Related papers: Topological Obstructions and How to Avoid Them

Topology optimization is used for the design of high-performance structures but remains fundamentally limited by its iterative nature, requiring repeated finite element analyses that prevent real-time deployment and large-scale design…

Many modern neural network architectures are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Sufficiently overparameterized neural network architectures in principle have the…

We study the problem of transfer learning, observing that previous efforts to understand its information-theoretic limits do not fully exploit the geometric structure of the source and target domains. In contrast, our study first…

Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information,…

We review a number a recent advances in the study of two-dimensional statistical models with strong geometrical constraints. These include folding problems of regular and random lattices as well as the famous meander problem of enumerating…

Disordered mechanical systems can deform along a network of pathways that branch and recombine at special configurations called bifurcation points. Multiple pathways are accessible from these bifurcation points; consequently, computer-aided…

We show that, in discrete models of quantum gravity, emergent geometric space can be viewed as the entanglement pattern in a mixed quantum state of the "universe", characterized by a universal topological network entanglement. As a concrete…

Josephson junctions in a two-dimensional electron gas with spin-orbit coupling are a promising candidate to realize topological superconductivity. While it is known that the geometry of the junction strongly influences the size of the…

We consider closed and orientable immersed hypersurfaces of translational manifolds. Given a vector field on such a hypersurface, we define a perturbation of its Gauss map, which allows us to obtain topological invariants for the immersion…

Flow matching has emerged as a powerful framework for generative modeling through continuous normalizing flows. We investigate a potential topological constraint: when the prior distribution and target distribution have mismatched topology…

Learning unknown dynamics under environmental (or external) constraints is fundamental to many fields (e.g., modern robotics), particularly challenging when constraint information is only locally available and uncertain. Existing approaches…

Topology Optimization seeks to find the best design that satisfies a set of constraints while maximizing system performance. Traditional iterative optimization methods like SIMP can be computationally expensive and get stuck in local…

Linear convergence of first-order methods is typically characterized by global optimization conditions whose constants reflect worst-case geometry of the ambient space. In high-dimensional or structured problems, these global constants can…

We propose to learn a hierarchical prior in the context of variational autoencoders to avoid the over-regularisation resulting from a standard normal prior distribution. To incentivise an informative latent representation of the data, we…

This paper revisits the origin of topology optimisation for fluid flow problems, namely the Poiseuille-based frictional resistance term used to parametrise regions of solid and fluid. The traditional model only works for true topology…

In this article we consider shape optimization problems as optimal control problems via the method of mappings. Instead of optimizing over a set of admissible shapes a reference domain is introduced and it is optimized over a set of…

Neural network training is commonly based on SGD. However, the understanding of SGD's ability to converge to good local minima, given the non-convex nature of loss functions and the intricate geometric characteristics of loss landscapes,…

Persistence diagrams (PDs) are now routinely used to summarize the underlying topology of complex data. Despite several appealing properties, incorporating PDs in learning pipelines can be challenging because their natural geometry is not…

We study traffic flow on roads with a localized periodic inhomogeneity such as traffic signals, using a stochastic car-following model. We find that in cases of congestion, traffic flow can be optimized by controlling the inhomogeneity's…

We consider the problem of embedding the nodes of a hypergraph into Euclidean space under the assumption that the interactions arose through closeness to unknown hyperedge centres. In this way, we tackle the inverse problem associated with…