The wind fields used for the present applications were retrieved from an implementation of the regional climate model (RCM) COSMO-CLM , the climate version of the operational forecast model COSMO-LM implemented over Italy and central Europe at a 0.0715° horizontal resolution (approximately 8 km, totalling 224 × 230 grid points) and forced by the general circulation model (GCM) CMCC-CM (Centro Euro-Mediterraneo sui Cambiamenti Climatici Climate Model; ). In that implementation, the analysed period spanned from 1971 to 2100, reproducing the CMIP5 historical experiment in the 1971–2005 period first and subsequently parting into two independent runs, representing, respectively, the IPCC RCP4.5 (intermediate) and RCP8.5 (severe) scenarios. The evaluation of the model showed particularly good skills in reproducing the climatic features of air temperature and precipitation over Italy . A subsequent focus on the wind fields over the Adriatic Sea , whose reproduction is a challenging task for hindcast and operational models as well due to the geometry of the basin and its complex coastal orography, highlighted outstanding skills for both intensity, although with some tendency to overestimate mean wind energy and direction. Most interestingly for ocean modelling applications, COSMO-CLM proved capable of capturing the bimodality of Bora (northeasterly) and Scirocco (southeasterly) in the northernmost part of the basin, which would be impossible to reproduce with previous climate models . In a recent work COSMO-CLM was also used to quantile-adjust near-surface wind speeds from the ECMWF ERA5 reanalysis, thus merging the accuracy of the former with the higher temporal resolution and the synchronisation with observed variability in the latter. For the wave modelling experiment described by and for the present work, the COSMO-CLM wind fields over the Adriatic Sea were retrieved for two 30-year periods in control conditions in the recent past (CTR; 1971–2000) and in the future in a severe RCP8.5 climate scenario (SCE; 2071–2100).

The modelling run described by that provides wave data for this application was thus implemented in SWAN, with reference to the Adriatic Sea in the CTR and SCE periods. SWAN provides a phase-averaged description of wind-generated sea states by solving a non-stationary wave action balance equation : 1∂N∂t+∂cxN∂x+∂cyN∂y+∂cσN∂σ+∂cθN∂θ=Swσ.

N represents the action density – namely, the wave energy density divided by the relative frequency – and t is time. The propagation of N (second to fifth term in Eq. ) is described in 2-dimensional space (x and y, expressible in both Cartesian and spherical coordinates, with speed, represented by cx and cy, respectively) and spectral space (radian frequency σ relative to a frame moving with the ocean current; angle θ normal to the wave crest; and speed cσ and cθ, respectively). Sw represents sources and sinks of wave energy density associated with generation, dissipation, and non-linear wave–wave interactions.

For the application presented here, the domain was discretised into an orthogonal curvilinear structured grid, with a horizontal resolution ranging from approximately 2 km in the northern region to 8–10 km in the southeasternmost part of the study area. Calm conditions were prescribed at the open boundary at the Otranto Strait, where waves generated within the basin were nonetheless permitted to radiate out of the domain. This assumption was made necessary by the lack of available wave fields consistent with the atmospheric forcing at the Mediterranean scale, but the validation confirmed that no major drawbacks in the results could be found beyond 100–200 km from the boundary. Wave spectra were discretised into 25 logarithmically distributed frequencies, ranging between 0.05 and 0.5 Hz, and 36 directional sectors, whereas the time step was set to 360 s. The bathymetry was reinterpolated from a 1 km resolution dataset used in previous applications and obtained by merging recent surveys in the shallow northern basin and in the southern continental margin into previous basin-scale information. Sea level rise between the CTR and SCE periods was taken into account by increasing the water depth in the latter scenario by 0.70 m, based on estimates by , which is, for the sake of simplicity, uniformly distributed throughout the domain. As wind forcings from COSMO-CLM were provided with 6-hourly time step, the same interval was maintained for the output, in which the main spectral parameters were saved for each grid point and the full spectra were saved for approximately 600 points along the Adriatic coast. The model validation was based on directional wave recordings from three observatories off the Italian coast along the main axis of the basin, namely the Acqua Alta oceanographic tower (AA, 45.31° N, 12.51° E; see ) and the Ortona and Monopoli buoys (respectively, 42.42° N, 14.51° E and 40.98° N, 17.38° E; Fig. ).

