esa-sceva/satcom-chunk-collection huggingface dataset:
# Computational Screening of New Perovskite Materials Using Transfer Learning and Deep Learning
## 3 Results and Discussions
### Screening Results Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Formula** & \(\mathbf{\tau}\) & **Formula** & \(\mathbf{\tau}\) & **Formula** & \(\mathbf{\tau}\) & **Formula** & \(\mathbf{\tau}\) \\ \hline YbTlBr\({}_{3}\) & \(-\)20.162 & CaTlCl\({}_{3}\) & \(-\)14.381 & CaTlI\({}_{3}\) & \(-\)14.0108 & PrBO\({}_{3}\) & \(-\)2.0396 \\ CaTlBr\({}_{3}\) & \(-\)14.3041 & NaHgCl\({}_{3}\) & \(-\)13.5492 & TlSnI\({}_{3}\) & 2.7082 & LiNbO\({}_{3}\) & \(-\)2.0303 \\ CrCoBr\({}_{3}\) & 1.55 & MgTlCl\({}_{3}\) & \(-\)6.3625 & TlFeI\({}_{3}\) & 2.9924 & LiTeO\({}_{3}\) & \(-\)1.945 \\ TlGeBr\({}_{3}\) & 1.6125 & TlNiCl\({}_{3}\) & 1.2386 & CsCrI\({}_{3}\) & 3.2934 & TaBeO\({}_{3}\) & \(-\)1.754 \\ TlFeBr\({}_{3}\) & 2.5339 & TlCoCl\({}_{3}\) & 1.4647 & CsInI\({}_{3}\) & 3.2934 & ThBeO\({}_{3}\) & \(-\)1.7537 \\ CsFeBr\({}_{3}\) & 2.8551 & TlGeCl\({}_{3}\) & 1.4794 & CsMgI\({}_{3}\) & 3.3089 & CeBeO\({}_{3}\) & \(-\)1.7537 \\ CsScBr\({}_{3}\) & 2.8556 & TlClCl\({}_{3}\) & 1.5469 & CsTiI\({}_{3}\) & 3.3142 & CrRhO\({}_{3}\) & \(-\)1.747 \\ CsPdBr\({}_{3}\) & 2.8561 & TlCuCl\({}_{3}\) & 1.9923 & CsSnI\({}_{3}\) & 3.3312 & PuPtO\({}_{3}\) & \(-\)1.6802 \\ CsPtBr\({}_{3}\) & 2.8713 & CsPdCl3 & 2.735 & CsGeI\({}_{3}\) & 3.404 & ZrBO\({}_{3}\) & \(-\)1.6005 \\ CsInBr\({}_{3}\) & 2.8777 & CsSnCl\({}_{3}\) & 2.7368 & RbTiI\({}_{3}\) & 3.4157 & PrBeO\({}_{3}\) & \(-\)1.5988 \\ CsGeBr\({}_{3}\) & 2.8964 & CsCuCl\({}_{3}\) & 2.739 & RbSnI\({}_{3}\) & 3.4195 & HfBO\({}_{3}\) & \(-\)1.5526 \\ CsNiBr\({}_{3}\) & 2.9272 & CsCoCl\({}_{3}\) & 2.7645 & CsYbI\({}_{3}\) & 3.4316 & TbSiO\({}_{3}\) & \(-\)1.4989 \\ RbCuBr\({}_{3}\) & 2.9466 & CsCrCl\({}_{3}\) & 2.7686 & CsTmI\({}_{3}\) & 3.4519 & SnBO\({}_{3}\) & \(-\)1.4453 \\ RbSnBr\({}_{3}\) & 2.9486 & CsInCl\({}_{3}\) & 2.7686 & RbVI\({}_{3}\) & 3.4526 & CuBO\({}_{3}\) & \(-\)1.3853 \\ RbVBr\({}_{3}\) & 2.9499 & BaNiCl\({}_{3}\) & 2.8079 & RbCrI\({}_{3}\) & 3.4599 & HgRhO\({}_{3}\) & \(-\)1.2912 \\ RbGeBr\({}_{3}\) & 2.9525 & RbGeCl\({}_{3}\) & 2.8194 & RbIrI\({}_{3}\) & 3.4599 & GeBO\({}_{3}\) & \(-\)1.0071 \\ RbCoBr\({}_{3}\) & 2.9532 & RbCuCl\({}_{3}\) & 2.8216 & RbCrI\({}_{3}\) & 3.4599 & CaCO\({}_{3}\) & \(-\)1.0055 \\ RbPdBr\({}_{3}\) & 2.9543 & RbSnCl\({}_{3}\) & 2.8251 & RbGeI\({}_{3}\) & 3.4601 & LiLaO\({}_{3}\) & \(-\)2.8329 \\ RbTiBr\({}_{3}\) & 2.9568 & RbPdCl\({}_{3}\) & 2.8331 & CsPbI\({}_{3}\) & 3.5094 & HgIrO\({}_{3}\) & \(-\)0.9862 \\ CsAuBr\({}_{3}\) & 3.0427 & RbFeCl\({}_{3}\) & 2.8369 & CsDyI\({}_{3}\) & 3.5292 & HgGeO\({}_{3}\) & \(-\)0.9036 \\ RbInBr\({}_{3}\) & 3.0441 & RbMgCl\({}_{3}\) & 2.8421 & CsCaI\({}_{3}\) & 3.5514 & CeAsO\({}_{3}\) & \(-\)0.8 \\ KGeBr\({}_{3}\) & 3.0488 & RbScCl\({}_{3}\) & 2.8635 & KSrI\({}_{3}\) & 3.5609 & TbBeO\({}_{3}\) & \(-\)0.7902 \\ CsYbBr\({}_{3}\) & 3.0711 & KCoCCl\({}_{3}\) & 2.9149 & KTiI\({}_{3}\) & 3.5761 & MnSiO\({}_{3}\) & \(-\)0.7002 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Top 200 predicted perovskites with predicted \(\tau\) values (tolerance factor).
# Multi-Tier Heterogeneous Beam Management for Future Indoor FSO Networks
## 2 Background and Motivation  
The number of mobile devices and the performance requirements of applications requiring network access continue to drive an unabated growth in wireless traffic [9]. The  
