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A method using radial basis function networks (RBFNs) to solve boundary value problems of mathematical physics is presented in this paper. The main advantages of mesh-free methods based on RBFN are explained here. To learn RBFNs, the Trust Region Method (TRM) is proposed, which simplifies the process of network structure selection and reduces time expenses to adjust their parameters. Application of the proposed algorithm is illustrated by solving two-dimensional Poisson equation.

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Background. The object of the research is the functional architecture of distributed computing systems with a variable (reconfigurable) structure characteristic of hybrid systems of cloud-network (grid) type. Despite the fact that Petri nets have long been studied both theoretically and practically, the methods of their interpretation continue to develop intensively. At present, the problem of embedding Petri nets in the architecture of distributed network applications used to implement global computing in modern mixed cloud, grid and cluster systems has not been sufficiently studied. It is shown that in modern studies, Petri nets are used mainly in the simulation of discrete systems and processes, and not as the basis for formalized specifications in the development of distributed applications. In this regard, the interpretation of Petri nets in applications to the functional architecture of distributed computing systems with a variable structure based on the network software of the intermediate class (middleware class) is relevant. The aim of the work was the integration of graphical representations of conceptual graphs, semantic networks, scenarios and Petri nets, which made it possible to create effective tools with graphical support for designing a functional architecture of distributed computing systems with a variable structure and, in particular, a cloudy architecture of the NCaaSoD type - Network Computing as a Service on Demand (network computing as a cloud service at the request of the user).Materials and methods. The conceptual models of distributed processes that are a graphical interpretation of the first-order predicate calculus are used. Conceptual graphs for distributed Petri nets of mixed type have been proposed, which allow describing computation processes in global computational networks with a view to their subsequent implementation. Results. Based on the integration of graphical representations of conceptual graphs, semantic networks, scenarios and Petri nets, conceptual representations of distributed reconfigurable Petri nets are proposed, allowing them to be directly integrated into the architecture of the computer network.Results. New conceptual-behavioral models based on conceptual graphs of distributed Petri nets have been proposed to determine the system and functional architectures of distributed computing systems with a variable structure provided to the user as a hybrid cloud-based network service; these models are distinguished by the possibility of operational reconfiguration and immediate execution.Conclusion. A method was proposed and formalized for embedding conceptual Petri nets into the architecture of cloud-networked computer systems such as NCaaSoD — network (cloud) computing as a service organized at the user's request. The rules for obtaining relations of connectivity between the positions and transitions of the Petri net, placed on the nodes of the physical computer network, are proposed.