The comparison to observational data (carried out in statistical terms, as climate models are not synchronised with observed variability) was focused on significant wave height (Hs), and mean direction showed overall satisfactory performances for the SWAN implementation. The reported tendency of COSMO-CLM to overestimate mean wind energy actually had a moderate impact on wave modelling and was partially compensated by other factors such as the southern boundary conditions and some residual limitations in reproducing orographic jets; its more marked effect was a partial overestimate of significant wave height statistics in the southernmost regions of the basin.

The end-of-century projections in a severe climate change condition outlined a composite scenario. While Hs in mean and stormy conditions appeared to decrease in most of the basin and for most directions, the effect of storms from the southern quadrant (southwest to southeast) on the northern Adriatic Sea was expected to intensify. This result, interpreted as a consequence of a northbound shift in the storm tracks in the Mediterranean region, was shown to have significant implications for the coastal regions. Besides the obvious impact of stronger storms where this will happen and besides the baseline sea level rise exacerbating the effect of storms even when their intensity is expected to decrease , the spatial variability in the impact of climate change will result in a modification of the patterns of energy fluxes onto and along the Adriatic coast, thus modifying the sediment transport rates and gradients and, ultimately, coastal morphodynamic processes.

The training and the application of DELWAVE were based on basin-wide wind fields from COSMO-CLM and pointwise wave time series at six locations exposed to different wave climates (Fig. ), including AA (45.31° N, 12.51° E), OB (42.42° N, 14.51° E), and MB (40.98° N, 17.38° E), which coincide with the observation points used for the SWAN model validation in and are representative of nearshore conditions, respectively, along the northern, central, and southern Italian coasts. Grado (45.68° N, 13.45° E) lies at the edge of the gulf of Trieste in the northernmost end of the Adriatic Sea, facing south, and is partially sheltered by the Istrian peninsula. OB2 (42.97° N, 15.35° E) and OB3 (43.37° N, 16.00° E) are located along an ideal transect off Ortona, respectively, in the middle of the basin and along the Croatian coast. Wind fields are provided as 6-hourly meridional and zonal components, whereas wave data, also 6-hourly, are given in terms of significant wave height (Hs), mean wave direction (d), and energy period (Tm-1,0). The model data from the control (CTR) period (1971–2000) were used for the network training, whereas future scenario (SCE; 2071–2100) data were used as a reference for assessing the network skills, particularly in terms of their capability to capture the features of the climate signal.

The data DELWAVE uses to conduct both training and inference are available in the form of a tensor, which contains three logically separate fields: spatial wind field, location encoding, and grid encoding. Each of this parts serves a specific purpose, and we elaborate on each in the following subsections.

DELWAVE is composed of three logically distinct components, each responsible for a specific processing task, as depicted in Fig. . The atmospheric encoder is responsible for encoding the input fields for each time step into high-dimensional vectors. These vectors are then passed to the temporal collapse block, where they are merged into a single vector and attenuated based on the temporal importance of the individual inputs, as explained in Sect.  below. Finally, the regression block transforms this vector into the three outputs required. Individual blocks are described in more detail in the following subsections.

Additionally, let us define the notion of an encoder. An encoder is a sequence of transformations, a sub-neural network, which maps a specific input to a, usually, high-dimensional vector. This vector is said to be an encoding of the provided input, carrying information about it, albeit in an obtuse way. The encoder–decoder structure , for example, is a common paradigm in machine learning that leverages this terminology.

The CTR period was used for training, while the SCE period was used for testing the final, developed model. The CTR data were further split into two parts: the actual training dataset (CTRtrn) and the validation dataset (CTRvld). CTRtrn contains the first 80 % of the training data, while CTRvld contains the remaining 20 %. The data for each location and for each variable (significant wave height, mean wave period, mean wave direction) are separately standardised to exhibit zero mean and a variance of 1. Prior to the standardisation, we log-transform significant wave heights as ln⁡(SWH+1). We give arguments for this transformation in the following paragraphs. Then, at each training iteration, a random batch of training samples is collected and the model loss function, the root mean squared difference, defined in Eq. (), is used to optimise DELWAVE's parameters, which are evaluated at these batches.