Figure 1: System-level view (**left**); and depiction of a single zone (**right**). _Instantaneous coverage_ is defined as the coverage for a specific orientation/emission configuration (beam width, angle, etc.). _Potential coverage_ is defined as the achievable coverage over all configurations.  
increasing device density and per-device data demand continues to drive requirements for area spectral efficiency [b/s/m2] or volumetric efficiency [b/s/m3]. The directionality and dynamic control of steerable wireless technologies in the mmWave, THz, and optical bands offer massive potential for improvement of both area spectral efficiency and realizable system capacity [10; 11; 12; 13]. In particular, exploiting narrow optical beams for local end-user wireless access can provide data densities and user-experienced data rates that far surpass 5G specifications. This is technically feasible based on the demonstrated potential of steered beams and the inherent signal density of focused light [6; 14; 15; 16; 17; 18; 19; 20]; however, many research challenges exist at the network and system levels. These challenges must be addressed in order to realize the potential of such indoor FSO networks in practice.  
Figure 2 demonstrates the breadth of characteristics for various wireless technologies and use-cases for current and future wireless devices. The left image represents the tradeoffs in directionality and coverage of various wireless technologies. Generally, the high resolution of directional communications offers the potential for massive gains in area spectral efficiency; however, the reduced instantaneous coverage also implies unique challenges when considering network provisioning for multiple dynamic users [21; 22; 23]. The right image of Figure 2 depicts the potential alignment of technologies with wireless devices that have various data demand and mobility characteristics. Applications such as interactive cloud-based services, video sourcing, augmented reality, and virtual reality have unique usage characteristics that vary greatly in terms of mobility and demand. These novel applications are changing the way that we interact with wireless networks. Furthermore, we expect unforeseen applications to contribute to the device and traffic dynamics in the coming years. This device heterogeneity motivates our vision of a heterogeneous multi-tier system, depicted in Figure 1, where the unique characteristics of various wireless technologies are optimally aligned with the characteristics of the wireless devices that they are connecting [24; 25; 26].
