In , HIRHAM5 is run at a spatial resolution of 0.05°×0.05°, with a snow layer of 10mw.e. and an internal albedo parameterization dependent on surface temperature, to determine the energy and moisture flux interactions at and below the surface . The resulting atmospheric fields are then used to force the firn model DMIHH offline.

DMIHH is a one dimensional model organized in 32 layers totaling 60mw.e. of firn. HIRHAM5 mass fluxes (snowfall, rainfall, deposition, sublimation), downwelling shortwave and longwave radiation (SW^↓, LW^↓), and latent and sensible heat fluxes (LHF, SHF) update DMIHH hourly at the surface. The surface state is determined via the SEB, with surface temperature being bound above by 0°C, and any excess energy producing surface melt: 1SEB=(1-α)SW↓+LW↓-LW↑+LHF+SHF+GHF, where GHF is the ground heat flux from the layers below, and LW^↑ the upwelling longwave radiation depending on the surface temperature. The surface albedo α is calculated internally based on surface temperature and snow depth, with α=0.85 for cold snow, decreasing to 0.65 as the surface warms toward the melting point, and dropping further as surface snow diminishes, with α=0.4 at lowest for bare ice exposure (i.e., zero snow fraction in the surface layer). While snowfall and rainfall are not directly part of the SEB, they have an indirect influence via the albedo: rainfall is simulated with a temperature of 0°C and thus warm the surface if its below the melting point; snowfall increases albedo if snow depth is low. The firn model runs offline, which means that atmospheric data applies forcing to the surface without feedback from the surface back to the atmosphere. Consequently, the latent and sensible heat fluxes are prescribed by the RCM without adjustment for actual surface characteristics.

The firn pack in the DMIHH simulation was spun up by repeatedly cycling 1980–1989 until decadal means of runoff and subsurface temperatures reached a steady state, then run continuously for 1980–2016, with daily aggregated SMB outputs and their associated driving HIRHAM5 outputs saved for the full period. This temporal aggregation potentially constrains melt emulation accuracy by smoothing sub-daily variability and short-lived extremes that control timing and peak rates of melt. However, daily resolution enables broader utility in future applications, aligning with the typical daily temporal resolution of both GCM and RCM output, as well as current ML downscaling approaches.

Although data from physical simulations are generally consistent, they may still contain extreme values and outliers caused by numerical instabilities, although these are very rare. While these individual outliers do not affect the overall assessment of modeled melt or other properties, they can disproportionately influence gradient-based optimization, distort the loss function, and impair stable convergence during ML model training. In order to decide how to treat these extreme values and outliers we must consider their source, their impact, and relation to other variables.

Aggregating rainfall and snowfall to daily values during postprocessing, some negative rainfall values arise as numerical artifacts. We set these values to zero for consistency. Furthermore, rainfall and snowfall show some suspiciously high values of up to about 700 and 1000 mm w.e. per day, respectively. Although they are likely caused by numerical instabilities in high relief topography, we do not correct these values to preserve consistency between the precipitation and the firn model output data used as target. However, we transform both rainfall and snowfall data by applying the logarithm (after adding 1, ensuring the transformation is well-defined for 0 values), which compresses these very high values. While this transformation neither adds nor removes information, it improves the data's usability for ML model training by preventing small values from vanishing in numerical noise relative to large outliers.

In rare instances, numerical instabilities also produce events of surface temperature runaway, leading to unrealistic surface temperatures approaching 0K due to excessive surface cooling from sensible heat flux as low as -400 Wm-2. We correct these heat flux values to a lower bound of -140 Wm-2, which was the lowest sensible heat flux observed in simulations unaffected by the runaway. The ranges for sensible and latent heat flux span several hundred watts per square meter, yet the majority of the data are concentrated in a very narrow range. To expand the narrow and compress the large ranges, we transform heat flux values x with a symmetric logarithm transformation symlog(x):=sgn(x)⋅log⁡(|x|/λ+1) according to , with λ=5 for LHF and λ=15 for SHF, corresponding to approximately half of their respective inter-quartile ranges.