Neural networks often have difficulties predicting extreme events in the tails of the distributions because these events are by definition rare and the network rarely encounters them in the training set. To learn and model underrepresented values of the target variables better, we increased their presence in the training set by employing the so-called random importance sampling. We illustrate random importance sampling in Fig. . If we observe the distributions of the three target variables in a randomly sampled batch (left panel in Fig. ), we can see that these are skewed. For example, the significant wave height is distributed similarly to a left-slanted gamma-like distribution with a very long tail. Therefore, the model is not exposed to the tail of the distribution frequently which inhibits efficient training in that part of the distribution. This results in systematic errors, where the regression accuracy for significant wave height drops with increasing height. This is understandable as samples with wave heights over 2 m constitute only a small fraction of the dataset, contributing less during training compared to samples with smaller wave heights.

Our implementation of importance sampling is conducted on the fly at batch acquisition. The reasons we do this on the fly as opposed to conducting this statically (oversampling before training and saving the new samples on disc) is the following: oversampling highly skewed distributions to a point of close uniformity would require a large number of additional samples. Since we had limited disc space, this was not an option. Therefore, we implemented single-variable importance sampling that oversamples one of the target variables at random for a given training batch. However, when we oversample one of the variables, the remaining usually remain biased. We can observe this effect in the skewed green histograms in Fig. , while the blue histograms are more uniform. To alleviate this issue, we alternated the sampling between the three target variables randomly to eliminate single-variable importance-sampling bias.

Furthermore, we alternated between regular sampling and importance sampling, where every second batch was randomly importance-sampled. This compromise offered the best performance out of the two approaches. We believe that this is due to the majority of data taking on only a small subset of values; thus, these values influence the loss more than the rare events. This is especially true for significant wave heights, where only 5 % of all samples across all locations exhibit wave heights over 2 m. Additionally, since fitting unbiased estimates of the tail of the distribution for significant wave heights was still challenging, we also penalised the network for misclassifying significant wave height twice as much as for the remaining two variables.

We conducted our training procedure in two stages. Since we trained our model on the Vega cluster , we were limited by the maximum time our training could take up. A single run could last up to 2 d maximum; therefore, we first trained our model using the Adam solver , with default PyTorch parameters, a learning rate of 10-3, and a weight decay of 10-6 for 2 d. Following this period, we extracted the model that performed on the validation dataset best, reinitialized the learning procedure with a reduced learning rate of 10-5, and retrained for 600 more epochs. We again took the model that performed on the validation dataset best and used it to compute the test dataset results we present in the following sections.

In this section, we present DELWAVE performance during the far-future period of 2071–2100, as benchmarked against SWAN simulations. In other words, SWAN simulations represent the ground truth DELWAVE aims to model. Figure  depicts DELWAVE–SWAN scatterplots of Hs, d, and Tm-1,0 at the locations of the Acqua Alta oceanographic tower (AA) and the Ortona and Monopoli buoys (OB and MB, respectively; see Fig.  for the locations). Results for other locations are provided in the Supplement.

We proceed by analysing DELWAVE performance using three related figures. Figure  depicts DELWAVE predictions for Hs, d, and Tm-1,0 compared to those from the SWAN model, obtained from the same wind fields, i.e. for the same forecasting time window. Figure  shows the overlaps of histograms of Hs, d, and Tm-1,0 from both DELWAVE and SWAN models. Note that close overlap of the distribution histograms from both models does not guarantee a good forecast since this overlap does not say anything about the synchronicity of both forecasts; one therefore needs to view Fig.  in conjunction with Fig. . Additionally, Fig.  illustrates how DELWAVE forecasting mean absolute errors change depending on which part of distribution we are modelling. Here, mean errors imply error averaging over all the forecasting samples in a specific distribution bin. Consequently, the error values are only well defined in the bins containing a large enough (e.g. over 100) number of samples. In what follows, we base our remarks on an interplay of messages from all three figures.

Location AA in the northern Adriatic (off the Venetian shore; see Fig.  for the location) is marked by an excellent performance in Hs and d prediction, indicated by the near-linear scatterplot displayed in Fig. . The same aspect of DELWAVE performance is illustrated via histogram distribution for the same three parameters in Fig. . Mean wave direction d (top row, right column of Fig. ) exhibits two maximums related to two dominant Adriatic winds, northeasterly Bora at roughly 75° and southeasterly Scirocco at roughly 135°. Short wave periods at the AA location, on the other hand, seem to be the hardest to predict, as can be seen from in the left column in either Fig.  or Fig. . This is, to some extent, expected: long wave periods correspond to longer waves and consequently windy atmospheric conditions. Short periods, on the other hand, correspond to calm conditions, where the network is essentially modelling low-amplitude, short-wavelength, stochastic sea surface behaviour.

