@article{SchneidereitBreuss2023, author = {Schneidereit, Toni and Breuß, Michael}, title = {Adaptive neural-domain refinement for solving time-dependent differential equations}, doi = {10.1186/s13662-023-03789-x}, year = {2023}, abstract = {A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent differential equations, one can apply the recently developed method of domain segmentation. That is, the domain may be split into several subdomains, on which the optimisation problem is solved. In classic adaptive numerical methods, the mesh as well as the domain may be refined or decomposed, in order to improve the accuracy. Also, the degree of approximation accuracy may be adapted. Therefore, it is desirable to transfer such important and successful strategies to the field of neural-network-based solutions. In the presented work, we propose a novel adaptive neural approach to meet this aim for solving time-dependent problems. To this end, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks employed are by default small, we propose a means to adjust also the number of neurons in an adaptive way. We introduce conditions to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary. Results are provided for several initial-value problems that illustrate important computational properties of the method.}, subject = {Neural forms; Differential equations; Physics-informed neural networks; Adaptive neural refinement; Domain decomposition; Neuronale Formen; Differentialgleichungen; Physikalisch informierte neuronale Netze; Adaptive neuronale Verfeinerung; Gebietszerlegung; Differentialgleichung; Neuronales Netz}, language = {en} } @phdthesis{Schneidereit2022, author = {Schneidereit, Toni}, title = {The solution of time-dependent ordinary differential equations using neural networks : collocation polynomial neural forms and adaptive neural domain refinement}, doi = {10.26127/BTUOpen-6389}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-63896}, school = {BTU Cottbus - Senftenberg}, year = {2022}, abstract = {Solving differential equations is still a topic of major interest, due to their appearance in many fields of science and engineering and a classic approach with neural networks builds upon trial solutions, the so-called neural forms. The latter are incorporated in a cost function that is subject to minimisation, to train the involved neural networks. Neural forms represent general and flexible tools for solving ordinary differential equations, partial differential equations as well as systems of each. However, the computational approach is in general highly dependent on a variety of computational parameters and the choice of the optimisation methods. Studying the solution of a simple but fundamental stiff ordinary differential equations with small feedforward neural networks and first order optimisation shows, that it is possible to identify preferable choices for parameters and methods. The neural network weight initialisation appears to be a sensitive topic, while having a major impact on the solution accuracy. Especially the use of non-random (deterministic) weights partially shows poor performance, but removes a stochastic component. Further research reveals, that a new polynomial representation of the neural forms can significantly increase the reliability of a deterministic initialisation (all weights have initially the same values assigned). In order to maintain smaller neural network architectures and solve the differential equation, even on fairly large domains, a new technique called domain segmentation (for initial value problems) is introduced. The solution domain splits into equidistant subdomains and the above-mentioned collocation polynomial neural forms are solved separately in each domain fragment. At the boundary of any subdomain, a new initial value is provided by the neural forms solution and directly incorporated in the adjacent one. In classic adaptive numerical methods for solving differential equations, the mesh as well as the domain may be refined or decomposed, respectively, in order to improve numerical accuracy. The subdomain distribution can also be connected with an adaptive refinement. That is, the neural network training status is combined with an adaptive subdomain size reduction in the new adaptive neural domain refinement algorithm. That is, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks are by default small, the number of neurons may also be adjusted in an adaptive way. Conditions are introduced to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary.}, subject = {Neural forms; Differential equations; Collocation polynomial neural forms; Domain segmentation; Adaptive neural domain refinement; Neuronale Formen; Differentialgleichungen; Neuronale Kollokationspolynome; Adaptive Neuronale Gebietsverfeinerung; Differentialgleichung; Kollokation; Numerisches Verfahren; Polynom; Differential equations; Collocation methods}, language = {en} }