For training a ML model we need a data set to train the model (training set), a dataset to monitor the progress during training and guide decisions during model development (validation set), and an until then unseen set for the final model evaluation (test set). When splitting data, internal dependence structures (e.g., temporal or spatial autocorrelation, data grouping/clustering) together with the modeling task need to be considered to avoid information leakage into the training set . For temporally structured data, this necessitates splitting along the time dimension in contiguous blocks (e.g., entire years) rather than individual samples (e.g., single days) to preserve temporal independence and ensure the validation and test sets remain truly distinct from the training set.

We split the data spanning from 1980 to 2016 into the three separate periods: a training period (1990–2013), a validation period (2014), and a test period (2016). The first 10 years of data (1980–1989) are included indirectly in the training set as decadal averages for the long-term module. To prevent information leakage between training and final evaluation, we introduced a one year gap between the validation and the test period, which is a common strategy for evaluating models using data with a structural dependency . Although the time windows for calculating the decadal mean conditions still overlap, this overlap is considered to have negligible impact on the validity of the tests because this study focuses on a relatively short period lacking significant trends. Therefore, the decadal means primarily capture location specific characteristics rather than temporal development.

Our data split prioritizes maximizing the training set to expose the model to a broad range of atmospheric conditions and interannual variability. Consequently, we allocated only single years for validation (2014) and test (2016) periods. While the representativeness of the test set is important to report unbiased final model performance, the representativeness of the validation set is important to make fair decision during model development. Expanding either set to two or three years would reduce training data without guaranteeing improved representativeness. To mitigate potential bias from using single-year periods, we instead verified that both 2014 and 2016 were non-anomalous years in terms of total melt extent (Appendix ).

For a model to successfully learn a specific task, we need not only a big quantity of data, but sufficient quantity of data that is actually relevant for the specific task we aim to solve. The spatial resolution of 0.05°×0.05° results in 58 391 grid cells for the GrIS, and thus just as many samples for every single day in the data set. While selecting a long training period of 23 years ensures large interannual variability in the training set, not all of this data contains information that is relevant for solving our task, since surface melt is zero, or very close to zero, for large areas of the ice sheet and a substantial part of the year. Additionally, neighboring grid cells often exhibit very similar behavior, meaning that large spatial datasets can contain considerable redundancy with respect to the melt patterns relevant to model learning.

To reduce the portion of low-relevance data samples, we apply strategic sub-sampling in time and space. The temporal sub-sampling reduces the number of no/low melt days by randomly sampling 100 d per year from a normal distribution centered around the 24 July (as peak of the melt season) and a standard deviation of 60 d. The spatial sub-sampling, on the other hand, favors grid cells in high melt areas over dry areas by selecting 5000 grid-cells according to predefined zone specific probabilities (see Appendix ). By sub-sampling 5000 grid cells and 100 d per year from the training period, we create a training set of 12 million samples, reducing training costs significantly compared to the full data set of 511 million samples. While only about 6 % of the full training set show melt above 1 mm w.e. per day, the temporal and spatial sub-sampling increase this ratio to approximately 26 %. This sub-sampling stabilizes the training process; without it, the network tends to converge to the trivial local minimum of constantly predicting zero melt. We monitor training progress on the likewise sub-sampled validation set, but conduct model comparisons on the entire validation period data set to inform design choices, and report final performance on the entire test period data set.

As inputs for our NN, we select atmospheric variables that contribute directly to the SEB, as defined in Eq. (), or indirectly: sensible heat flux SHF, latent heat flux LHF, downwelling longwave LW^↓, downwelling shortwave radiation SW^↓, rainfall R, snowfall S, and near-surface (2 m) air temperature T. We denote these input variables at daily resolution Xd. In addition, we include the cyclic features cos⁡(2⋅π365DOY) and sin⁡(2⋅π365DOY) (with DOY the day of year) to encode seasonality in our model. Lastly, long-term history is represented by aggregates of the input variables over the previous 10 year average, denoted by Xl. The target variable is daily surface melt. Since absorbed shortwave radiation is highly sensitive to surface albedo α, we also test model setups incorporating α. As a last preprocessing step, all the data is standard scaled to zero mean and unit variance with respect to the sub-sampled training data.

