Abstract
The paper uses a general neural network approach to solving partial differential problems. The solution is the output of a neural network with one hidden layer and linearly and nonlinearly adjustable input parameters (weights) during training. Network training is considered as a multi-criteria optimization problem solved by constructing a Pareto front and then choosing the optimal solution in some sense. An evolutionary algorithm for constructing solutions based on the Pareto front is proposed. The method is tested on two reference problems: the Laplace equation in the unit square with a discontinuous Dirichlet boundary condition (Motz’s problem) and without initial conditions, using measurement data of varying accuracy. Optimal solutions were obtained in accordance with two different selection criteria.
This work was supported by the Russian Science Foundation under grant no. 22-21-20004, https://rscf.ru/project/22-21-20004/..
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Lazovskaya, T. et al. (2023). Investigation of Pareto Front of Neural Network Approximation of Solution of Laplace Equation in Two Statements: with Discontinuous Initial Conditions or with Measurement Data. In: Kryzhanovsky, B., Dunin-Barkowski, W., Redko, V., Tiumentsev, Y. (eds) Advances in Neural Computation, Machine Learning, and Cognitive Research VI. NEUROINFORMATICS 2022. Studies in Computational Intelligence, vol 1064. Springer, Cham. https://doi.org/10.1007/978-3-031-19032-2_42
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DOI: https://doi.org/10.1007/978-3-031-19032-2_42
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