In recent years, graph neural networks (GNNs) have gained increasing popularity and have shown very promising results for data that are represented by graphs. The majority of GNN architectures are designed based on developing new convolutional and/or
pooling layers that better extract the hidden and deeper representations of the graphs to be used for different prediction tasks. The inputs to these layers are mainly the three default descriptors of a graph, node features $(X)$, adjacency matrix $(A)$, and edge features $(W)$ (if available). To provide a more enriched input to the network, we propose a random walk data processing of the graphs based on three selected lengths. Namely, (regular) walks of length 1 and 2, and a fractional walk of length $gamma in (0,1)$, in order to capture the different local and global dynamics on the graphs. We also calculate the stationary distribution of each random walk, which is then used as a scaling factor for the initial node features ($X$). This way, for each graph, the network receives multiple adjacency matrices along with their individual weighting for the node features. We test our method on various molecular datasets by passing the processed node features to the network in order to perform several classification and regression tasks. Interestingly, our method, not using edge features which are heavily exploited in molecular graph learning, let a shallow network outperform well known deep GNNs.
Differential algebraic Riccati equations are at the heart of many applications in control theory. They are time-depent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been u
sed heavily computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton-Kleinman iteration. Approximating the space-time problem in low-rank form requires fewer applications of the discretized differential operator and gives a low-rank approximation to the overall solution.