A NN is composed of k layers, which represents a concatenation of functions such that f=fk∘fk-1∘…∘f1. Each fi takes the previous layer's outputs, combines them linearly, applies a nonlinear activation function, and passes the results forward to the next layer. The coefficients of these linear combinations are the parameters that are optimized during training of the NN. The output layer k yields the final prediction, while the previous layers are called the hidden layers. Each layer consists of neurons which represent the number of outputs of each function fi, and NN structures are often summarized by denoting the number of neurons of each (hidden) layer, e.g., a network with three layers containing 10 neurons each is written as “10-10-10” . The hidden layers typically share the same non-linear activation function, e.g., Rectified Linear Unit (ReLU). We use LeakyReLU activation function in the hidden layers, a computationally efficient variant of ReLU that mitigates the drawbacks of classical ReLU . The output layer often uses a different activation function than the hidden layers and is based on the task. Since our task is a regression task, we use no activation function. The whole network operates on scaled data; during inference the model predictions are reverse-scaled and clipped at zero to ensure non-negative melt predictions.

We designed the NN in three separate modules: two feature extraction modules and one regression module, as depicted in Fig. . The first extraction module, the short-term module, takes the daily inputs Xd of days t-N,…,t as inputs to determine the current forcing on the surface layer. Furthermore, it incorporates the seasonal encodings for day t, intended to approximate the firn cold content through seasonal indicators.

The second module, the long-term module, uses the long-term inputs Xl. These long-term inputs are motivated by the spin-up procedure common to firn models and are meant to describe the prevailing firn characteristics at a site. We include these inputs alongside the seasonal encoding to provide location-specific information on firn cold content and bare-ice exposure proneness, which are factors that affect surface albedo and, consequently, surface melt.

The outputs of the two modules are then concatenated and fed into the final regression module which outputs the melt prediction SM^(t) for that day. While our proposed model configuration, Modular NN, consists of these three modules only, the network can be extended further by an autoregressive element, or by additional target variables. The autoregressive element (dashed arrow in Fig. ) feeds the melt of the previous day back into the daily module of the network to include the self–enhancing effect of surface melt. Alternatively, we use albedo as additional target variable since simultaneously learning albedo might improve melt predictions . In this case, Eq. () holds true not only for melt SM but simultaneously for albedo. While the weights of the NN f are shared for predicting melt and albedo throughout most of the network, the regression module branches before its final layer, with separate last hidden layers for the two output neurons predicting melt and albedo (indicated by the dashed neuron connections for albedo output in Fig. ).

Given this network design described above, we develop our emulator in two stages. First, we optimize the network configuration while keeping the atmospheric input variables fixed. Then, using the best configuration, we study how using different subsets of atmospheric variables as inputs affect model performance by retraining on multiple variable subsets.

To determine the necessary yet sufficient network modules and elements, we developed our configuration iteratively, by sequentially tuning the network configurations listed in Table . During this process we keep the atmospheric variables used for inputs fixed: As daily inputs Xd we use all the relevant atmospheric variables SHF, LHF, LW^↓, SW^↓, R, S, and T, together with the seasonal encoding. As proxy inputs for the long-term conditions Xl we use the 10 year averages of S and T.

The simplest configuration, Regression NN, consists only of the short-term module, using climate conditions of only the current day t as input (i.e. N=0), representing a pure regression model without any short-term or long-term historical information. This results in 9 input features (7 climate variables + 2 seasonal encoding variables), and we choose the hidden layers of the network to be 64-128-128-64-32-16-16, terminating in a single output neuron for melt prediction.