## 4 Solutions to the Problem
### Solutions for Velocity Field Based on Caputo Fractional Derivative  
The momentum equations (Equation (13)) are transformed in terms of the Caputo fractional derivative as:  
\[\begin{split}&\phi_{0}\big{[}^{C}D_{\tau}^{\alpha}f(\xi,\tau)+ \lambda_{1}\,^{C}D_{\tau}^{2\alpha}f(\xi,\tau)\big{]}=\phi_{1}\Big{\{}\frac{ \partial^{2}f(\xi,\tau)}{\partial\xi^{2}}+\lambda_{2}\,^{C}D_{\tau}^{\beta} \frac{\partial^{2}f(\xi,\tau)}{\partial\xi^{2}}\Big{\}}\\ &-\phi_{2}M\Big{\{}f(\xi,\tau)+\lambda_{1}\,^{C}D_{\tau}^{\alpha} f(\xi,\tau)\Big{\}}-\frac{\phi_{1}}{K}\Big{\{}f(\xi,\tau)+\lambda_{2}\,^{C}D_{ \tau}^{\beta}f(\xi,\tau)\Big{\}}+\phi_{3}Gr\Phi(\xi,\tau)\end{split} \tag{33}\]For convenience in the solution, Equation (33) is re-arranged and organized as:  
\[\begin{array}{l}\frac{\phi_{0}}{\phi_{1}}\big{[}1+\lambda_{1}{}^{C}D_{\tau}^{ \alpha}\big{]}{}^{C}D_{\tau}^{\alpha}f(\xi,\tau)+\frac{\phi_{2}}{\phi_{1}}M \big{[}1+\lambda_{1}{}^{C}D_{\tau}^{\alpha}\big{]}f(\xi,\tau)\\ =\big{\{}1+\lambda_{2}{}^{C}D_{\tau}^{\beta}\big{\}}\frac{\partial^{2}f(\xi, \tau)}{\partial\xi^{2}}-\frac{1}{k}\big{[}1+\lambda_{2}{}^{C}D_{\tau}^{\beta} \big{]}f(\xi,\tau)+Gr_{0}\Phi(\xi,\tau)\end{array} \tag{34}\]  
where:  
\[Gr_{0}=\frac{\phi_{3}}{\phi_{1}}Gr.\]  
Employing the LT in Equation (34), in accordance with Equation (18), yields:  
\[\begin{array}{l}\frac{\phi_{0}}{\phi_{1}}[1+\lambda_{1}q^{\alpha}]\,q^{ \overline{s}}\overline{f}(\xi,q)+\frac{\phi_{2}}{\phi_{1}}M[1+\lambda_{1}q^{ \alpha}]\overline{f}(\xi,q)=\\ \big{\{}1+\lambda_{2}q^{\beta}\big{\}}\frac{\partial^{2}\overline{f}(\xi,q)}{ \partial\xi^{2}}-\frac{1}{k}\big{\{}1+\lambda_{2}q^{\beta}\big{\}}\overline{ f}(\xi,q)+Gr_{0}\overline{\Phi}(\xi,q)\end{array} \tag{35}\]  
Incorporating Equation (26) into Equation (35) leads to the following form:  
\[\frac{d^{2}\overline{f}(\xi,q)}{d\xi^{2}}-\frac{(1+\lambda_{1}q^{\alpha})(\phi _{2}M+\phi_{0}q^{\alpha})}{\phi_{1}(1+\lambda_{2}q^{\beta})}\overline{f}(\xi, q)-\frac{\overline{f}(\xi,q)}{K}=-\frac{Gr_{0}(1-e^{q})}{q^{2}}e^{-\xi\sqrt{ \alpha_{0}q^{\alpha}}}, \tag{36}\]  
with the corresponding transformed boundary condition:  
\[\overline{f}(0,\tau)=\frac{(1-e^{-q})}{q^{2}},\,\overline{f}(\infty,\tau)=0. \tag{37}\]  
Using the boundary conditions from Equation (37), the exact analytical solution of Equation (36) in the LT domain is given by:  
\[\overline{f}(\xi,q)=\bigg{\{}\frac{(1-e^{-q})}{q^{2}}+\Theta_{0}(q)\bigg{\}}e ^{-\xi\sqrt{\frac{(1+\lambda_{1}q^{\alpha})(\phi_{2}M+\phi_{0}q^{\alpha})}{ \phi_{1}(1+\lambda_{2}q^{\beta})}+\frac{1}{K}}}-\Theta_{0}(q)e^{-\xi\sqrt{ \alpha_{0}q^{\alpha}}}, \tag{38}\]  
where:  
