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# Faster than real-time, phase-resolving, data-driven model of wave propagation and wave-structure interaction

Jeffrey Harris

► To cite this version:

Jeffrey Harris. Faster than real-time, phase-resolving, data-driven model of wave propagation and wave-structure interaction. Applied Ocean Research, 2025, 154, pp.104291. 10.1016/j.apor.2024.104291. hal-04871928

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Submitted on 7 Jan 2025

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# Faster than real-time, phase-resolving, data-driven model of wave propagation and wave-structure interaction

Jeffrey C. Harris

LHSV, ENPC, Institut Polytechnique de Paris, EDF R&D, Chatou, France

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## ARTICLE INFO

Dataset link: http://github.com/jcharris/mews

**Keywords:**

Wave propagation

Wave-structure interaction

Machine learning

## ABSTRACT

A machine learning time-series prediction approach is proposed for wave propagation and wave load prediction. Under unidirectional wave conditions and variable bathymetry, given a wave gauge upstream, a model is shown to reproduce wave elevation or wave forces downstream under irregular steep and either non-breaking or breaking conditions. Attempts to perform the opposite calculation, predicting upstream conditions from downstream measurements, results in higher error, likely due to information loss under breaking conditions. For choice of machine learning approach, comparisons show that the Time-series Dense Encoder (TiDE) approach results in a good balance between model complexity, stability, computational time, and error. Over a flat bottom, time-series of wave elevation can be predicted up to 10 wavelengths away, though with a degraded accuracy compared to shorter distances. Similar results are shown for time-series of forces on a vertical cylinder, showing better results than a simple Morison approach, as used in engineering tools such as OpenFAST, but with a similarly fast computational time. Generalizations show that training on irregular wave data permit extrapolations to periodic wave cases. Finally, the same method also is also demonstrated at field-scale, comparing results between two offshore buoys.

## 1. Introduction

With rapid growth in offshore developments, it is often necessary to be able to predict wave transformation in the coastal environment, as well as to predict wave forces on offshore structures. From early works by Havelock (1940), MacCamy and Fuchs (1954), or Morison et al. (1950), relatively simplified calculations, to more recent ones (Suja-Thauvin et al., 2020) including higher-order kinematics and even structure movement, ultimately there remain open questions about the forces on vertical cylinders due to water waves. The problem is challenging in part because the largest forces are not always related to the largest wave crests (Kjeldsen et al., 1986).

Despite significant improvements in classical numerical modeling techniques, using either linear or nonlinear potential flow, or using a Navier-Stokes solver or computational fluid dynamics (CFD) approach (Aggarwal et al., 2016), this can often be time-consuming. In the case of potential flow, linear wave modeling is often real-time, and can be included in engineering design tools such as OpenFAST (Wang et al., 2022), and even second-order effects through efficient representations of quadratic transfer functions (Bredmose and Pegalajar-Jurado, 2021), but third-order or higher terms become more complicated. Fully nonlinear potential flow approaches can require complex algorithms for maintaining reasonable computational time (Harris et al., 2022), and treating breaking waves within potential flow may add uncertainty

from dissipation models (Mohanlal et al., 2023), and for the resulting wave impacts, open questions remain about semi-empirical load model development (Renaud et al., 2023). While many of the engineering formulations are based on potential flow, it is known that vortex formation around a cylinder can also be important (Yang and Rockwell, 2004), which cannot be easily incorporated into a potential formulation. Alternatively, CFD models can even include the important and more complex process of breaking wave impacts, and the computer hardware they run on continually improve in speed, but this approach is far from real-time. With hundreds of processor cores, computational times are often measured in hours, days, or even weeks for reproducing seconds or minutes of an experiment (Li and Fuhrman, 2022; Balle Martin et al., 2023). Meanwhile, with new developments in digital twins for offshore structures (Branlard et al., 2023), there is particular interest in real-time or faster simulations.

Many studies have shown the power of machine learning (ML) applications for water waves which, after sufficient model training, can be extremely rapid in comparison to a traditional model. For example, Kuehn et al. (2023) made a super-resolution model of coastal waves for accelerating calculations. Meanwhile, Eeltink et al. (2022) developed an ML model for breaking that combined with a physics-based evolution model for matching wave tank experiments, and Liu et al.

<footnote>E-mail address: jeffrey.harris@enpc.fr</footnote>
(2024) further developed this approach, an important look of implementing breaking in phase-resolved wave models. A direct replacement for SPH (smoothed particle hydrodynamics) with a variety of materials was even shown by Sanchez-Gonzalez et al. (2020) using graph neural networks. Other major developments include the development of physics-informed neural networks by Raissi et al. (2019), imposing physical knowledge into trained models. Alternatively, Brunton et al. (2016) has been able to turn equation discovery through ML into a nearly automated process through the SINDy (Sparse Identification of Nonlinear Dynamics) approach. Just among time-series prediction, ML applications are vast (Lim and Zohren, 2021). These developments are substantially more sophisticated in terms of trained parameters to neural networks in papers from just a couple decades previous (Deo and Jagdale, 2003), where neural networks were used with 15 parameters. In coastal or ocean engineering, indeed the number of applications for ML are now numerous, with much larger networks (Kim and Lee, 2022; Juan and Valdecantos, 2022). Similar to other large language models (LLMs), there are now LLMs devoted to time-series prediction (Jin et al., 2023; Gruver et al., 2024).