After having established this regression baseline, we investigate the impact of historical information by tuning Modular NN consisting of a short-term and a long-term module. The short-term module takes daily inputs from both the current and several previous days. The long-term module has two input neurons, and we choose two hidden layers of 32 neurons each. The hidden layers of the short-term module comprise 128-128-256, and the regression module combining the extracted feature from the two modules comprises 256-128-64-32-16-16. We tuned the number of the of preceding days N∈[1,10] and found that N=9 resulted in the best performance on the validation set. To investigate the necessity of the long-term module, we fix the number of past days at N=9 and tune the network without the long-term module (Short-term NN). This results in a network with 72 input features and hidden layers 128-128-256-256-128-64-32-16-16.

To investigate the impact of the surface albedo α, we train a configuration Modular NN w. α, where α is included in the daily inputs Xd. While this model cannot be used for firn emulation since α itself is an output of the firn model, this model serves as upper benchmark. In contrast, the model Multitarget NN does not use α as additional input but as additional target, so that the model can learn α and its impact on melt production. Multitarget NN results in two output neurons, one for melt and one for albedo. Next, we test whether incorporating an autoregressive step in the Modular NN improves model performance. Autoreg NN is thus also based on the configuration of Modular NN, but the melt of the previous day t-1 is used as additional input. Autoreg NN is tuned using the true previous melt as input during training (called teacher-forcing), although we also experimented with autoregressive learning approaches using the previous prediction directly during training. More details on the tuning process can be found in Appendix .

We perform a systematic input selection study on our identified best-performing configuration Modular NN. In ML, such studies are called input ablation or sequential input selection: input variables are iteratively added or removed (ablated), and model performance is assessed after retraining. This approach assesses the necessity of each input variable for solving the task, contrasting with post-hoc feature importance analyses (e.g., SHAP ) that evaluate how much a given feature contributes to an already trained model's prediction . The goal is to include all necessary variables while excluding redundant ones, as they inflate model complexity and can damage model performance, generalization, and interpretability. Finding the relevant features is not always straightforward and should not rely solely on feature correlation, but include domain knowledge .

Since we know the direct drivers and mechanisms of melt production, i.e., the SEB terms according to Eq. (), we can systematically aggregate and remove inputs based on domain knowledge. Our input selection study is summarized in Table , and is twofold: The first part is to sum up the energy input terms LW^↓, LHF, and SHF, since they all contribute equally to the SEB. The NN needs only to learn from total energy input rather than the individual components, improving model robustness while reducing complexity. In contrast, SW^↓ needs to be handled separately, because albedo strongly determines the energy finally absorbed by the surface. We define Modular NN EBMT_d, which uses daily inputs of the energy balance variables EB (with LW^↓, LHF, and SHF summed up, and SW^↓ separate), the mass variables M (S and R), and the near-surface temperature T. The long-term inputs remain the same as in the default Modular NN (10 year averages of S and T).

The second part is a classical input selection study, sequentially adding and removing input variables. For this we start with a more extensive model, Modular NN EBMT, using all the atmospheric variables from the short-term module also in the long-term module. This allows us to determine the importance of long-term energy input rather than relying on temperature T as a proxy. We then sequentially remove variables: removing T yields Modular NN EBM (energy balance and mass terms only); removing R yields Modular NN EBS (energy balance and snowfall); removing S yields Modular NN EBR (energy balance and rainfall).

Across all models, the RMSE substantially exceeds MAE, indicating considerable under- and overestimation of melt across melt events of all magnitudes, as shown by the dark blue colored hexagonal bins in Fig. . The superior performance of Modular NN, Autoreg NN, and Multitarget NN compared to Regression NN and Short-term NN stems primarily from improved predictions for a majority of data points for melt events up to 50 mm w.e. per day (narrower dark red band in Fig. c, e, f compared to a and b). For Modular NN, 64 % of the RMSE is attributable to absolute residuals up to 5 mm w.e. per day, a further 34 % to absolute residuals between 5 and 15 mm w.e. per day, and only 2 % of the RMSE to absolute residuals exceeding 15 mm w.e. per day.

