Numerical approximation of inverse problems for PDEs via neural network augmentation

In this thesis, we consider the numerical approximation of inverse problems for linear and nonlinear elliptic PDEs by augmenting them with a neural network to predict unknown or uncertain model coefficients. The neural network acts as a prior for the coefficient and attempts to reconstruct its value from observations of some output quantities of interest related to the PDE solution or from the solution itself. While neural network augmentation for inverse problems has recently been proposed in the literature, we extend the idea to several test cases dealing with diffusion problems and nonlinear elasticity problems. We demonstrate that, under certain conditions, neural networks are highly effective at recognizing many different types of coefficients for these problems. However, we also show that the computational cost of performing these simulations is potentially prohibitive for large-scale applications. To resolve this problem, we attempt to combine neural network augmentation with the reduced basis method with the aim of enhancing computational efficiency. We then discuss the limitations of such an approach and provide some ideas for future applications and further research.

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