Whatever ML approach is trained, or even for validating a classical model, in addition to the numerous published works, more and more datasets have become available in recent years. For example, the De-Risk database (Pierella et al., 2021) provides numerically generated nonlinear wave kinematics, including breaking dissipation, suitable for offshore design work. For laboratory wave gauge results, the FLOAT-ECH database (Kim et al., 2024) includes unidirectional and multidirectional wave conditions, and the CCP-WSI group even produces blind tests for model validation (Ransley et al., 2019). Field data initiatives such as the FOWD database (Häfner and Jochum, 2021), the CDIP buoy network (Behrens et al., 2019), the ONR test bed for wave models (Ris et al., 2003) or lidar wave gauge data from the Field Research Facility in North Carolina (Irish et al., 2006) are all large datasets available for use.

The developments of time-series prediction within ocean engineering are numerous, from wave prediction (Xu et al., 2021) to prediction of tsunami inundation from offshore observations (Mulia et al., 2022). The results are not just academic, but for example improvements in wave prediction have direct application in floating wind turbine controllers as shown by Ma et al. (2018) or Fontanella et al. (2021), if ML solutions can be applied to deployed offshore applications. Real-time prediction of structure movement has already been done (Guo et al., 2021) or indeed the inverse, computing the sea-state from the movements of a structure (Grilli et al., 2018). Ship motion simulations can also be improved by including an ML component (Silva and Maki, 2022), and even marine structures with mooring lines have been modeled using neural networks (Christiansen et al., 2014).

Often ML applications are focused on real-time applications with complex geometries where other methods are too computationally complex, but some work has also been done for loads on vertical cylinders, which can be valuable both for direct application but also to compare with benchmarks that are well understood. Ma and Sclavounos (2021) used a support vector machine approach to model forces on cylinders, whereas Tang et al. (2024) used a Gaussian process based model for a similar problem. For more complex structures, Wang and Ti (2024) considered other ML approaches for symmetric cylinders of arbitrary cross-sections, and Yu et al. (2022) used an M5 model tree approach to compute forces on composite bucket foundations. Alternatively, Pena and Huang (2021) used a generative adversarial network, Wave-GAN, for producing results for wave-structure interaction similar to a Navier-Stokes solver, in far less time.

For wave prediction, the problem has already seen substantial development in terms of deterministic, stochastic, and ML solutions. In 1DH (one horizontal dimension), Blondel et al. (2010) has laid out guidelines for wave prediction from a probe, essentially noting that information travels with the group velocity of the waves. Desmars et al. (2020) recently used a weakly nonlinear Lagrangian wave model for this type

of deterministic approach. Stochastic approaches (Mérigaud and Ringwood, 2018) can also take advantage of information that deterministic models cannot. Similar advantages exist with ML approaches, seen in 1DH by many (Silva and Maki, 2022; Xu et al., 2021; Chen et al., 2023). In 2DH (two horizontal dimensions), ML solutions are to date less common, with classical methods being typically considered (Qi et al., 2018; Simpson et al., 2020; Kim et al., 2023). For example, Fisher et al. (2021) looked into the forecast horizon and number of point measurements required for different accuracies, including the advantage of phase-resolved versus spectral measurements. Alternatively, ML approaches have been applied by Le et al. (2022) and Li et al. (2023). Indeed, a vast number of works have been produced in the last few years relating to ML solutions, due not just to the ease of constructing models with sparse data measurements, but the rapid computational time (Mohaghegh et al., 2021).

In this paper, the construction of a machine learning solution is proposed, first testing three different ML approaches. Given the vast number of recent works on the subject, a model is developed prioritizing simplicity and clarity, so the scope is limited here to unidirectional long-crested waves, and an upstream wave gauge is used to predict either a downstream wave elevation (over potentially variable bathymetry) or the force on a structure. Results are shown for both non-breaking and breaking waves over a submerged bar, as well as irregular wave impacts on a vertical cylinder. In order to show the generalizability of the result, the model trained on irregular waves is then applied to regular waves, with reasonable agreement with experimental data.

## 2. Methodology

### 2.1. Problem description

While most numerical models are based on discretizing completely a section of space and time (Fig. 1a), this requires suitable initial and boundary conditions. Alternatively, one may often be faced with an ill-defined problem (Fig. 1b), where a sparse measurement or set of measurements may be known for a certain amount of time, and even an approximate solution is acceptable. For example, in the case of unidirectional waves, which go through an a priori unknown but physical transformation (after propagating a fixed distance over some bathymetry), it should be possible to estimate the downstream wave elevation. Although apparently a black-box approach, which obviously has limitations, a finite number of parameters may describe the dynamical system to an acceptable approximation. Generally the duration of the input time-series required can be related back to the group speed of the incoming waves.

