## 7 Use Case 1: CNN for Damage Location Regression and Classification  
The first use case uses the data delivered by the GUW simulation introduced in the previous section. In total, there were 43 different data sub-sets, each related to a specific position of the circular damage in the plate structure, including the baseline measurement without damage. A classical CNN model was chosen to predict the damage positions (_x,y_) and provide binary damage classification. The CNN input was a down-sampled and low-pass filtered GUW signal (128 data points). The outputs are two continuous variables, \(p_{\mathrm{x}}\) and \(p_{\mathrm{y}}\), normalized to the full range of the damage location in the x- and y-direction with a 10% margin, i.e., the minimum location coordinate corresponds to 0.1, and the maximum value corresponds to 0.9. If \(p_{\mathrm{x}}\) < 0.1 and \(p_{\mathrm{y}}\) < 0.1, then no damage was detected (i.e., classification output). The model architecture and its parameters are shown in Definition 3. The CNN was implemented with the ConvNetJS framework [17] and trained with the typical 500 epochs at a default learning rate of \(\alpha\) = 0.01 using the adagrad trainer (batch size was chosen as 1 due to the low sample count). **Definition 3**. : _Architecture and parameters of the CNN model (ConvNetJS) using 1 dim convolution and pooling operations._  
Convolutional Neural Network ================= ================= ================= =================  Input : [128,1,1]  Output : [2]  Layers: [L8 P13]  [1] input : out[128,1,1]  [2] conv : in[128,1,1] out=[124,1,4] k=[5,1] filters=4 stride=1  [3] relu : out=[124,1,4]  [4] pool : in-[124,1,4] out=[62,1,4] k=[2,1] filters=4 stride=2  [5] conv : in-[62,1,4] out=[60,1,4] k=[3,1] filters=4 stride=1  [6] relu : out=[60,1,4]  [7] pool : in-[60,1,4] out=[30,1,4] k=[2,1] filters=4 stride=2  [8] fc : in-[120] out=[16]  [9] tanh : out=[16]  [10] fc : in-[16] out=[8]  [11] tanh : out=[8]  [12] fc : in-[8] out=[2]  [13] regression : in-[2] out=[2]  Predictors: 128
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A typical GUW signal and its low-pass-filtered and down-sampled version are shown in Figure 14. The low-pass filter was a simple exponential filter with a filter coefficient of \(k\) = 0.2. Figure 13: GUW signal simulation using a 2 dim viscoelastic wave propagation model. (**Left**) Simulation set-up. (**Right**) Some example signals with and without damage (blue areas show damage features). To capture training variations, the original model was trained 100 times, each training starting with a randomly initialized model but always with the same training and test data set. Figure 15 shows the comparison of the prediction results of the continuous Foo and discretized and scaled FooFP model for the regression tasks. The prediction delivers the damage position coordinates, and a non-damage detection is given by a (0,0) value pair (or close to). The RMSE and maximal position prediction errors are computed. Results for static and dynamic scaling were compared. The summary of results is as follows:

1. The total value range of all input, output, intermediate, and parameter values depends on the particular training of the original model but is mostly in the range [\(-\)4, 4] (see Figure 15, \(V\) column). Therefore, a static or preset scaling of \(s_{0}\) is in the range [3000, 10,000]. 2. The discretization error of integer arithmetic is neglectable for all linear operations but depends slightly on the discretization parameters of the non-linear functions (FP1/FP2/FP5 in Figure 15 represents LUT resolution with \(1/\Delta X=1/2/5\)). 3. The dynamic scaling, compared with static scaling, shows no significant improvement in the model accuracy (maximal 5%) but is unexpected with a larger variance (see Figure 15). 4. The overall discretization error depends on a particular model parameter set, i.e., with nearly the same floating-point accuracy, the integer model accuracy can differ significantly (RMSE and \(E_{\text{max}}\)).