Modular NN (Fig. c), Modular NN with α (Fig. d), and Autoreg NN (Fig. f) show some notable overestimation pattern of melt events above 50 mm w.e. per day. This systematic deviation originates from an unusually early melt event in April in the SW basin, where severe overestimation occurs on only one single day (Fig. ). While the model successfully predicts the unusually early surface melt for most days during this event, one particular day exhibits sensible heat flux values up to 460 Wm-2. While such extreme sensible heat flux values also appear in the training set, the combination of an anomalously large heat flux with unusually early melt creates conditions that are effectively out-of-sample.

But what causes these over- and underestimations? For this, we look at a typical day from the peak melt season in the test set in more detail. Figure  shows true and predicted melt, alongside residuals, for different models for a day in July. The melt maps are given together with the daily melt extent (ME) in km², whereby we only consider melt > 1 mm w.e. per day, and its median and IQR in mm w.e. per day.

Modular NN predicts spatially over-smoothed melt fields, evident in both the predicted field itself and the residual map in Fig. b, which also leads to a larger ME. Autoreg NN in teacher-forced mode (Fig. d) shows substantial improvement in the spatial structure, with more pronounced contours in the predictions. However, the true previous melt used to make teacher-forced predictions is not available when applying the emulator to new climate data, where the model is run in inference mode, using its own prediction of the previous day. While the predicted melt field in inference mode (Fig. c) still exhibits sharper contours compared to Modular NN, the residual plot shows that there is uncertainty on where exactly these sharp contours should be.

Modular NN with α as an additional input shows that knowledge of surface albedo improves the spatial patterns significantly (Fig. e). This suggests that the autoregressive element's importance lies mainly in its role as a proxy for surface albedo, which in turn is also an indicator of melt presence. However, as noted above, this configuration cannot be used for predictive applications. The alternative model Multitarget NN, which uses albedo as an auxiliary target instead does not lead to noteworthy improvement of melt predictions (Fig. h). Unfortunately, the predicted albedo fields are also over-smoothed (Fig. f, g), with the residual map closely mirroring the spatial patterns of the melt residuals (Fig. h).

Daily maps for additional days throughout the year are shown in Appendix , with the maps for a day in June (Fig. ) and August (Fig. ) showing similar spatial structures as those for July in Fig. . These results reveal the fundamental challenge: the simultaneous over- and underestimations arise from the model's inability to accurately reconstruct the sharp spatial structures of the surface state, which are present at the daily scale. Without explicit knowledge of surface albedo, the model produces smoothed fields that systematically overestimate melt in some locations while at the same time underestimating it in others, creating the characteristic spatial error pattern observed in the residuals.

In the remainder of this section we investigate the performance of Modular NN EBM spatially and temporally. This analysis includes predictions from both the test year 2016 to show the error patterns of a single independent year and the entire period 1990–2016 to investigate persistent systematic biases of our model.

Figure  shows the seasonally aggregated melt for the test set, starting from December 2015 to November 2016. The overestimated melt extent and lower median of melt predictions compared to true melt in the daily maps for Modular NN (Fig. , Appendix ) is also visible in the seasonal aggregates of Modular NN EBM across all seasons. The south-west coast shows some underestimation in surface melt in spring (MAM, Fig. b), which intensifies in summer (JJA, Fig. c). In contrast the north and north-west coastal region shows overestimation in summer. However, Fig.  shows that this melt overestimation in JJA is a behavior more specific to the test year, rather than a characteristic of the model Modular NN EBM. The residual plot for JJA averaged across 1990–2016 shows systematic underestimation of melt within the whole ablation zone, and some overestimation at the transition between ablation and percolation zone especially in the north and north-west (compare GrIS zones Fig. ). Together with the superior performance of Modular NN with α this indicated that the underestimation stems from the lowered albedo for shallow snow depths and bare ice. The long-term variables using 10 year averages in combination with the daily variables for in total 10 days are not enough to derive snow depth and thus decreased albedo due to bare ice exposure. Including variables at a monthly or quarterly scale leading up to the prediction day might help in approximating snow depth and associated albedo.

