## 4 Discussion  
In this work, we investigated the impact of two methods for the fitness landscape manipulation, namely surf [12] and DFs [6], on the PE problem of stochastic biochemical models. Following the results obtained in [5], we addressed a common limitation of these two methods, that is, the need for expert knowledge regarding the characteristics of the fitness landscape. This was achieved by modifying the surf algorithm to co-evolve the optimal value of the \(\gamma\) hyperparameter and by performing a separate optimization process to evolve an optimal DF for the problem at hand [25]. Our results show that the PE employing the evolved DF has a performance that is comparable with the one employing the analytical DF, further corroborating their application for the PE of biochemical models and in contexts where there is no knowledge about the fitness landscape [25]. The co-evolution of the \(\gamma\) hyperparameter in surf also proved to be an effective strategy by improving the convergence speed and the final results with respect to a PE not employing surf surrogates, notably with one less hyperparameter to be manually set by the user when compared with the original surf algorithm [12]. Figure 11 shows the distribution of the optimal \(\gamma\) values automatically identified by our approach. As can be observed from the plot, the mode of the distribution is \(\gamma=7\) and, overall, the optimization of this parameter led always to values between 5 and 8. This confirms the adequacy of the search space for the parameter \(\gamma\) (i.e., [2, 10]) employed in our tests. However, these improvements were observed only on fitness landscapes that were dilated with the analytical DF exploiting expert knowledge. In our tests, the application of surf prevented the convergence to optimal solutions on landscapes dilated with the evolved DF, suggesting that not all DFs are beneficial to the construction of surrogates by means of surf, even when a high value of \(\gamma\) is employed. As a future extension of this work, we plan to further investigate this issue by evolving DFs that take into account the transformations of the landscape applied by surf. Moreover, we plan to improve the evolution of DFs by identifying different dilations for each dimension of the considered PE problem. Another limitation that affects surf regards the algorithm used to generate the surrogate models, which is characterized by a high time and space complexity. As already discussed in [13], surf relies on DFT, which can calculate the multi-dimensional spectra of the fitness landscape from equispaced samples. That is, surf interpolates the samples over a multi-dimensional regular lattice, an operation characterized by exponential complexity. In practical terms, surf has reasonable time and space complexity up to five or six variables, but the computation time required makes it unsuitable for more than four variables. Thus, we plan to investigate different approaches to computing the Fourier transform using, for example, sparse samples. This will greatly improve the performances of surf, in terms of computation time. We will also explore alternative strategies, not involving the computation of a Fourier transform, to generate surrogate models with a less-than-exponential complexity, thus allowing for the synergistic application of DFs and surf to high-dimensional optimization problems, such as the PE of large RBMs with several missing parameters [30]. **Author Contributions:** Conceptualization, M.S.N., D.M.P. and L.M. ; methodology, M.S.N., D.M.P. and L.M. ; software, M.S.N., D.M.P. and L.M. ; formal analysis, M.S.N., D.M.P., S.S., P.C. and L.M. ; investigation, M.S.N. and D.M.P. ; data curation, M.S.N. ; writing--original draft preparation, M.S.N., D.M.P., S.S., P.C. and L.M. ; writing--review and editing, M.S.N., D.M.P., S.S., P.C. and L.M. ; visualization, M.S.N. and D.M.P. ; supervision, M.S.N. and L.M. ; project administration, L.M. All authors have read and agreed to the published version of the manuscript.