Similar observations can be made for OB and MB locations. SWAN Hs is modelled very reliably with DELWAVE. Multi-modal direction histograms at all locations are also reproduced to a high degree of accuracy, as can be seen from the middle column of Fig. . On the other hand, the network seems to be struggling to reproduce northerly directions (roughly 0° ± 10°) at this location. This leads to horizontal strips of incorrect predictions displayed in the scatterplot of the right column, second row in Fig.  and to a bump in mean absolute error in the histogram displayed at the same location in Fig. .

Figure  also hints at quantitative estimates of DELWAVE performance. When it comes to Hs predictions (middle column), errors at all locations grow with significant wave height from errors below 5 cm for Hs below 1 m to errors on the order of 10–15 cm for Hs over 3 m. DELWAVE predictions of mean wave direction d (right column) exhibit the smallest errors in the directional bins corresponding to prevalent wind patterns. In general, directional errors are below 25° and even lower at AA. High directional errors at 0 and 360° stem at least partly from the algorithm's false distinction between 0 and 360° directions.

Wave period Tm-1,0 predictions are illustrated in the left column of Fig. . At all locations, periods below 6 s are captured well by DELWAVE, with prediction errors below 0.25 s. Longer periods, however, likely corresponding to an incoming swell, exhibit more diverse behaviour. MB location wave periods seem to be captured more accurately in the long-period limit, with the forecast error dropping below 0.1 s. At the OB location, the errors in the long-period limit slightly rise from 0.2 to 0.3–0.5 s. The AA location, on the other hand, indicates a sharp rise in the Tm-1,0 prediction error, which reaches 1 s for the period above 8 s.

The error behaviour at the AA location is possibly explained by the differing roles played by the basin geometry, the local wind sea, and swell. Location AA is prevalently exposed to northeasterly Bora (blowing from roughly 75°) and to southeasterly Scirocco (blowing from 135°). In the case of Bora, the fetch is quite limited since Bora is a cross-basin wind. Therefore, we do not expect swell to play a major role at the AA location during Bora conditions: the wave field at the AA location must be determined by local wind conditions. The case of Scirocco is very different. Scirocco is an along-axis wind, with the largest fetch in the Adriatic Basin. This means that during Scirocco, the swell field at AA is determined to a large extent by non-local wind patterns in the south of the basin. Local wind conditions at the AA location are furthermore often a poor proxy for winds in the south Adriatic. Bora in the north (promoting short fetch and shorter wave periods) coinciding with Scirocco in the south (promoting long-period swell arriving at AA) is, for example, not unusual. These circumstances likely pose a challenge for the DELWAVE deep network, resulting in growing errors during longer wave periods (which most likely occur during Scirocco).

This explanation can be further substantiated by comparing wave period MAE to Hs, Tm-1,0, and wave power. The latter is computed from the linear theory to be 13P=ρg264πHs2Tm-1,0, with ρ being the water density and g the acceleration due to gravity. This comparison is depicted in Fig. , which corroborates this interpretation and constrains DELWAVE limitations in capturing the basin-scale dynamics.

The concentration of the highest values of MAE at low values of Hs and P (left and right columns, respectively) confirms that largest errors tend to be associated with low-energy, nearly random sea states, even in the presence of relatively long waves along the main basin axis (Scirocco at AA), and thus with limited impacts on possible practical applications. It is further worth mentioning that a separate analysis, carried out by independently considering the rising and declining phases of the sea states (not shown), did not exhibit any preferential concentration of the higher values of MAE in either phase. Wave period error is therefore not systematically larger during either onset or calming of the storm, suggesting that it is not directly related to the sequential and monotonous temporal encoding of inputs within DELWAVE. Had this not been the case, we would have expected some error asymmetry with regard to the timing of the storm.