In the remainder of this section we investigate the performance of Modular NN EBM at the basin level. To enable performance comparison across the six basins, we present the mean annual melt and performance scores for each basin and the whole ice sheet in Table . The melt amount and the scores are given for the test year 2016, together with the mean ± standard deviation of the single years for the entire period 1990–2016. Figure  shows basin-wise integrated true and predicted melt for the test year alongside their climatologies, together with daily over- and estimation throughout the annual cycle for the test year (middle rows) and averaged over all years 1990–2016 (bottom rows).

Basins SE and SW show the highest melt with respect to their basin area, and also the highest year-to-year fluctuations in melt amount, followed by basin N (Table ). Correlation R2 between true and predicted melt is high across all basins, with the northern basin having slightly lower R2 (0.95–0.97) than the southern basins (0.98). The Ranom2 computed on anomalies from the climatology is highest for the southern basins SE and SW (0.93 and 0.92), with low standard deviation of 0.02. In contrast, the northern basins N and NE show Ranom2 of 0.78 and 0.75 respectively, paired with higher standard deviations of 0.07 and 0.11 respectively. This indicates that the emulator captures variability with respect to the climatology particularly well in the SW and SE basins, which have stronger anomalies that the model can learn, while the other basins show lower anomaly skill.

In accordance with that, the RMSE mean and standard deviations are highest for basins N and NE compared to the other basins. Also, as with the whole ice sheet, MAE is significantly lower than RMSE for all basins, indicating that a few large residuals persist across all regions. The MAE-to-RMSE ratio indicates that basins NE and NW are affected most by a small number of high residuals, while basins SW and SE are the least dominated by such outliers, and more by small and medium errors.

Basin SE is the only one with a positive bias on average (0.01), while the other basins show zero or slightly negative MBE of -0.02 at most on average across the years. However, the respective standard deviations across the years are bigger than the absolute means of the biases, i.e. no basin has a systematic bias across the years. Basins N and SW show the largest fluctuations in bias with a standard deviation of 0.05, while they also show high average melt together with high melt variability relative to mean melt amount (27 %), and thus these basins are particularly variable and hard to learn.

Figure  shows that not only is there no systematic bias across the years, but also that over- and underestimating melt is largely synchronous, thus there are also no pronounced systematic temporal biases. However, basins N and NW show a tendency toward early-season underestimation and late-season overestimation, while basin NE shows a dominance of underestimation throughout the entire year.

2016 is a relatively high melt year for each basin, but within one standard deviation from the respective means. Basins N and NW show anomalous positive MBE of 0.06 and 0.05 respectively, while basin SW shows strong negative bias of -0.06, such that the biases cancel each other out across the whole GrIS. Figure  shows enhanced melt at peak melt season for basins N and NE, causing more melt overestimation than normal, and higher Ranom2. Basin NW also shows high melt at peak season, but then a sudden decrease end of July, and thus the melt amount deviates significantly from the climatology which leads to a Ranom2 1.5 standard deviation lower than for the average year. Also SW and to a lesser extent CE and SE have lower Ranom2 than the average year as they show a lot of temporal fluctuation in the melt (top panels in Fig.  SW and SE), also starting with some early melt. For SW and SE the RMSE and MAE are also more than a standard deviation above the average.

These deviations in 2016 highlight the model's challenges in capturing anomalous melt patterns that deviate substantially from the climatological mean, particularly in basins with pronounced seasonal deviations such as NW's abrupt late-summer decline and early melt onset in basins SW and SE. However, while individual basin biases in 2016 are notable (positive for N and NW, negative for SW), these deviations appear to be year-specific rather than systematic. Averaging across the entire 1990–2016 period reveals that the model exhibits no persistent systematic biases, with year-to-year fluctuations in error patterns largely balanced over time. Although based on the performance including the training set, this temporal stability suggests that the emulator's performance is robust across the multi-decadal timescale.