## 1 Introduction
### Theoretical Background
#### 1.1.3 Using Neural Networks for the Simulation of Physical Processes in a Virtual Laboratory  
It is quite difficult to work remotely with a real (non-virtual) laboratory facility when several groups of students are supposed to do the assignment simultaneously because the connection can be bad and expensive equipment can be damaged. Therefore, it makes more sense to organize students' work with a local digital twin of a laboratory facility. This kind of digital twin requires a high-quality simulation of physical and any other processes in the facility. Today simulation of physical processes is usually based on solving differential equations with boundary conditions. Generally, it is impossible to formulate exact solutions to such problems, so approximate solutions serve as a model. Differential equations, boundary conditions, etc. approximately conform with the simulated object, process, or phenomenon. A great number of conventional, proven approaches are available that make it possible to solve such problems. They include various versions of [PERSON]'s method, finite-difference method, finite element method, boundary integral equations, asymptotic expansions, etc. [21; 22; 23; 24; 25]. Most of them are various types of numerical methods that allow obtaining pointwise or local solution approximation. Formulating an analytical form on the basis of a pointwise solution is a separate problem. In conventional methods, a mathematical model in the form of differential equations, boundary conditions, etc. is regarded as a source object. If such a model describes the real object quite accurately and thoroughly, such idealization is justified, but it does not hold true for the majority of technical applications, therefore, apart from them, observations are made. The latter should be taken into consideration in the course of the simulation. In this situation, the application for neural network modeling is justified [26]. The neural-network-based approach [27; 28; 29; 30; 31; 32; 33; 34; 35; 36] makes it possible to obtain the solution immediately in the form of a function meeting the required smoothness conditions. It is even more important that neural networks are not affected by data errors and that natural parallelization of computations is possible. It is necessary to mention a certain type of problems which require to develop a model for the whole interval at which the values of certain parameters change rather than for the fixed values of parameters. For a virtual laboratory, this kind of mathematical statement of a problem allows modifying the conditions in which virtual experiments are conducted without changing the model of the experimental setup. If the model is developed in a standard way, on a finite set of parameter changes, a lot more calculations are required. Anecdotal evidence suggests [37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49] that applying neural networks appears to be more promising for such problems. The gist of this approach can be explained by the simplest (in the general case non-linear) boundary problem (1). The specificity of this problem is that its statement includes the parameter \(\mathbf{r}=(r_{1},\,\ldots,\,r_{k})\), which change at some intervals \(r_{i}\in(r_{i}^{-};r_{i}^{+}),\ i=1,\ldots,k\):  
\[A(u,\mathbf{r})=g(\mathbf{r}),u=u(\mathbf{x},\mathbf{r}),\mathbf{x}\in\Omega( \mathbf{r})\subset R^{p},\ B(u,\mathbf{r})\big{|}_{\Gamma(\mathbf{r})}\ =f(\mathbf{r}), \tag{1}\]  
where \(A\) is a differential operator; \(B\) is the operator that allows establishing boundary conditions; \(\Gamma\) is the boundary of domain \(\Omega\).