Dynamical systems theory (DST) is fundamental for many areas of science and engineering. It can provide deep insights into the behavior of systems evolving in time, as typically described by differential or recursive equations. A common approach to facilitate mathematical tractability and interpretability of DS models involves decomposing nonlinear DS into multiple linear DS combined by switching manifolds, i.e. piecewise linear (PWL) systems. PWL models are popular in engineering and a frequent choice in mathematics for analyzing the topological properties of DS. However, hand-crafting such models is tedious and only possible for very low-dimensional scenarios, while inferring them from data usually gives rise to unnecessarily complex representations with very many linear subregions. Here we introduce Almost-Linear Recurrent Neural Networks (AL-RNNs) which automatically and robustly produce most parsimonious PWL representations of DS from time series data, using as few PWL nonlinearities as possible. AL-RNNs can be efficiently trained with any SOTA algorithm for dynamical systems reconstruction (DSR), and naturally give rise to a symbolic encoding of the underlying DS that provably preserves important topological properties. We show that for the Lorenz and Rössler systems, AL-RNNs derive, in a purely data-driven way, the known topologically minimal PWL representations of the corresponding chaotic attractors. We further illustrate on two challenging empirical datasets that interpretable symbolic encodings of the dynamics can be achieved, tremendously facilitating mathematical and computational analysis of the underlying systems.

This broad applicability complicates their evaluation.

Message-passing has proved to be an effective way to design graph neural networks, as it is able to leverage both permutation equivariance and an inductive bias towards learning local structures in order to achieve good generalization. However, current message-passing architectures have a limited representation power and fail to learn basic topological properties of graphs. We address this problem and propose a powerful and equivariant message-passing framework based on two ideas: first, we propagate a one-hot encoding of the nodes, in addition to the features, in order to learn a local context matrix around each node. This matrix contains rich local information about both features and topology and can eventually be pooled to build node representations. Second, we propose methods for the parametrization of the message and update functions that ensure permutation equivariance. Having a representation that is independent of the specific choice of the one-hot encoding permits inductive reasoning and leads to better generalization properties. Experimentally, our model can predict various graph topological properties on synthetic data more accurately than previous methods and achieves state-of-the-art results on molecular graph regression on the ZINC dataset.

Learning a graph topology to reveal the underlying relationship between data entities plays an important role in various machine learning and data analysis tasks. Under the assumption that structured data vary smoothly over a graph, the problem can be formulated as a regularised convex optimisation over a positive semidefinite cone and solved by iterative algorithms. Classic methods require an explicit convex function to reflect generic topological priors, e.g. the $\ell_1$ penalty for enforcing sparsity, which limits the flexibility and expressiveness in learning rich topological structures. We propose to learn a mapping from node data to the graph structure based on the idea of learning to optimise (L2O). Specifically, our model first unrolls an iterative primal-dual splitting algorithm into a neural network. The key structural proximal projection is replaced with a variational autoencoder that refines the estimated graph with enhanced topological properties. The model is trained in an end-to-end fashion with pairs of node data and graph samples. Experiments on both synthetic and real-world data demonstrate that our model is more efficient than classic iterative algorithms in learning a graph with specific topological properties.

Large Vision-Language Models (LVLMs) typically align visual features from an encoder with a pre-trained Large Language Model (LLM). However, this makes the visual perception module a bottleneck, which constrains the overall capabilities of LVLMs. Conventional evaluation benchmarks, while rich in visual semantics, often contain unavoidable local shortcuts that can lead to an overestimation of models' perceptual abilities. Here, we introduce TopoPerception, a benchmark that leverages topological properties to rigorously evaluate the global visual perception capabilities of LVLMs across various granularities. Since topology depends on the global structure of an image and is invariant to local features, TopoPerception enables a shortcut-free assessment of global perception, fundamentally distinguishing it from semantically rich tasks. We evaluate state-of-the-art models on TopoPerception and find that even at the coarsest perceptual granularity, all models perform no better than random chance, indicating a profound inability to perceive global visual features. Notably, a consistent trend emerge within model families: more powerful models with stronger reasoning capabilities exhibit lower accuracy. This suggests that merely scaling up models is insufficient to address this deficit and may even exacerbate it. Progress may require new training paradigms or architectures. TopoPerception not only exposes a critical bottleneck in current LVLMs but also offers a lens and direction for improving their global visual perception. The data and code are publicly available at: https://github.com/Wenhao-Zhou/TopoPerception.

The process by which Large Language Models (LLMs) acquire complex capabilities during training remains a key open question in mechanistic interpretability. This project investigates whether these learning dynamics can be characterized through the lens of Complex Network Theory (CNT). I introduce a novel methodology to represent a Transformer-based LLM as a directed, weighted graph where nodes are the model's computational components (attention heads and MLPs) and edges represent causal influence, measured via an intervention-based ablation technique. By tracking the evolution of this component-graph across 143 training checkpoints of the Pythia-14M model on a canonical induction task, I analyze a suite of graph-theoretic metrics. The results reveal that the network's structure evolves through distinct phases of exploration, consolidation, and refinement. Specifically, I identify the emergence of a stable hierarchy of information spreader components and a dynamic set of information gatherer components, whose roles reconfigure at key learning junctures. This work demonstrates that a component-level network perspective offers a powerful macroscopic lens for visualizing and understanding the self-organizing principles that drive the formation of functional circuits in LLMs.

"One major advantage of the sliced divergences is the dimension-independent sample complexity.

Safe navigation within a workspace is a fundamental skill for autonomous robots to accomplish more complex tasks. Harmonic potentials are artificial potential fields that are analytical, globally convergent and provably free of local minima. Thus, it has been widely used for generating safe and reliable robot navigation control policies. However, most existing methods do not allow customization of the harmonic potential fields nor the resulting paths, particularly regarding their topological properties. In this paper, we propose a novel method that automatically finds homotopy classes of paths that can be generated by valid harmonic potential fields. The considered complex workspaces can be as general as forest worlds consisting of numerous overlapping star-obstacles. The method is based on a hybrid optimization algorithm that searches over homotopy classes, selects the structure of each tree-of-stars within the forest, and optimizes over the continuous weight parameters for each purged tree via the projected gradient descent. The key insight is to transform the forest world to the unbounded point world via proper diffeomorphic transformations. It not only facilitates a simpler design of the multi-directional D-signature between non-homotopic paths, but also retain the safety and convergence properties. Extensive simulations and hardware experiments are conducted for non-trivial scenarios, where the navigation potentials are customized for desired homotopic properties. Project page: https://shuaikang-wang.github.io/CustFields.

Dynamical systems theory (DST) is fundamental for many areas of science and engineering. It can provide deep insights into the behavior of systems evolving in time, as typically described by differential or recursive equations. A common approach to facilitate mathematical tractability and interpretability of DS models involves decomposing nonlinear DS into multiple linear DS combined by switching manifolds, i.e. piecewise linear (PWL) systems. PWL models are popular in engineering and a frequent choice in mathematics for analyzing the topological properties of DS. However, hand-crafting such models is tedious and only possible for very low-dimensional scenarios, while inferring them from data usually gives rise to unnecessarily complex representations with very many linear subregions. Here we introduce Almost-Linear Recurrent Neural Networks (AL-RNNs) which automatically and robustly produce most parsimonious PWL representations of DS from time series data, using as few PWL nonlinearities as possible.