Naturally one can create new neural network architectures, but given the rapid pace of ML developments, it can be helpful to map the physical problem to an established problem in image or video processing, or in time-series prediction, in order to benefit from the latest developments. For a long-term multivariate forecasting problem, predicting a variable y as a function of known history and some dynamic covariate x can be written as:

$$[y_{i:i+H}] \approx f([x_{i-L:i-1}], [y_{i-L:i-1}]) \quad (1)$$

using slice notation, where predicted values of y from time index i to i + H are assumed to be a function of earlier values of both x and y, between i - L and i - 1. Values of H refer to the horizon of future time-steps, and L refer to the look-back of past time-steps. In the problems considered here, no static covariates will be considered.
<figure><img src="image_4.png" /><figcaption>Fig. 1. Schematic of problem. (a) Classical modeling approach, where at each time-step values depend on the state of the domain at the previous step, in combination with initial conditions (ICs) and boundary conditions (BCs), and often points are separated in space by a grid spacing Δx and in time by Δt. (b) Wave prediction approach, where H points are computed in a batch for unidirectional water wave propagation (the horizon), given that L points of a time-series (the look-back). Filled circles denote known values, and open circles denote computed conditions.</figcaption></figure>

## 2.2. ML approaches

At each point in time, the future state can be related back to a subset of the input time-series (Fig. 1b). If this is the case, a neural network can be used for this function due to the universal approximation theorem (Cybenko, 1989; Hornik, 1991).

Although there are many parameters that can be modified, for each case below, to provide a baseline for future improvement, the error, computational time, and number of trainable parameters are reported. In order to compute the error (the loss function), the mean-squared error (MSE) is used for a variable y and observations ŷ:

$$MSE = \frac{1}{N} \sum_{i=1}^{N} (y_i - \hat{y}_i)^2 \quad (2)$$

and the fit of a particular solution is judged in terms of minimizing this value. It is worth noting that Mentaschi et al. (2013) showed that other loss functions, such as that of Hanna et al. (1985), may be better than the mean square error for some wave problems.

For comparing methods, the DARTS library (Herzen et al., 2022) is used, making use of its implementation of various time-series prediction algorithms, with parameters described below. All were trained using the Adam optimizer (Kingma, 2014), and calculations were performed on a NVIDIA T1200 GPU. Three algorithms were considered:

**LSTM model.** Long short-term memory (LSTM) neural networks (Hochreiter and Schmidhuber, 1997) are a type of recurrent neural network that have been previously shown to be useful for solving marine dynamics problems (Silva and Maki, 2022; Liu et al., 2022).

**N-BEATS model.** Next, we consider the neural basis expansion analysis for interpretable time series forecasting (N-BEATS) model proposed by Oreshkin et al. (2019). This is a more complex deep neural network, but which seems to outperform classical statistical time-series forecasting approaches in various benchmarks, unlike earlier architectures, including LSTM (Makridakis et al., 2018).

**TiDE model.** After the plethora of different neural network architectures have been developed in recent years, Das et al. (2023) proposed the Time-series Dense Encoder (TiDE), a comparatively simple neural network architecture that could outperform some considerably more complex models.

Taking the same inputs as the other models (Fig. 1), the data in this network (Fig. 2) passes through four different residual block networks.

Each residual network (Deng et al., 2009) is composed of linear (fully-connected) transformations, ReLU activation function (Hahnloser et al., 2000) activation functions (i.e., $f(x) = max(0, x)$), and a Monte Carlo dropout (Gal and Ghahramani, 2016). In the present case, the dropout rate for all residual blocks was set to 0.1.

From this diagram we can trace data through each calculation. For example, if the look-back consists of L = 65 points, and is used to predict a horizon of H = 20 points, the covariate projection residual block has 780 trainable parameters, and is applied (with the same parameters) to each of the L points of the covariate (in a wave propagation problem below, this would be the upstream measurements of the wave elevation). The encoder block would have 99968 trainable parameters, and pass 128 parameters onto the decoder, which would have a similar size, with 99072 trainable parameters. This would then pass to the temporal decoder, with 594 trainable parameters, which, like the projection step, would be applied multiple times (here H times), to decode the output time-series. Finally, there is a linear skip connection, with 1320 parameters; similar to the residual block design, this is thought to improve the training of the model.

Note that in all approaches tested here, as is typical for ML studies, the inputs and outputs are scaled to a range between 0 and 1 of the training data, and likewise predictions are inverted to obtain values in terms of physical dimensions.

## 3. Wave prediction over submerged bar

The wave prediction problem can essentially be defined as searching for a function that, given discrete observations of a wave gauge $η_0(t)$ upstream, computing the downstream wave gauge elevation $η_1(t)$, thereby using Eq. (1), thus in a nondimensional form with $x = η_0/h_0$ and $y = η_1/h_0$ for some reference distance $h_0$, where discrete times $t_i$ are separated by a fixed timestep $Δt$, and where the values L and H can be fixed from knowledge of wave theory. The physical system is formally not well-defined, given that no conditions have been imposed on the velocity field (not to mention vorticity), etc., but as seen below, the errors are often still reasonable.

In order to test the different wave prediction algorithms, the test case of wave propagation over a submerged bar, as measured experimentally by Beji and Battjes (1993), is considered. The wavemaker depth is 0.4 m deep, the top of the bar is 0.1 m deep, and there is a 1/20 front-slope and 1/10 back-slope (Fig. 3). Test cases were considered for
<figure><img src="image_5.png" /><figcaption>Fig. 2. Diagram of variant of TiDE architecture used here. Each line is labeled with the vector length transmitted between individual blocks, and each dense (or fully connected) layer is labeled with the number of trainable parameters, both in terms of weights and bias.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Schematic of submerged bar test case of Beji and Battjes (1993) showing the locations of the 8 wave gauges. Waves are generated on the left and propagate towards the right, and for even sinusoidal incident waves, the waves pass through a nonlinear and dispersive transformation over the bar.</figcaption></figure>

regular and irregular waves, two different period waves, and for non-breaking, spilling, and plunging waves (Table 1). For the irregular wave cases, a JONSWAP spectrum (Hasselmann et al., 1973) was imposed. Measurements were made at 8 wave gauges, sampling at 10 Hz, at x = 6, 11, 12, 13, 14, 15, 16, and 17 m, which we refer to as WG1 to WG8. For reference, note that in 0.4 m deep water, the group speed of 1.0 s waves is 0.89 m/s, and of 2.5 s waves is 1.74 m/s (which decreases to 0.81 m/s and 0.96 m/s, respectively, when moving to 0.1 m).

