## 3 Testing with a Simple Room Case
### Test Method  
Inverse prediction tests are carried out using computational fluid dynamics (CFD) simulations. Inverse prediction is similar to finding the inverse function in that there may be multiple solutions for a single target value. Although we experimentally obtained data for some physical quantities under a certain boundary condition, we may obtain multiple boundary conditions by making inverse predictions based on some of the data. Using more data reduces the likelihood of multiple solutions, and when there is enough data, the solution should be unique, because we know that these data are obtained from our forward-based experiments and we believe that our simulations are accurate. [PERSON] [4] used velocity and temperature data from all locations and his test method was  
1. Construct the objective/cost function as in Equation (8) (minimization, which can be seen as a \"distance\"), where \(u\) refers to the velocity, subscripts \(x\), \(y\) refer to the velocity direction, \(t\) refers to the temperature, and \"\(\thicksim\) refers to the simulation results. Detailed information can be found in reference [4]; \[F=\frac{1}{3}\sqrt{\sum\frac{\left(\dot{u}_{y}-u_{y}\right)^{2}}{\sum{u_{y}}^{2 }}+\sum\frac{\left(\dot{u}_{x}-u_{x}\right)^{2}}{\sum{u_{x}}^{2}}+\sum\frac{ \left(\dot{t}-t\right)^{2}}{\sum{t}^{2}}}\] (8)
2. Set the following parameter range: 0 \(<V_{in}<\) 1 m/s, 0 \(<T_{in}<\) 20 \({}^{\circ}\)C;
3. Minimize the objective function with a single-objective genetic algorithm and output the corresponding parameters.  
In fact, [PERSON], [PERSON] considered all physical quantities to have the same weight after the same dimensionless process when performing step 1. It was also a process of converting from multiple-objective to single-objective. Since experiments are subject to errors and different CFD models differ in their solution results, especially in the near-wall region [56], it is difficult to obtain exactly the same results as experiments in simulations. It leads to  
Figure 4: Illustration of the non-isothermal convection case.  
the possibility that there is no one combination of parameters that makes the simulation results identical to the experimental results, so that the optimization algorithm can only obtain a set of parameter combinations (possibly not identical to the parameters used in the experiments) that make the simulation results \"closest\" to the experimental results at the defined \"distance\" (objective function). [PERSON], [PERSON] encountered this problem when making inverse predictions based on experimental data. In this paper, the physical quantities are separated to form an objective function with respect to \(u_{y}\) and an objective function with respect to \(t\). The following conditions are required to use NSGA-II, where \(D\) is a sample set consisting of some parameter combinations and their response values obtained based on the simulation results and calculation functions.  
1. Construct the objective function as in Equation (9). Using MSP criteria as an example, \[\begin{array}{l}F_{1}=\hat{g}\big{|}OK_{uy}(D)\\ F_{2}=\hat{g}\big{|}OK_{t}(D)\end{array}\] (9) where \(OK_{uy}(D)\) and \(OK_{t}(D)\) are OK models obtained by interpolating from \(D\). Their response values \(y_{n}\) at known points \(x_{n}\) are calculated as Equation (10); subscripts 1 and 2 refer to the numbers of the objective functions. The two objective functions \(F_{1}\) and \(F_{2}\