## 2 Materials and Methods
### General Framework for the Automatic Explanation of the Obtained Results by the (CAE) ANN  
It is important to note that the predicted values of conventional ANN models (and the CAE ANN model as well) for the parameter under consideration (i.e., yield strength of AA6082) are typically mean values. Conventional ANN models generally do not calculate quantities other than the average value for the output parameters. With minimum supplementation, the other ANN models could calculate the local standard deviation and local data density of the results. Consequently, using the proposed approach, the fundamental framework for interpreting/explaining the obtained results can be applied to a wide range of conventional ANN models. As part of the traditional scientific approach, which is founded on an impartial mathematical explanation, the observed phenomenon is divided into the smallest feasible components, for which the fundamental equations are recorded. Such an approach typically yields differential equations or systems of (partial) differential equations, which are subsequently solved by the application of well-established mathematical and/or numerical techniques. Using fundamental rules (like [PERSON]'s laws of mechanics) or previously established mathematical concepts (like [PERSON]'s equations, [PERSON]'s principle, [PERSON]'s stability criterion, etc.) can frequently assist us, too. The primary aim is to (uniquely) mathematically characterize the entire considered phenomenon by using the fewest significant parameters and their derivatives (first or higher). Specifically, by examining the found solutions, we can comprehend and interpret the observed phenomenon if we can mathematically (i.e., objectively) characterize it. It should be noted that a graphical presentation of the (main) results of the described scientific process reveals the most important outcomes of the observed phenomena, like the mass spectrogram in materials, environmental science, and medicine (e.g., [36]), the response spectrum in structural analysis (e.g., [37]), etc. Thus, in general, the explanation involves more than merely summarizing/describing the outcomes; rather, it involves extracting the most crucial details from the data. The authors are well aware that the proposed framework is only a trial aimed at attempting to derive new principles from the observed material phenomena using AI. To explain the results, we observe one predicted output parameter (i.e., \(k\)-th output parameter, \(\hat{y}_{k}\)) at a time, with two changing input parameters \(x_{i}\) and \(x_{j}\) (\(i\
eq j\)). All other input parameters are either ignored or fixed at the selected value. We can also display such data graphically, allowing us to combine the graphs together with the processed explanation. There are numerous approaches to obtaining an explanation:  
**First approach:** We calculated approximations of the first derivatives (gradients of functions) for various quantities (mean value, local standard error, local data density) near the selected value of the observed output parameter. These approximations revealed the relative influence of individual input parameters. The crucial point here is that estimating the local standard error provides an estimate of variability (the size of epistemic and aleatory uncertainties). The assessment of the local data density provides an estimate of the prediction's reliability: with a higher data density, as indicated by a higher value and darkerareas in the graphic displays, forecasts are generally more reliable, and with a lower data density, less reliable! Gradients in the directions of input parameters indicate their influence, with larger values indicating greater influence and vice versa. Keeping in mind that all genuine phenomena are probabilistic rather than deterministic, empirical models [19] bring us closer to a true representation of the observed phenomenon. Remember that the CAE model is nearing its limitations if the prediction/simulation results are accompanied by small \(\hat{\rho}\) vales!