\[\Theta_{0}(q)=\frac{Gr_{0}\phi_{1}K(1+\lambda_{2}q^{\beta})(1-e^{-q})}{q^{2} \big{\{}a_{0}\phi_{1}q^{\alpha}K(1+\lambda_{2}q^{\beta})-K(1+\lambda_{1}q^{ \alpha})(\phi_{2}M+\phi_{0}q^{\alpha})-\phi_{1}(1+\lambda_{2}q^{\beta})\big{\}}}.\]
## 3 Results and Discussion
### Adsorption Batch Experiments  
#### 3.3.1 Effect of Adsorbent Dose and Particle Size  
An adsorption batch experiment was conducted to investigate the adsorption and stabilization efficiency of employed adsorbents for Cd and Pb as affected by adsorbent dose and adsorbent particle size. As previously described in the Material and Methods section, all adsorbent materials were converted to micro- and nanoparticle sizes and were applied at 0.5, 1, 2, 4, and 8.0% (\(w\)/\(w\)) application rates. Adsorbents with nanosize particles showed higher sorption efficiency for Cd and Pb (Figure 5A,D) against microparticle-sized adsorbents. Higher removal efficiency from nanosized adsorbents could be due to enhanced net external surface area (Table 4), which generated more surface porosity and ultimately induced more active sites available to fix heavy metals (Cd, Pb) on them [51]. Furthermore, higher surface area and more available active sites shortened the diffusion path of metals, which consequently increased the removal efficiency of nanosized adsorbents [52]. An increase in removal percentage was noted by increasing adsorbent application rates. An anticipated increase in removal efficiency of Cd and Pb by increasing adsorbent dose was mainly due to the abundance of more available active sites by increasing adsorbent application rate. Also, higher specific surface areas, which enhanced surface porosity and abundance of attached surface functional groups, could be responsible for higher removal of tested heavy metals [53]. Among all tested adsorbents, LDH-N, NZ-N, and HCl-N performed excellently and showed the highest removal efficiency and adsorption rate for Cd and Pb. The removal efficiency and adsorption rate were found in the following order: LDH-N > NZ-N > HCl-N. Comparatively more removal and adsorption of heavy metals by LDH-modified zeolite adsorbent could be due to the abundance of diverse surface functional groups, especially the addition of NO\({}_{3}\)\({}^{-}\), which ultimately generated more electrostatic interaction between cationic heavy metal ions and the zeolite adsorbent surface [41]. Additionally, the highest surface area among all tested adsorbents also favored more removal and adsorption of heavy metals by the LDH-modified nanozoolite adsorbent. The best-performing adsorbents (LDH-N, NZ-N, and HCl-N) were selected and tested for Cd and Pb removal and adsorption at variable initial pH.  