Although preliminary tests show that results hold for any combination of wave gauges, for simplicity of presentation, we focus here on the transformation over the bar, between WG3 and WG5, where physically waves are steep and potentially breaking, and where waves propagate

approximately one wavelength, a typical distance for short-term wave prediction problems (e.g., Kim et al., 2024).

### 3.1. Training

First, for a straightforward way the numerical errors, consider the JLN test case (Table 1), for irregular non-breaking waves. This case can be used to verify estimates of the parameters L and H based on the linear group velocity, c_g, following Fig. 1, knowing the linear dispersion relation and phase speed (Dean and Dalrymple, 1991):

$$ \omega^2 = gk \tanh kh \quad (3) $$
<figure><img src="image_7.png" /><figcaption>Fig. 4. Comparison of input window parameters *L* and *H* from Eq. (1) versus error for JLN test case for predicting wave gauge 5 from wave gauge 3, training on 30 minutes of data sampled at 10 Hz, and testing on 15 minutes of data.</figcaption></figure>

<caption>**Table 1**</caption>

Conditions for wave propagation over a submerged bar studied by Beji and Battjes (1993). Wave height refers to upstream conditions, at the toe of the bar (WG 1). For irregular waves, wave height and period refers to significant wave height and peak period, respectively.

<table><caption>**Table 2**</caption><thead><tr><th>Case</th><th>Waves</th><th>Height</th><th>Period</th><th>Condition</th><th>Time</th></tr></thead><tbody><tr><td>SLN</td><td>Sinusoidal</td><td>2.2 cm</td><td>2.5 s</td><td>Non-breaking</td><td>5 min</td></tr><tr><td>SSN</td><td>Sinusoidal</td><td>3.7 cm</td><td>1.0 s</td><td>Non-breaking</td><td>5 min</td></tr><tr><td>SLS</td><td>Sinusoidal</td><td>3.4 cm</td><td>2.5 s</td><td>Spilling</td><td>5 min</td></tr><tr><td>SSS</td><td>Sinusoidal</td><td>5.4 cm</td><td>1.0 s</td><td>Spilling</td><td>5 min</td></tr><tr><td>SLP</td><td>Sinusoidal</td><td>4.1 cm</td><td>2.5 s</td><td>Plunging</td><td>5 min</td></tr><tr><td>SSP</td><td>Sinusoidal</td><td>6.6 cm</td><td>1.0 s</td><td>Plunging</td><td>5 min</td></tr><tr><td>JLN</td><td>JONSWAP</td><td>2.1 cm</td><td>2.5 s</td><td>Non-breaking</td><td>45 min</td></tr><tr><td>JSN</td><td>JONSWAP</td><td>2.8 cm</td><td>1.0 s</td><td>Non-breaking</td><td>45 min</td></tr><tr><td>JLS</td><td>JONSWAP</td><td>3.1 cm</td><td>2.5 s</td><td>Spilling</td><td>45 min</td></tr><tr><td>JSS</td><td>JONSWAP</td><td>4.1 cm</td><td>1.0 s</td><td>Spilling</td><td>45 min</td></tr><tr><td>JLP</td><td>JONSWAP</td><td>3.8 cm</td><td>2.5 s</td><td>Plunging</td><td>45 min</td></tr><tr><td>JSP</td><td>JONSWAP</td><td>4.8 cm</td><td>1.0 s</td><td>Plunging</td><td>45 min</td></tr></tbody></table>

Comparison of methods for wave propagation problem on a subset of 15 min of the JLN test case for passing from wave gauge 5 from wave gauge 3, trained on 30 min of data.

<table><thead><tr><th>Model</th><th>Param.</th><th>Training</th><th>Pred. time</th><th>RMS error</th></tr></thead><tbody><tr><td>LSTM</td><td>3.4k</td><td>130 s</td><td>0.18 s</td><td>2.9 mm</td></tr><tr><td>N-BEATS</td><td>6.5M</td><td>620 s</td><td>1.97 s</td><td>2.8 mm</td></tr><tr><td>TIDE</td><td>161k</td><td>86 s</td><td>0.18 s</td><td>2.4 mm</td></tr></tbody></table>

due to capturing the slower, higher-frequency and shorter waves. As a result, we find indeed that the linear wave theory limits are reasonable.

Considering the time-series produced with the TIDE model, a good agreement is seen for the JLN test case (Fig. 5a). A reasonable point once non-breaking irregular wave prediction is established is to verify that the same thing can be done for breaking waves, and indeed for the plunging wave case, JLP, we again find a good representation (Fig. 5b). This is notable if for no other reason that unlike a potential flow model that requires additional breaking wave model to be added, like Mohanlal et al. (2023), here the only input is the data, and the dynamics of the system can be computed as a result.

