again.

Next we examine the effect of the ANN on a 48 h forecast. Here we compare the LR simulation with (LRANN) and without (LR) the use of the ANN. Both simulations start from the same initial conditions as the model truth. The evolution of RMSE‾ is presented in Fig. . The RMSEs corresponding to the regular case are higher than for the chaotic case. This is because the regular case exhibits a repeating pattern of long-lived, high-amplitude convective events. In comparison, the chaotic case produces short-lived perturbations with very small amplitude, leading to smaller climatological variability.

For both orographies, the ANN has a clear positive effect on the forecast until the error of LR saturates, after which the error of LRANN continues to grow. For the chaotic case, this leads to a detrimental impact of the ANN after about 1 d. Also, the SDtotal of LRANN is rapidly exceeding that of LR. This is because, in contrast to LR, the shaded region for LRANN includes the variability due to the ANN realizations which significantly contributes to the total variability. This is seen in Fig.  and discussed further in the next section.

It is not surprising that LRANN deteriorates as the forecast lead time increases, since the ANNs are not perfect (as opposed to the data they were trained on) and the resulting errors accumulate over time, leading to biases. This is clearly visible in Fig. , where it is seen that the SME of h and u diverge, which is in contrast to LR. The SME of r for LR is the result of a negative bias in the amount of rain produced (see Fig. ), which is caused by the coarse graining of the orography. This bias is significantly reduced by the ANNs. The divergence of the SME of h is the result of applying ANNs that, in contrast to the model, do not conserve mass. This leads to accumulated mass errors, causing biases in the wind field due to momentum conservation and a change in probability for the fluid to rise above Hc and Hr. We therefore investigate if reducing the mass error, by adding a penalty term to the loss function of the ANN, can increase the forecast skill further.

Instead of including mass conservation in the training process of the ANN, it is natural to first try to correct the mass violation by post processing the ANN corrections. We tested two approaches, i.e., homogeneously subtracting the spatial mean of the h corrections and multiplying the vector of positive (negative) h corrections with the appropriate scalar when the mass violation is positive (negative). Neither of these simple approaches led to improvements. We, therefore, included mass conservation in a weak sense in the training process of the ANN, as described in Eq. (). We trained ANNs with mass conservation weightings of wmass=1,10,100,1000. These weightings result in a contribution to the loss function of roughly 0.2 %, 0.7 %, 2 %, and 5 % throughout the training process, respectively (not shown). Note that the ANNs presented in the previous section correspond to wmass=0.

Figures  and show the single time step predictions for the regular and chaotic cases, respectively. Clearly, the mass conservation penalty term in the loss function has the desired effect of reducing the mass error for both orographies. Also, the error bars of the mass bias go down. A clear, convincing correlation between the reduction in SME and bias for h and any other field and/or metric is not detected, with the possible exception of the SME for u in the chaotic case. A tradeoff between increasing RMSE and decreasing MSE for increasing wmass was expected but is not observed. The RMSE even tends to decrease a minimal amount for the chaotic case.

Figure  presents the mean RMSE of the 48 h forecasts for all weightings. The weak mass conservation constraint has the desired effect on the forecast skill. For the chaotic case, more than 15 h in forecast quality is gained. For the regular case, the number is unclear since the RMSE is still lower than LR and has not yet saturated after 48 h. However, we can say that it is at least 30 h. As hypothesized, Fig.  indicates that the divergence of the domain mean error of the wind u is delayed as the weighting wmass is increased. This, in turn, positively affects the domain mean of the rain r. To support these claims, we look at Fig. , which shows the correlation between the bias in h and the bias in wind u and rain r, respectively. In the single step predictions, these correlations were not conclusively detected. However, as the forecast evolves, the wind bias becomes almost completely anticorrelated to the mass bias. A strong correlation between the mass bias and the rain bias is also established after a few time steps, likely when the change in the probability of crossing the rain threshold resulting from the mass bias has taken effect. We also note that the larger wmass, the weaker the correlations. We hypothesize that, as the mass bias weakens, other causes for introducing domain mean biases in the wind and rain field become more significant. Such other causes may, for example, depend on the orography or the state of u and r.

Next we look at the variability in the forecast errors in terms of SDtotal, SDtime, and SDANN in Fig. . For small weightings, the variability caused by ANN realizations dominates the total variability. However, as the weighting increases, the variability due to the initial conditions takes over. This again confirms the benefits of adding the mass penalty term to the loss function, as it demonstrates a decrease in the sensitivity of the forecast to the training process of the ANN.

A visual examination of the animations of the forecast evolution suggests that convective events produced in the LR run are wider and shallower than in the coarse-grained HR run. This behavior mimics the collapse of convective clouds towards the grid length that is typical of kilometer-scale numerical weather prediction models, as noted in the introduction. This then leads to a lack of rain mass but also, via conservation of momentum, a drift in the wind field. The convective events in the LR simulations are therefore also increasingly misplaced as the forecast lead time increases. The ANNs are capable of sharpening the gradients of the convective events, leading to highly accurate forecasts of convective events up to 6–12 h. After this, spurious, missed, and misplaced events start to occur, although the forecast skill remains significant a while longer, which is in contrast to the LR simulations where the forecast skill dissolves after just a few hours. A snapshot of the state for the chaotic case is presented in Fig. . The main rain event is misplaced for LR due to the bias in the wind field. Also, LR misses the small neighboring events which the LRANNs do catch. Furthermore, it is also clear that LRANN for wmass=100 is closer to the truth for all variables than for wmass=0.