Figure 4: TGA analysis of natural and modified zeolite sediments. (**A**) NZ-M, natural zeolite with microparticle size; LDH-M, LDH-zeolite with microparticle size; HCl-M, HCl-zeolite with microparticle size; NaOH-M, NaOH-zeolite with microparticle size; NaCl-M, NaCl-zeolite in the microparticle size. (**B**) NZ-N, natural zeolite with nanoparticle size; LDH-N, LDH-zeolite with nanoparticle size; HCl-N, HCl-zeolite with nanoparticle size; NaOH-N, NaOH-zeolite with nanoparticle size NaCl-N, NaCl-zeolite in the nanoparticle size.
# Survey on Q-Learning-Based Position-Aware Routing Protocols in Flying Ad Hoc Networks
According to our earlier discussion in this section, we can say that the topology controller (formation controller) iteratively updates the mobility of each UAV within a swarm by using the mobility information of its one-hop neighbors. Additionally, the output of the topology controller decides the topology of the UAV swarm by predicting the present and future mobility information for each UAV (acceleration, velocity, position, and flying direction) [21]. Thus, we can say that relative trajectory knowledge given by the formation controller and link stability is highly coupled [29]. It can ensure stable connectivity between UAVs during flocking. The formation controller updates the mobility information for each UAV in the next timeslot based on the mobility information in the current timeslot, which indicates the similarity with the Markov property. This is because the Markov property states that the next states of the process depend only on the current state of the process. As a result, QL-based PMDP formulation can be adopted to make routing decisions to find the most stable path in FANETs. Owing to this relationship, researchers have used the QL technique to select the optimal relay nodes for forwarding data in FANETs by designing a multi-objective reward function. Because the reward function reinforces the action policy of an RL agent and accelerates the algorithm convergence for optimal decision making, a good reward function considering multiple objectives (delay, relay node energy, and distance progress toward the destination node) gives better routing performance in FANETs. Consequently, this joint consideration of multiple objectives significantly improves the packet delivery ratio (PDR), throughput, end-to-end delay, and balances the energy consumption in FANETs. Considering the high mobility, constraint energy, and memory resources of UAVs, the QL method is more suitable for FANET routing decision making than deep reinforcement learning because it is computationally more expensive and requires a large memory to store training samples and a history of action-reward pairs. The relationship between QL and position-based forwarding is discussed further in Section 3. In FANETs, UAVs can utilize the GPS to localize them in global coordinates. In a GPS-denied environment, UAVs can use range-free and range-based cooperative localization techniques to identify self-location and the location of neighbors [14; 19]. Consequently, position-based routing can be effectively used in dynamic FANETs. In this article, we only extensively survey existing QL-based position-aware routing protocols for FANETs. We also discuss open issues and challenges and their potential research directions related to QL-based position-aware routing in FANETs. In the following subsections, we address related studies and summarize the contributions of this study.
## 5 Ontology Access Management Framework Development
### Analysis of Existing Ontological Resources  
Access management is a relatively new field that aims to secure digital infrastructures against vulnerabilities or threats [31]. Although knowledge of access management in smart cities issues is primarily held by those in the ICT industry, due to the widespread use of ICT, access management knowledge should be disseminated to the general public [31]. Furthermore, the range of contributions provided by diverse professionals in this sector has steadily established a wide knowledge base of access management across different disciplines. Some of these efforts have led to the development of ontologies to help define, organize and signify a vocabulary of concepts relating to a particular specialist [32]. The NeOn method suggests that pre-existing non-ontological and ontological tools should be reused for the purpose of domain ontology construction. The semantic model produced draws upon ontologies which have been defined and, consequently, is in accordance with alternative tools considered to be context ontological. Efforts have been made to refine the analysis of reusable tools by producing lists of the possible users and uses, as well as by suggesting competency questions. There is the potential for the different concepts dis-cussed for future ontologies to be categorized. As such, a core aspect of the NeOn approach being applied in the current work is to re-use existing ontological resources where possible. A notable advantage of reusing ontologies is the fact that they are preformulated which saves both money and time when developing ontologies. In addition, utilizing pre-existing ontologies adheres to the principle of creating integrated knowledge bases. If those developing ontologies are given free rein, the majority will opt to re-use an existing ontology if it is practical to do so. What follows is a description of the prevailing ontological resources which are known to contribute to the access management ontology of built environments. The ontology reflects the semantic integration required to achieve access management in smart cities. This includes resources and policies but also permission between physical built environment assets, IoT devices that are related to the built environment, cyber-physical systems, current built environment services (smart parking, attendance management, access door system and smart air conditioning system), existing security standards, digital twin and BIM datasets, as well as newly developed user interfaces those who use them.
