## 2 Method  
In this work, we present the integration of the optimization framework with the PM of musical instruments (Figure 1). In brief, this method enables the modification of the PM parameters through correction factors. It is an iterative process to solve an optimization problem. In every iteration, the optimizer tries a new set of values for the correction factors (\(CF_{new}\), Figure 1), resulting in new synthesized signals. Next, the signals are evaluated according to the predetermined criteria. The goal is to determine the optimal set of correction factors derived by the solution of an optimization problem. An optimal set of correction factors is the one that when applied on the relative parameters of the PM, will result in the synthesized signal with the highest evaluation score (Output, Figure 1). We would like to note here that in this work we used the known PM techniques, for the simulation of the musical instrument, thus, any modeling challenges (e.g., nonlinearities) concerning the PM were taken from the relevant literature (see Section 3.1). Although nonlinearities concern the PM and not the adopted optimization framework, it is a fact that they affected our approach by imposing the need to use a different set of correction factors for every note produced rather than for a single set. The first step is to define the details of the synthesis part (Input 1, Figure 1). It includes the determination of (a) the modifiable parameters, (b) the parameters' limits, and (c) the specific modification type of every parameter. In this step, the designer should ensure that unwanted alterations of the core elements in the PM algorithm are avoided. Thus, setting the modifiable parameters (determination point a) and their limits (determination point b) is essential to avoid results with no physical meaning, such as placing the position of a tonehole outside the body of a wind instrument. In principle, all the parameters of the algorithm can be potentially considered tunable parameters. However, if parameter A depends on parameter B (i.e., A = f(B)), and B only affects A (i.e., there is not a parameter C for which C = g(B)), then modifying both A and B is unnecessary. Based on the above, we chose the tunable and locked parameters. Type of modification (determination point c) in the proposed method is the mathematical expression which, with the use of a correction factor (\(CF\)), tweaks a parameter \(P\): (e.g., \(P_{modified}=P_{initial}+CF\), \(P_{modified}=P_{initial}\cdot CF\), \(P_{modified}=P_{initial}^{CF}\), etc.). The second step includes the evaluation of the generated signal. The goal in our approach essentially is the following: the synthesized signal, generated by a PM according to building details (Input 2, Figure 1), to be as similar as possible to a reference. In our case, the reference is derived from a recorded signal generated by a musical instrument (Input 3, Figure 1). The proposed model, apart from recordings, works with other reference signals as well. For example, the goal signals could be generated by digital synthesis techniques (e.g., additive) and even signals consisting of raw numbers describing sound parameters (e.g., inharmonicity, deviation, durations, and other acoustic features found, for example, in [10]). In every iteration, the optimizer assigns new values to the correction factors' set (\(CF_{new}\), Figure 1), which modifies correspondingly the Physical Modeling resulting in a new synthesized signal (Figure 1). The synthesized signal is then compared with the recording. An objective function calculates the resemblance of these two signals (in our case, the function is cross-correlation, see Section 3.2). It enables the quantification of the degree of their similarity (\(DS\), Figure 1). The optimizer determines which specific set of correction factors (\(CF_{k}\), Figure 1) resulted when applied to the PM, in the maximum degree of similarity (\(DS_{k}\), Figure 1) between the synthesized and the recorded signal (Figure 1). This particular set is the optimizer's output (Figure 1).