## 3 Theory and Calculation
### Proposed Solution by CAE ANN  
CAE ANN is a special type of multi-dimensional, non-parametric regression and represents a kind of probabilistic neural network [28]. Developed by [PERSON] and [PERSON] [29], it enables relatively simple empirical modelling of different phenomena. The description in this paper enables a simple and efficient application of CAE ANN in real bridge design. For the estimation of vector \(\hat{\mathbf{R}}\) in Equation (4), the conditional probability density function was applied [29]. The estimator of the optimal solution of the problem can be expressed as:  
\[\hat{r}_{k}=\sum_{i=1}^{N}A_{m}\cdot r_{nk}, \tag{5}\]  
where  
\[A_{m}=\frac{a_{m}}{\sum_{i=1}^{N}a_{i}}. \tag{6}\]  
In Equations (5) and (6), the Gaussian density function with width \(w\) is chosen as a weight function. It is centered at each \(m\)-th model vector in order to determine the influence of the \(m\)-th model vector at the point of the prediction vector as:  
\[a_{m}=\frac{1}{(2\pi)^{(D/2)}w_{m}^{D}}\exp\left[-\sum_{l=1}^{D}\frac{(q_{l}- q_{ml})^{2}}{2w_{m}^{2}}\right], \tag{7}\]  
where \(D\) represents the number of input parameters taken into account. The quality (generalization) and smoothness (ease of explanation) of the solution depend on the parameter \(w\) (smoothing parameter), which is not easy to determine. In the presented study, the problem of determining \(w\) was converted into determining the minimum of a function of a one-dimensional problem [30]. A representative and random database is the fundamental presumption in the usage of CAE ANN, which is rarely met in practice. Consequently, in CAE ANN solutions, information on the relative density of the data, \(\hat{\rho}\), is provided as additional information, which can be used to provide information on the reliability (accuracy) of the prediction. This information is calculated as follows [30]:  
\[\hat{\rho}=\frac{1}{N}\sum_{i=1}^{N}a_{m}. \tag{8}\]  
In CAE ANN Equations (2)-(6), normalized values of input parameters are used as input values. The normalization equation for \(i\)-th input parameter \(\overline{q}_{i}\) is expressed as:  
\[\overline{q}_{i}=\frac{q_{i}-q_{i,min}}{q_{i,max}-q_{i,min}}. \tag{9}\]  
When designing bridges built by the incremental launching method, Equation (7) for a simplified design case (i.e., two design input parameters, namely, the ratio between cross-section height and characteristic length, \(h\)/\(L\), and the cross-section width, _width_) can be re-written as:\[a_{n}=\frac{1}{2\pi\,w_{h/L}\,w_{\rm width}}\exp\Biggl{[}-\frac{\left(\left(h/L \right)-\left(h/L\right)_{n}\right)^{2}}{2w_{h/L}^{2}}-\frac{\left(width-width _{n}\right)^{2}}{2w_{width}^{2}}\Biggr{]}. \tag{10}\]  
The input parameters without an index correspond to the newly designed bridge under consideration, while the input parameters with an index correspond to the \(n\)-th bridge from the database. Equation (5) for the design output parameter, i.e., total prestressing force, S, can be re-written as (note that in the case of one output parameter \(k\) is equal to 1):  
\[\hat{S}=\sum_{i=1}^{N}A_{n}\cdot S_{n}. \tag{11}\]  
Determination of Epistemic and Aleatory Uncertainties for Extended Database by LHS Method 3.3.1. On the (Types of) Uncertainties  
Uncertainties in the data add complexity to different engineering problems at all levels, i.e., [31]. Generally, if uncertainties are reduced, the confidence and reliability of the (measured) data and predictions increase. Two main categories of uncertainties [26; 27] must be considered when creating the predictive model (by, i.e., CAE ANN): aleatory uncertainties, also known as inherent variability or randomness, and epistemic uncertainties, also known as variability, brought on by a lack of knowledge. All natural and physical phenomena have inherent uncertainties known as aleatory uncertainties that are unaffected by the observer or the method of observation.