# Distributed Interference-Aware Power Optimization for Multi-Task Over-the-Air Federated Learning
## 4 Proposed Method
### Centralized Scheme
#### 4.1.2 Device Transmit Power Optimization
8:else
9:\(\varsigma^{\text{low}}\leftarrow\varsigma\). 10:endif
11:endwhile
12:Obtain \(\theta_{m}^{*}\) based on (30). 13:Output: Optimal solutions \(\{p_{k}^{*},\theta_{m}^{*},\varsigma^{*}\}\). ```  
**Algorithm 1** Centralized scheme for solving problem (P1)  
While the centralized scheme achieves globally optimal solutions under full CSI, it requires an aggregation center to collect information from all APs, including intra- and inter-cell channels. This introduces significant signaling overhead and limits the system's scalability in real-world implementations. To address this, the next subsection proposes a distributed power control algorithm based on IT. In this framework, each AP requires only local CSI and interference thresholds, thereby enabling decentralized optimization while still approximating the global Pareto boundary through iterative coordination.
## 5 Experiments and Assumptions on Abnormal Behavior Detection and Noise Removal Performance
### Parameters Used in Ensemble Model for Attack Detection and Noise Reduction
#### 5.3.6 Logistic Regression  
Datasets with different attack patterns may require more iterations for the logistic regression model to converge. We set the iteration to 1000, which is ten times the default setting, for coverage on datasets where various attack patterns exist. Stopping criteria set the model to converge at a regular rate using the default value of 0.0001. In addition, the inverse of regularization strength, which means the reliability of the training data, was set to 0.8 instead of the default value of 1.0 to flexibly cope with the noise data existing in the training dataset.
## 3 Optical Model Comparison and Evaluation of Influences on the Incident Irradiance on the PV Module
### Optical Model Comparison
It was obtained that the deviation could be as high as 40% [28], as shown in Figure 7, where the average front and rear irradiance values measured by specular points of a five-module string are plotted, at a height of 1.7 m, ground albedo of 0.1, and tilt angle of 30\({}^{\circ}\). Focusing on this aspect, this is characterized by the rear irradiance non-uniformity \(NU\), calculated as  
\[NU=\frac{max(G_{\text{rear}})-min(G_{\text{rear}})}{(max(G_{\text{rear}})-min(G_ {\text{rear}}))/2} \tag{5}\]  
where \(max(G_{\text{rear}})\) and \(min(G_{\text{rear}})\) are the maximum and minimum values of \(G_{\text{rear}}\) along the module, respectively [57, 58, 59, 60]. [PERSON] et al. in [35] investigated \(NU\) along the module as a function of the distance between the PV module and a reflective surface placed behind the module. In Figure 8a are depicted the measurement points. In Figure 8b, instead, are shown the rear irradiance magnitude and \(NU\), considering different measurement points, as a function of the distance between the PV module and the reflective surface. The testbed consisted