Next, in order to avoid problems with the fitting, it is worth recalling that input time-series are divided into training and test sections, and we may need to estimate the amount of training data that is required. In order to do this, taking the same JLN test case, between WG3 and WG5, and with the selected values of horizon and look-back, we can vary the amount of training data (Fig. 6). Interestingly, not many waves seem to be required to provide reasonable accuracy, but naturally more training data reduces the overall error. Indeed, for approximately 161k trainable parameters, even with 30 min of data, and not considering experimental noise, the dynamical system still remains underdetermined.

## 3.2. Inverse problem

Once the forward modeling is treated, one may consider if the inverse is possible; that is, to compute the input wave as a result of downstream wave gauge measurements. In the non-breaking cases, we again see good agreement (Fig. 7a). In some sense the predictability of the problem for non-breaking waves is unsurprising, since the equations are symmetric with respect to time.

For breaking models, one may expect that this reversibility is broken, making reference to the Saint-Venant or shallow water equations, where a hydraulic jump will cause an information loss. Indeed, for plunging waves, we see that the performance is worse (Fig. 7b), with numerous peaks not reproduced correctly. It would require more experimental data to know if the underestimated peaks are those which were breaking, but the possibility of a wave-by-wave breaking determination is intriguing.

$$c_p = \sqrt{\frac{g}{k} \tanh kh} \qquad (4)$$

$$c_g = \frac{1}{2} c_p \left( 1 + kh \frac{1 - \tanh^2 kh}{\tanh kh} \right) \qquad (5)$$

and then the horizon and look-back can be computed by integrating the inverse of the group velocity:

$$H \Delta t \approx \int \frac{1}{c_{g,max}} dx \qquad (6)$$

$$L \Delta t \approx \int \frac{1}{c_{g,min}} dx \qquad (7)$$

where suitable ranges of frequency is chosen.

As a first estimate, we use limits from $\omega_p/4 < \omega < 4\omega_p$, for peak angular frequency $\omega_p = 2\pi/T_p$, and then applying the above equations with the distance between WG3 and WG5 and the depth over the bar, corresponds to values of $H = 20$ and $L = 34$. Once the values of $L$ and $H$ are established, one can compare the results between each of the ML approaches considered, and we find that all of the methods do a reasonable job of predicting the time-series (Table 2). Indeed, though some exceptions may be found, generally the ML solution is similar between all of the approaches. As a result, the defining feature is the computational performance, and we see that although LSTM provides the smallest number of parameters, it is neither the fastest or the most accurate. N-BEATS allows for more parameters, but similarly, is not as efficient or accurate as TIDE, so this model will be retained.

To verify this initial estimate, we can vary the values of $L$ and $H$ and see the variation in error (Fig. 4). We see that although simulations with $H = 1$ are possible, advancing one timestep at a time, this is not the most accurate nor the fastest for predictions. If $H$ exceeds some value, however, indicating a prediction farther in the future than is predictable due to the group speed of new waves, the error increases again. Similarly, increasing the value of $L$ generally improves accuracy,
<figure><img src="image_8.png" /><figcaption>Fig. 5. Forward prediction of JLN (left; RMS error of 2.4 mm) and JLP (right; RMS error of 7.1 mm) test cases, comparing experimental results and TiDE results, using wave gauge 3 as the input.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Comparison of training time versus accuracy for JLN test case, predicting wave gauge 5 from wave gauge 3.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. 7.** Inverse prediction of JLN (left; RMS error of 2.4 mm) and JLP (right; RMS error of 8.5 mm) test cases, comparing experimental results and TiDE results, using wave gauge 5 as the input.</figcaption></figure>

### 3.3. Generalizability

Finally, let us consider what happens when waves are trained on very different datasets. For some cases, it may be necessary to estimate the behavior for nonlinear irregular waves but only have data for sinusoidal cases. Many numerical models allow for straightforward computation of various regular wave cases, for example, but would be computationally expensive in terms of running irregular waves. Testing this, using all six regular wave test cases as training data, we see that the performance is severely inadequate (Fig. 8). This may be due to the fact that only two periods for the regular waves were used, but it seems likely that even with a large number of different wave periods that the results might be poor.

Next, we consider the sinusoidal test case SLN, but train it on the JLN test case. This may correspond for example to an experimental setup where irregular waves may be more easy to generate than pure regular waves, and the behavior of the system to regular waves is desired. We see that indeed the results are quite reasonable, even if trained starting from WG1 to all of the others (Fig. 9).

Notably, while this is admittedly a black-box interpretation of the dynamical system, this does model the nonlinear dispersive problem to a reasonable approximation far faster than real-time. This ratio is different for each wave gauge and test case, but may be as high as a prediction 5000 times faster than real-time (Table 2). Though it is difficult to fairly compare with quickly evolving computer systems, this is substantially faster than a numerical model of the system in the traditional sense (Engsig-Karup et al., 2012; Ferrand and Harris, 2021; Mohanlal et al., 2023).

### 4. Wave prediction over flat bottom

Thus far, results have been shown for propagation over a small number of wavelengths, which next we extend to larger distances. Using the data from Eeltink et al. (2022), irregular unidirectional waves in a 30 m long tank, with 0.7 m deep water, generated by a piston wavemaker, were considered. Wave gauge time-series at 24 different positions are available, over a total distance of 19.1 m, and the sampling frequency was 32 Hz.

The test case of a peak period of 0.80 s and significant wave height 54 mm, with significant wave breaking reported along the wave tank, was considered. Under these conditions, the linear wavelength of the peak period waves is close to 1 m.

