Line Graph Vietoris-Rips Persistence Diagram for Topological Graph Representation Learning

While message passing graph neural networks result in informative node embeddings, they may suffer from describing the topological properties of graphs. To this end, node filtration has been widely used as an attempt to obtain the topological information of a graph using persistence diagrams. However, these attempts have faced the problem of losing node embedding information, which in turn prevents them from providing a more expressive graph representation. To tackle this issue, we shift our focus to edge filtration and introduce a novel edge filtration-based persistence diagram, named Topological Edge Diagram (TED), which is mathematically proven to preserve node embedding information as well as contain additional topological information. To implement TED, we propose a neural network based algorithm, named Line Graph Vietoris-Rips (LGVR) Persistence Diagram, that extracts edge information by transforming a graph into its line graph. Through LGVR, we propose two model frameworks that can be applied to any message passing GNNs, and prove that they are strictly more powerful than Weisfeiler-Lehman type colorings. Finally we empirically validate superior performance of our models on several graph classification and regression benchmarks.

On the stability of Lipschitz continuous control problems and its application to reinforcement learning

We address the crucial yet underexplored stability properties of the Hamilton--Jacobi--Bellman (HJB) equation in model-free reinforcement learning contexts, specifically for Lipschitz continuous optimal control problems. We bridge the gap between Lipschitz continuous optimal control problems and classical optimal control problems in the viscosity solutions framework, offering new insights into the stability of the value function of Lipschitz continuous optimal control problems. By introducing structural assumptions on the dynamics and reward functions, we further study the rate of convergence of value functions. Moreover, we introduce a generalized framework for Lipschitz continuous control problems that incorporates the original problem and leverage it to propose a new HJB-based reinforcement learning algorithm. The stability properties and performance of the proposed method are tested with well-known benchmark examples in comparison with existing approaches.

Sobolev Training for Operator Learning

This study investigates the impact of Sobolev Training on operator learning frameworks for improving model performance. Our research reveals that integrating derivative information into the loss function enhances the training process, and we propose a novel framework to approximate derivatives on irregular meshes in operator learning. Our findings are supported by both experimental evidence and theoretical analysis. This demonstrates the effectiveness of Sobolev Training in approximating the solution operators between infinite-dimensional spaces.