of a bi-PV with an active area of 1.53 m\({}^{2}\) mounted on a rack that allowed the placement of a reflective panel (reflectivity > 94%) at different distances behind the module plane. It can be seen that \(NU\) was 27% at a distance of 5 cm, 50% at 20 cm, and when the distance was increased to 60 cm, \(NU\) decreased to 25%. Comparing the incident \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**PVysyst**} & \multicolumn{2}{c}{**Fraunhofer ISE Tool**} \\  & **RMSE** & **MBE** & **RMSE** & **MBE** \\ \hline Front irrad. A & 11.1\% & 4.6\% & 10.5\% & 6.3\% \\  & (46.2 W/m\({}^{2}\)) & (19.3 W/m\({}^{2}\)) & (43.7 W/m\({}^{2}\)) & (26.4 W/m\({}^{2}\)) \\ Front irrad. B & 8.9\% & 4.9\% & 6.5\% & 2.9\% (9.5 W/m\({}^{2}\)) \\ Rear irrad. A & 31.4\% & –21.5\% & 13.8\% & –3.0\% \\  & (33.8 W/m\({}^{2}\)) & (–23.1 W/m\({}^{2}\)) & (14.9 W/m\({}^{2}\)) & (–3.2 W/m\({}^{2}\)) \\ Rear irrad. B & 34.7\% & –23.2\% & 19.8\% & –3.5\% \\  & (15.0 W/m\({}^{2}\)) & (–10.0 W/m\({}^{2}\)) & (8.6 W/m\({}^{2}\)) & (–1.5 W/m\({}^{2}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: RMSE and MBE obtained by PVysst and Fraunhofer ISE tools on the two PV plants, A and B.  
Figure 7: Measurement of the solar incident radiation non-uniformity on the rear side of bifacial PV modules. Image retrieved from [28]. irradiance magnitude and \(N\)\(U\) with the distance module panel, both trends showed a single peak, at 50 cm and 20 cm, respectively.
## 3 Empirical Gramians
### State-Space Empirical Gramians
#### 3.1.1 Empirical Controllability Gramian  
The (linear) controllability (The term **controllability** is used instead of **reachability** as in [2; 4; 56].) Gramian quantifies how well the state of an underlying linear system is driven by the input and is defined as:  
\[W_{C}:=\int_{0}^{\infty}\mathrm{e}^{At}\,BB^{\intercal}\,\mathrm{e}^{A^{ \intercal}t}\,\mathrm{d}t=\int_{0}^{\infty}(\mathrm{e}^{At}\,B)(\mathrm{e}^{ At}\,B)^{\intercal}\,\mathrm{d}t.\]  
The empirical variant is given by the following definition based on [2; 35].  
**Definition 1** (Empirical Controllability Gramian).: _Given non-empty sets \(E_{u}\) and \(S_{u}\), the **empirical controllability Gramian**\(\widehat{W}_{C}\in\mathbb{R}^{N\times N}\) is defined as:_  
\[\widehat{W}_{C} =\frac{1}{|S_{u}|}\sum_{k=1}^{|S_{u}|}\sum_{m=1}^{M}\frac{1}{c_{ k}^{2}}\int_{0}^{T}\Psi^{km}(t)\,\mathrm{d}t,\] \[\Psi^{km}(t) =(x^{km}(t)-\bar{x}^{km})(x^{km}(t)-\bar{x}^{km})^{\intercal}\in \mathbb{R}^{N\times N},\]  
_with the state trajectories \(x^{km}(t)\) for the input configurations \(\hat{u}^{km}(t)=c_{k}e^{m}\circ u(t)+\bar{u}\), and the offsets \(\bar{u}\), \(\bar{x}^{km}\)._  
For an asymptotically stable linear system, delta impulse input \(u_{i}(t)=\delta(t)\) and an arithmetic average over time as offset \(\bar{x}=\frac{1}{T}\int_{0}^{T}x(t)\,\mathrm{d}t\), the empirical controllability Gramian is equal to the controllability Gramian \(\widehat{W}_{C}=W_{C}\)[2]. The numerical computation of this empirical Gramian requires \((|S_{u}|\cdot M)\) trajectories and is related to POD.