Plots of time-series (Fig. 10) show that the ability of the TiDE model to predict wave propagation at 10 wavelengths is qualitatively reduced compared to near-field predictions. That said, this is a large distance, with a relatively short training period (approximately 13 min), and a large number of trainable parameters (3.5M), such a behavior may be expected. Further, given the large look-back, in terms of 2045 time-steps for the probe at 9.98 m, it may be possible in the future to develop a ML architecture that scales better with distance.

### 5. WSI model

Next we are interested in representing the forces on a vertical cylinder as some unknown function of a finite set of discrete wave elevation observations, where now in Eq. (1), the known covariate is again the wave elevation, $x = \eta/h_0$, but the output can be taken as the combination of $y = [F_x, M_y]/(\rho g D^3)$, for unknown horizontal force $F_x$ and overturning moment $M_y$.
<figure><img src="image_11.png" /><figcaption>**Fig. 8.** Prediction of (or failure to predict) JLN test case, comparing experimental results and TiDE results, using wave gauge 3 as the input, but trained on the six regular wave test cases.</figcaption></figure>

<table><caption>**Table 3**
Test conditions considered by Chaplin and Rainey (2012).</caption><thead><tr><th>Storm</th><th>D (mm)</th><th>H<sub>s</sub> (mm)</th><th>T<sub>p</sub> (s)</th></tr></thead><tbody><tr><td>1</td><td>25.0</td><td>68.3</td><td>0.845</td></tr><tr><td>2</td><td>25.0</td><td>75.5</td><td>0.930</td></tr><tr><td>3</td><td>25.0</td><td>85.9</td><td>1.014</td></tr><tr><td>4</td><td>20.8</td><td>75.3</td><td>1.026</td></tr><tr><td>5</td><td>20.8</td><td>80.4</td><td>1.080</td></tr><tr><td>6</td><td>16.7</td><td>66.1</td><td>1.028</td></tr><tr><td>7</td><td>16.7</td><td>68.0</td><td>1.097</td></tr><tr><td>8</td><td>16.7</td><td>66.0</td><td>1.159</td></tr><tr><td>9</td><td>16.7</td><td>63.7</td><td>1.215</td></tr><tr><td>10</td><td>16.7</td><td>54.5</td><td>1.242</td></tr></tbody></table>

For testing, long-duration experiments performed by Chaplin and Rainey (2012) were performed in a wave flume of Southampton University, 18 m long, 0.42 m wide, and 0.7 m deep. Over 200,000 waves were observed over 10 different wave conditions as described in Table 3, using a JONSWAP spectrum with a peak enhancement factor of 3.3. Here we focus on storm 3, due to the number of large wave force peaks observed in the 6 hour time-series. Wave gauges were positioned 1706 mm, 1262 mm, 765 mm, and 0 mm upstream of the cylinder. In this work, we consider only the wave gauge alongside the cylinder. As a result of the distance, there may be an influence of the (25 mm diameter) cylinder on the recorded wave, though this should not affect the overall connection of the wave-induced forces and the wave elevation. Forces were measured using two sensors, at the top and bottom of the cylinders. Time-series were recorded at 200 Hz, after filtering at 100 Hz.

For non-breaking waves, engineering methods are well established when the problem is not too steep; DNV (2010) proposes that irregular wave kinematics can be predicted from several different methods,

including that of Grue et al. (2003), that of Wheeler (1970), or that of a second-order kinematic model. Many other approaches also exist, such as a local Fourier approximation (Sobey, 1992). In order to compare an ML solution with a traditional approach, here we consider a Wheeler stretching approach, following a simple but established engineering practice, and then the TiDE ML approach.

## 5.1. OpenFAST prediction

For engineering purposes, particularly because diffraction is unlikely to be a major contributor here, strip-theory can be used, integrating the forces over the length of the cylinder as a function of the velocity and acceleration. Multiple methods exist to get the kinematics of the waves from the free-surface elevation, including Wheeler (1970) stretching, a local Fourier approximation Sobey (1992), and more.

In order to benchmark state-of-the-art result for this test case, OpenFAST with the latest hydrodynamic developments (in the module HydroDyn) are tested (Wang et al., 2022), using Wheeler stretching. This approach, due to the simplicity, is widely used, essentially based on Morison's equation, with simplified wave kinematics. Notably this does not include any parameterization for breaking wave impacts, which were reported to be significant in the experiments by Chaplin and Rainey (2012). Given that the waves are in deep water (Fig. 11), we can write the equations as essentially:

$$F_{\text{Wheeler}} \approx \rho D \int_{-h}^{0} \left( \frac{1}{2} C_D u |u| + \frac{\pi}{4} C_M D \frac{du}{dt} \right) dz \quad (8)$$

$$\mathcal{F}(u) = |\omega| \exp kz\mathcal{F}(\eta) \quad (9)$$

$$\mathcal{F}\left(\frac{du}{dt}\right) = i\omega|\omega|\exp kz\mathcal{F}(\eta) \quad (10)$$

where $\mathcal{F}(\cdot)$ is the Fourier transform, and where constants $C_D = 1.2$ and $C_M = C_p + C_A = 2.0$ are chosen to match Chaplin and Rainey (2012). Notably, one important parameter is the frequency cutoff, $f_c$; following DNV (2010) the cutoff used here is $f_c = 4f_p$, or four times the peak frequency. This comes from the fact that high-frequency components vary rapidly close to the free-surface and may add numerical errors but physically do not contribute much to the applied force.

