## 7 Use Case 1: CNN for Damage Location Regression and Classification
Multiple trained models should be analyzed with respect to the final discretization error, selecting the best model. 5. The prediction error of the discretized model (RMSE and \(E_{\text{max}}\)) differs only slightly. 6. There is no increase in the classification error compared with the continuous model, showing an overall stable prediction behavior. To conclude, the discretization, even with a moderate static scaling, does not degrade the prediction accuracy. Figure 14: Down-sampled GUW signal from simulation and low-pass-filtered rectified (envelope approximation) signal as input for the CNN (damage at position x = 100, \(y=100\)). The MLISA REXA VM Forth program of the discretized model is shown in Appendix A in Algorithm A1. The model code occupies 1746 dynamic and 2166 static words in the CS (i.e., occupying about 8k Bytes of RAM). The entire textural code size is 18,452 Bytes. The forward computation requires the execution of 1280 words, which is equivalent to 85 ms/MHz (assuming 15k/words/s/MHz [11]). ## 8 Use Case 2: ANN Polynomial Models

Based on the previous use case evaluation indicating non-linear functions as the primary source for approximation errors, we want to force high non-linearity in an ML surrogate model for a polynomial of degree \(n\):

\[F(x)=k_{0}+k_{1}x+k_{2}x^{2}+k_{3}x^{3}+\ldots+k_{n}x^{n} \tag{8}\]

The ANN model architecture for the implementation of such a surrogate model is shown in Definition 4. It consists of only 17 neurons. A polynomial of degree 4 was chosen with the following parameters:

\[ \begin{array}{l}k_{0}=100\\ k_{1}=0.5\\ k_{2}=1\\ k_{3}=-0.5\\ k_{4}=-0.1\\ x=[0,10]\end{array}  \tag{9}\]

The model was trained using the polynomial model (500 epochs); 100 independent models were trained using the same training data with 1000 randomly selected function samples. After model training, the Foo transformation process and a model analysis were performed. **Definition 4**. : _Architecture and parameters of the ANN model (ConvNetJS) as a surrogate regression model for highly non-linear analytical model functions._

Figure 15: Foo/FooFP model analysis of the GUW regression CNN model. The classification error was always zero. Artificial Neural Network ================================s  Classes: undefined  Input: [1,1,1]  Output: [1]  Layers: [L5 P9]  [1] input : out=[1]  [2] fc : in=[1] out=[4]  [3] tanh : out=[4]  [4] fc : in=[4] out=[8]  [5] tanh : out=[8]  [6] fc : in=[8] out=[4]  [7] tanh : out=[4]  [8] fc : in=[4] out=[1]  [9] regression : in=[1] out=[1]  Predictors: l ```
The following discretization parameters for the activation function were chosen, as shown in Table 6.

The statistical analysis of the prediction results of the continuous Foo and the discretized FooFP model is shown in Figure 16. The summary of the results is as follows:

1. The prediction errors of the discretized model (RMSE and \(E_{\text{max}}\)) are significantly higher compared to the continuous model. 2. The discretization error results from the non-linear _tanh_ function, which is clearly highlighted if a scaled float-point alternative is used in the discretized model, with errors similar to the continuous model. 3. Choosing the _tanh_ and underlying _log10_ discretization parameters leads to the prediction error. Modification of the LUT partitioning and interval coefficients can improve the _RMSE_ as well as the maximum error \(E_{\text{max}}\). 4. Dynamic scaling shows no significant improvement in the model accuracy (maximal 5%). 5. As shown at the bottom of Figure 16, the discretization error is not constant; instead, it introduces discontinuity. The MLISA REXA VM Forth program of this model is shown in Appendix A in Algorithm A2. The model code occupies 18 dynamic and 89 static words in the CS. The entire textural code size is 921 Bytes. The forward computation requires the execution of 70 words, which is equivalent to 5 ms/MHz.