The analysis of the storms was carried out by comparing the DELWAVE results to the SWAN time series during the period of 2071–2100. For each time series, the storms were identified following the method proposed by Boccotti – namely, (i) finding the events with Hs larger than 1.5 times the mean value Hs‾ of each respective series, (ii) merging the events parted by less than 10 h, and (iii) discarding those overall shorter than 12 h. Figure  compares SWAN and DELWAVE peak Hs and directions for each storm at AA, OB, and MB (the same is shown for the other locations in the Supplement), considering entire sets of storms occurring during the period separately and the annual maxima for each series. While the former provides a broader view on how DELWAVE reproduces the whole meteo-marine climate at each location, the latter aims at assessing its capability of addressing extreme events. The picture is flanked by a quantification of the DELWAVE precision (how many DELWAVE-predicted storms are actually present in the SWAN time series) and recall (how many SWAN-modelled storms are retrieved by DELWAVE). These two metrics are computed as 14precision=TPTP+FP,recall=TPTP+FN, where TP, FP, and FN denote true positive (storm present in SWAN and predicted by DELWAVE), false positive (storm predicted by DELWAVE but not present in SWAN), and false negative (storm present in SWAN but not predicted by DELWAVE) classifications. Figure  shows an example of the application of Boccotti's method in SWAN and DELWAVE storms and the occurrence of false negatives and false positives.

All considered sets exhibit a satisfactory performance with very high scores (precision and recall ≥ 0.95) when all the storms are considered. When only annual maxima are taken into account, precision and recall are lower, though fairly high (≥ 0.8) and without an evidently prevailing directional offset. Considering the whole storm sets, most of the false negatives and positives are generally clustered among the weakest events. This can be explained by considering that, for particularly weak or short events, small absolute errors can mean large relative errors. Therefore, in a small Hs limit, a small error in the reproduction of Hs can already significantly impact whether the criteria for the identification of storms are met or not (Fig. ).

This result seems to be in contradiction with the results for the yearly maxima sets, where prediction and recall scores decrease and the number of false negatives and positives increases. This contradiction is, however, only apparent and related to the propagation of Hs prediction errors downstream into the identification of the yearly maxima. More precisely, in this case, the mismatch does not seem related to the classification of an event as a storm but rather to its classification as a yearly maximum: in fact, a slight error in predicting the peak height of storm events can introduce some noise in the ranking of the events and, in particular, in the identification of the yearly maxima, leading to a mismatch between DELWAVE and SWAN. Nonetheless, as long as small errors in the prediction of the peak Hs are the cause for this mismatch, even if the events retrieved by DELWAVE are not exactly the ones resulting from the SWAN time series, their properties (or at least their peak Hs) should be quite close, which should be sufficient for most practical applications.

One of the main scopes of DELWAVE is to provide a computationally cheap model emulation system capable of providing large ensemble predictions for wave climate at a multi-decadal scale. This kind of applications is, to some extent, complementary to the event-scale analysis of single storms and requires a specific assessment of the network capability of capturing the main statistical features of the climate signal. Figure  provides a twofold comparison of the climatological normals of the monthly mean, median, and 99th percentile of Hs at AA, OB, and MB (the same values for the other locations are provided as a Supplement) provided by SWAN and reproduced by DELWAVE. The statistics resulting from SWAN and from the DELWAVE time series are compared to each other in the end-of-century scenario (2071–2100; SCE), and both are compared to the statistics from the control condition (CTR), available only for SWAN in the 1971–2000 period. The good agreement between DELWAVE and SWAN is also confirmed when considering climatological statistics, with a small (≤ 5 %), though systematic, underestimate of 99th-percentile values, reflecting what was discussed in Sect. . Compared to the CTR climatologies, the mismatch between DELWAVE and SWAN is generally small compared to the difference between SCE and CTR conditions, suggesting that the noise possibly introduced by the model mimicking is weaker than the climate change signal in the considered locations. Not surprisingly, the only way in which the performance seems partially affected by seasonality is through the modulation of significant wave height and the tendency of the network to underestimate higher (and therefore wintry) values. Nevertheless, Fig.  shows that the potential modelling errors, introduced by the DELWAVE model, are substantially smaller than the difference between the scenario (2070–2100) and control periods (1970–2100).

Following a similar approach for the directional wave climate, the linearised wave roses in Fig.  show that the agreement between DELWAVE and SWAN allows us to capture important impacts of climate change in the wave regime not only in absolute terms, but also in response to projected shifts in the wind regimes. This is, for instance, the case of the slight weakening of Bora (NE) storms associated with an intensification of Scirocco (SE) events in the northern Adriatic Sea in the broader framework of a tendency towards an overall decrease in the storminess in most of the basin, suggested by and confirmed by the DELWAVE projections.