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Regularization of Elliptic Partial Differential Equations using Neural Networks

  • This thesis presents a method for interpolating data using a neural network. The data is sparse and perturbed and is used as training data for a small neural network. For severely perturbed data, the network does not manage to find a smooth interpolation. But as the data resembles the solution to the one-dimensional and time-independent heat equation, the weak form of this PDE and subsequently its functional can be written down. If the functional is minimized, a solution to the weak form of the heat equation is found. The functional is now added to the traditional loss function of a neural network, the mean squared error between the network prediction and the given data, in order to smooth out fluctuations and interpolate between distanced grid points. This way, the network minimizes both the mean squared error and the functional, resulting in a smoother curve that can be used to predict u(x) for any grid point x.

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Author:Hanna Wulkow
Document Type:Master's Thesis
Tag:elliptic partial differential equations; neural networks
Granting Institution:Freie Universität Berlin
Advisor:Christof Schütte, Martin Weiser
Date of final exam:2020/01/23
Year of first publication:2020
Page Number:63
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