On the Use of Artiﬁcial Intelligence to Deﬁne Tank Transfer Functions

: Experimental test facilities are generally characterised using linear transfer functions 1 to relate the wavemaker forcing amplitude to wave elevation at a probe located in the wavetank. 2 Second and third order correction methods are becoming available but are limited to certain 3 ranges of waves in their applicability. Artiﬁcial intelligence has been shown to be a suitable 4 tool to ﬁnd even highly nonlinear functional relationships. This paper reports on a numerical 5 wavetank implemented using the OpenFOAM software package which is characterised using 6 artiﬁcial intelligence. The aim of the research is to train neural networks to represent non-linear 7 transfer functions mapping a desired surface-elevation time-trace at a probe to the wavemaker 8 input required to create it. These ﬁrst results already demonstrate the viability of the approach 9 and the suitability of a single setup to ﬁnd solutions over a wide range of sea states and wave 10 characteristics. 11


INTRODUCTION
Waves are of particular interest in marine research as they play a key role in un- 15 derstanding fluid-structure interactions. Wave tank test facilities are used to generate 16 desired water waves in order to model certain sea-states in a controlled environment. 17 Both physical and simulated wavetanks exist, each with their own merits. However, 18 in all cases wavemakers are required and some method to control wavemaker action 19 to obtain a desired sea state or time trace of surface elevation in the tank. Early work 20 employed analytical solutions relating piston or paddle motion amplitude to surface 21 elevation [1]. Industry standard is now the characterisation of a wavetank using tank 22 transfer functions (TTF) [2], describing the ratio of surface elevation to wavemaker 23 amplitude and phase for each frequency component. Even for a flat bathymetry, non-24 linear extensions are often required for anything beyond very small amplitude waves to 25 take into account bound waves and wave-wave interaction. Significant progress was 26 achieved by extension to second order [3], but despite further progress these analytical 27 solutions are still limited in their range of applicability and no single method encompass-28 ing all features relevant in applied ocean and coastal research exists [4][5][6]. Recently, large 29 collaborative projects in offshore wind [7] and wave power [8] independently identified    the short "memory" or "vanishing gradient", which has been investigated by [11]. The 60 influence of an item in sequence weakens as the sequences goes on. This is problematic 61 for learning long sequences, as long patterns and sparse temporal dependencies may 62 not be recognised.

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Neural networks methods have been used by other researchers in the field. [12] 64 uses Recurrent Neural Networks (RNNs) to predict statistical properties of water wave 65 amplitude time traces as a function of the wavemaker parameters (blower RPM and 66 frequency). While they only predicted statistical properties of the wave produced and 67 not a specific time-series, this still shows the promise of RNNs in predicting water waves.

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A novel neural network modelling approach has been proposed by [13]. This proposal 69 presents the idea of Physics Informed Neural Networks to solve differential equations, 70 giving an example of wave propagation. In this method, the laws of physics are invoked 71 to constrain calculated values to a suitable range. This method may be suitable for 72 application to the Navier-Stokes equations.

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The layout of this paper is as follows. In the next section we explain the method- However, we have a vector v which is a set of n observations (h i , t i ) i = 1, n at the probe to which we can associate a vector U of n wavemaker inputs from a previous time interval. We know that these inputs ad outputs are related by equation (1) The universal approximation theorem for neural networks asserts that we can use 87 the input and output vectors u, v to find an approximation for the unknown function  order to generate sufficient data with which to train the neural network models. Figure   101 2 presents the workflow for this step.

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Random wavemaker input traces were generated by summing up individual contributions for each frequency f using the formula, A Tukey filter was applied to the resulting time trace to ensure wavemaker input began 103 with zero amplitude, avoiding shock-waves in the numerical wavetank.

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This process was repeated multiple times to define a set of samples. Each sample, 105 u i was then used as input to the OpenFOAM program for the wavetank configuration  forward neural network is capable of approximating any well-behaved function [17]. 126 We thus aim to replace classical TTF with trained NN, which, once trained, should yield 127 any required surface elevation trace without further iterative calibration steps. . It is customary to use k-fold cross validation in order to remove bias in the training 130 data. In our work we split the data samples on an 8 : 2 ratio of training to test samples.

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The members of each set are chosen randomly. This process is repeated multiple times 132 and the model giving the best performance is chosen.

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A further check on the accuracy of our method is illustrated in figure 3. Once 134 a neural network model has been chosen, we can take the wavemaker input that is 135 predicted as required to generate a given time profile at the probe and feed this as input 136 to the numerical wavemaker. The computed time profile at the probe can be compared 137 to that which was fed to the neural network to produce the wavemaker input.

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We prepared 1000 samples of wavemaker input, each of 30s duration, using equation  of the input to the wavemaker and subsequent computed surface elevation of the probe.

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The flat line to the left of the surface elevation plot represents the time delay between  Table 2 shows the set of hyper parameters that we identified after an extensive  Following the validation process discussed in the previous section and presented in  The values for each wave are non-dimensionalised with gravitational acceleration g 180 and wave period τ as pairs of H gτ 2 and h gτ 2 and plotted as a point cloud over the well 181 known plot highlighting the validity of different wave theories [19], Fig 6. The majority 182 of waves is highly non-linear and can best be described with 5 th order stream function 183 theory, reaching into 2 nd and 3 rd order stokes theory. Some few waves seem to exceed 184 the breaking limit which requires further investigation but might well be due to the 185 highly irregular nature of the timetrace.

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The key feature to note here is, that the waves are highly nonlinear and, while 187 direct comaprison is still outstanding, could be expected to pose a formidable challenge 188 to conventional wave calibration techniques. Figure 7 presents two example results 189 for surface elevation using the predicted wavemaker input besides their target data.

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These results were achieved using the same settings found to be optimal for the small can be observed at the end of the trace, but overall the agreement is still remarkable. Figure 6. Wave parameters encountered in the larger wave cases represented as dots over the validity ranges of several theories for periodic water waves, according to [19]. Original figure adapted from Wikimedia Commons, the free media repository