For the purposes of designing a proper ML solution, it is worth noting that the Wheeler stretching result, in the case of deep water, can be rewritten as:

$$F_{\text{Wheeler,deep}} \approx -\frac{\pi}{4} C_M D^2 \rho g \mathcal{H}(\eta) + \frac{1}{2} C_D \rho g D \eta |\eta| \quad (11)$$

where $\mathcal{H}(\eta) = \frac{1}{\pi} p.v. \int_{-\infty}^{\infty} \eta(\tau)/(t - \tau)d\tau$ is the Hilbert transform of the free-surface elevation. Semi-empirical relations connecting force to elevation depending on the Hilbert transform are not new; Tang et al. (2024) and Taylor et al. (2024) both also use this. Although these other works focus on different physical regimes, we see that the force at one point in time depends on the wave elevation over some finite interval, both before and after in time. Physically, this relates to waves around the cylinder, or both upstream and downstream in a small area.

## 5.2. ML predictions

For the WSI problem, we again consider a TiDE solution, though as mentioned just above, since there is not a distance between the wave gauge and the cylinder, the estimation of $H$ and $L$ are not the same as in Fig. 1 or Eq. (1). Instead, following the above remark that the inputs should be roughly symmetric in time about the point of interest, we need to include a shift of the wave elevation record, such that:

$$[y_i: i+H] \approx f([x_{i+S-L: i+S-1}], [y_{i-L: i-1}]) \quad (12)$$

with a shift $S$. For the results presented here, $H = 40$, $L = 600$, and $S = 300$ (which corresponds to an input window of 600 timesteps, or
<figure><img src="image_12.png" /><figcaption>Fig. 9. Prediction of SLN test case, comparing experimental results and TiDE results, using wave gauge 1 as the input, but trained on the JLN test case.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 10. Comparison between predicted and experimentally measured wave time-series from Eeltink et al. (2022), at three different distances, up to nearly 10 wavelengths, to show qualitatively how accuracy is lost with distance.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** Physical regimes of WSI problem from Chaplin and Rainey (2012). (a) Steepness and relative depth of the 10 different storms considered, plotted along with the breaking criterion of Miche (1944). Though this criterion is designed for periodic waves and not irregular waves, we see that the tests are for strongly nonlinear waves in deep water. (b) Diameter relative to wavelength and depth to distinguish physical regimes for flow around a cylinder in deep water, following Chakrabarti (1987), showing that inertia and drag are expected to be dominant terms.</figcaption></figure>

<table><caption>**Table 4**
Comparison of methods for WSI problem for simulating 5 h of storm 3 of Chaplin and Rainey (2012), training on approximately 1 h.</caption><thead><tr><td>Model</td><td>Param.</td><td>Training</td><td>Pred. time</td><td>RMS error</td></tr></thead><tbody><tr><td>OpenFAST</td><td>2</td><td>N/A</td><td>410 s</td><td>0.042 N</td></tr><tr><td>TiDE</td><td>1.1M</td><td>1286 s</td><td>36.2 s</td><td>0.036 N</td></tr></tbody></table>

3.0 s). Preliminary tests showed that results were relatively insensitive to the exact values.

Like all comparisons of computational performance, it is difficult to fairly compare the ML solution with OpenFAST, due to OpenFAST running on a CPU and the DARTS library using a GPU. Nevertheless, Table 4 show that TiDE provides a reasonable result, if just comparing the prediction time and RMS error between the two. Note that neglecting the training time likely is fair, given that the time to precompute the wave kinematics in the SeaState module of OpenFAST is neglected here. Both are significantly faster than real-time, being useful for engineering applications.

Comparing time-series alone can be difficult, as noted by Tang et al. (2024), as for particular waves Wheeler stretching may provide an accurate solution, or for example, may under-predict or over-predict forces. As a result, we present one example in Fig. 12, showing that in general all non-breaking wave force time-series agree, but given that OpenFAST does not include a breaking model, it underpredicts some peaks significantly. Interestingly, the ML solution was unable to capture the mechanical oscillations of the cylinder, but this resonance between 20 Hz and 30 Hz was not well resolved in time in this dataset. In any case, next we consider the statistics of these irregular wave forces.

### 5.3. Force peaks

One of the things most important to wave impacts on a cylinder are the maximum forces, which is the maximum or minimum force (or moment) for a particular wave cycle. Following Suja-Thauvin et al. (2020), here the maximum and minimum of the force is taken between two consecutive zero down-crossings of the wave elevation at the wave gauge next to the cylinder. From this, the peaks both measured and predicted can be compared (Fig. 13), showing that the ML approach outperforms OpenFAST, though at times can overpredict the load as well, which the OpenFAST solution only rarely does.

The fact that some peaks are still under-predicted by the ML approach may be related to the fact that the wave elevation was taken

from the gauge next to the cylinder, instead of upstream. This would make it difficult for the model to distinguish between a non-breaking and an already-breaking wave. Indeed, as noted by Battelle Martin et al. (2023), breaking and the distance from breaking is an important part of wave loading.

Finally, looking at the probability of exceedance (Fig. 14) based on the force peaks, we see that the observed behavior, as expected, has some large, but rare, impacts, whereas the OpenFAST results does not have exhibit the same behavior due to the missing breaking wave impact model. The TiDE result, due to being trained on a subset of the observed data, although does not perfectly match the extreme wave impact statistics, shows a better match.

