Karl Larsson

Erik Burman, Daniel Elfverson, Peter Hansbo et al.

We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples.

Jonatan Vallin, Karl Larsson, Mats G. Larson

We develop a geometric approximation theory for deep feed-forward neural networks with ReLU activations. Given a $d$-dimensional hypersurface in $\mathbb{R}^{d+1}$ represented as the graph of a $C^2$-function $φ$, we show that a deep fully-connected ReLU network of width $d+1$ can implicitly construct an approximation as its zero contour with a precision bound depending on the number of layers. This result is directly applicable to the binary classification setting where the sign of the network is trained as a classifier, with the network's zero contour as a decision boundary. Our proof is constructive and relies on the geometrical structure of ReLU layers provided in [doi:10.48550/arXiv.2310.03482]. Inspired by this geometrical description, we define a new equivalent network architecture that is easier to interpret geometrically, where the action of each hidden layer is a projection onto a polyhedral cone derived from the layer's parameters. By repeatedly adding such layers, with parameters chosen such that we project small parts of the graph of $φ$ from the outside in, we, in a controlled way, construct a network that implicitly approximates the graph over a ball of radius $R$. The accuracy of this construction is controlled by a discretization parameter $δ$ and we show that the tolerance in the resulting error bound scales as $(d-1)R^{3/2}δ^{1/2}$ and the required number of layers is of order $d\big(\frac{32R}δ\big)^{\frac{d+1}{2}}$.

Erik Burman, Mats G. Larson, Karl Larsson et al.

We consider an inverse problem involving the reconstruction of the solution to a nonlinear partial differential equation (PDE) with unknown boundary conditions. Instead of direct boundary data, we are provided with a large dataset of boundary observations for typical solutions (collective data) and a bulk measurement of a specific realization. To leverage this collective data, we first compress the boundary data using proper orthogonal decomposition (POD) in a linear expansion. Next, we identify a possible nonlinear low-dimensional structure in the expansion coefficients using an autoencoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train an operator network to map the expansion coefficients representing the boundary data to the finite element (FE) solution of the PDE. Finally, we connect the autoencoder's decoder to the operator network which enables us to solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method (FEM) in the linear setting and establish an optimal error estimate in the $H^1$-norm. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.