Stabilizing and Solving Inverse Problems using Data and Machine Learning

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 auto-encoder, which provides a parametrization of the dataset in a lower-dimensional latent space. We then train a neural network to map the latent variables representing the boundary data to the solution of the PDE. Finally, we solve the inverse problem by optimizing a data-fitting term over the latent space. We analyze the underlying stabilized finite element method in the linear setting and establish optimal error estimates in the $H^1$ and $L^2$-norms. The nonlinear problem is then studied numerically, demonstrating the effectiveness of our approach.

Implicit Hypersurface Approximation Capacity in Deep ReLU Networks

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 $\phi$, 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 $\phi$ 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 $\delta$ and we show that the tolerance in the resulting error bound scales as $(d-1)R^{3/2}\delta^{1/2}$ and the required number of layers is of order $d\big(\frac{32R}{\delta}\big)^{\frac{d+1}{2}}$.

The Geometric Structure of Fully-Connected ReLU-Layers

We formalize and interpret the geometric structure of $d$-dimensional fully connected ReLU-layers in neural networks. The parameters of a ReLU-layer induce a natural partition of the input domain, such that in each sector of the partition, the ReLU-layer can be greatly simplified. This leads to a geometric interpretation of a ReLU-layer as a projection onto a polyhedral cone followed by an affine transformation, in line with the description in [doi:10.48550/arXiv.1905.08922] for convolutional networks with ReLU activations. Further, this structure facilitates simplified expressions for preimages of the intersection between partition sectors and hyperplanes, which is useful when describing decision boundaries in a classification setting. We investigate this in detail for a feed-forward network with one hidden ReLU-layer, where we provide results on the geometric complexity of the decision boundary generated by such networks, as well as proving that modulo an affine transformation, such a network can only generate $d$ different decision boundaries. Finally, the effect of adding more layers to the network is discussed.