Note that this behavior around rare, extreme events could be improved in future work with either more training data (including at higher sampling frequencies than are available in this dataset), or a modification of the loss function. Here the mean-squared error is used, which produces the best average behavior, but a modification could be used to prioritize matching extreme events, or separating the load model into a non-breaking, Wheeler stretching approach, and a breaking, ML approach.

## 6. Field-scale prediction

Finally, it may be useful to apply the method not only in a laboratory setting, but also to field-scale cases. This requires measurements at two (or more) locations nearby, ideally several wavelengths. The Coastal Data Information Program (CDIP) buoys (Behrens et al., 2024) off Hawaii can be used, as the Datawell Waverider buoys at the station 225 (at 21°28.644'N, 157°45.410'W, in 84 m depth) and at the station 198 (at 21°28.656'N, 157°45.114'W, in 81 m depth) are approximately 512 m apart, and the local wave field is often visible through Sentinel-2 (Drusch et al., 2012) satellite imagery (Fig. 15).

In the field, there are a few additional complexities to consider, including added physics (e.g., wind forcing, currents), multi-directional waves, and missing data. For multi-directional waves, a particular advantage of the CDIP data is that Datawell Waverider buoys record motion over three axes (x-, y-, and z-displacement), so at least for long-crested conditions, if all three displacements are used as inputs, it may be possible to predict, e.g., the vertical displacement at another station.

Using roughly 11.5 h of contiguous data, from 23 January 2023, with measurements every 0.781 25 s, training on the first half of the time-series, and comparing predictions over the second half, moderately good agreement was found (Fig. 16). Through trial and error (given
<figure><img src="image_15.png" /><figcaption>Fig. 12. Comparison of time-series of horizontal force from storm 3 of Chaplin and Rainey (2012) between measured, OpenFAST, and TiDE results.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. Comparison of peaks in horizontal force from storm 3 of Chaplin and Rainey (2012) between measured, OpenFAST, and TiDE results. Black lines indicate an over- or under-prediction of the peak by 50%.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 14. Comparison of probability of exceedance from storm 3 of Chaplin and Rainey (2012) between measured, OpenFAST, and TiDE results.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Fig. 15.** False color image at location of CDIP Waverider buoys at stations 225 and 198, off Hawaii, from 23 January 2023.</figcaption></figure>

*Source:* Contains modified Copernicus Sentinel data (2023).

<figure><img src="image_19.png" /><figcaption>**Fig. 16.** Test of TiDE model on field-scale measurements, using the CDIP 225 x-, y-, and z-displacements as input, and CDIP 198 z-displacement as the output, on 23 January 2023.</figcaption></figure>

that the problem is not one-dimensional), a look-back of 340 points (266 s), and a horizon of 50 points (39 s), was chosen. Although the results are considerably less precise than the laboratory results in the earlier sections, this does show the possibilities and limitations of the final model.

## 7. Conclusions

Given all of the ML solutions proposed in recent years, it can be valuable to understand the possible performance on well-known benchmarks. Equally, given increasingly complex second-order, third-order, and higher-order solutions (with a large number of degrees of freedom) proposed for engineering solutions, it can be useful to test what models could be feasible, presently in a more black-box solution,

but could later be in a more interpretable model (with a smaller number of trainable parameters). From this perspective, the question posed by ML approaches is one of knowing what information is required as input, and what output is expected, for the dynamical system.

In this paper, a proof-of-concept is proposed for real-time wave and wave-structure interaction model that includes nonlinear, dispersive, and breaking effects. This model depends on an upstream wave condition, and is for a long-crested unidirectional wave. Notably while providing results similar to more complex numerical models that require specialized knowledge, the present setup permits moderate accuracy results with only knowledge of the linear group speed of the waves in the area of interest.

Results agree with classical results, predicting wave gauge results based on time-series of upstream wave gauges, or alternatively predicting forces on a vertical cylinder from wave gauge time-series. Although less interpretable in its present form than a Navier-Stokes model output, the model provides results faster than real-time after training, and if training data is available, may be adapted to any problem geometry. In testing multiple ML approaches, the TiDE approach appears to provide a reasonable balance of accuracy versus computational time when rapid calculations are required.

While there are some limitations to the present results, it is worth noting that there are a number of applications, including for computer graphics visualization, rapid prototyping, or real-time control, where exact wave prediction may not be required, but where simple linear relations may not be sufficient. As well, the models here were trained with a standard root mean square error, but tested with metrics based on peak forces, while better performance may be achieved by adjusting the loss function. While interesting results can be shown here, further work will be required to provide a given accuracy to any particular test case.

For generalizing this ML approach in the future, it is possible that one needs to place the upstream wave gauge sufficiently upstream that it is before any local wave breaking begins. For multi-directional, short-crested waves, inclusion of multiple wave gauges will be necessary, planned for future work, and it is recognized that structural responses are important for design, and this method could easily be extended to flexible structures if suitable data is available.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgments

The author would like to thank Sendar Beji for making available data of the submerged bar test case results, and John Chaplin for providing a copy of the wave load measurements on a vertical cylinder.

## Data availability

The code, MEWS (Minimum Effort Wave Simulation), is available at http://github.com/jcharris/mews. Datasets used for training will be made available upon reasonable request.

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