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Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.

We investigate the effect of the well-known Mycielski construction on the Shannon capacity of graphs and on one of its most prominent upper bounds, the (complementary) Lov\'asz theta number. We prove that if the Shannon capacity of a graph, the distinguishability graph of a noisy channel, is attained by some finite power, then its Mycielskian has strictly larger Shannon capacity than the graph itself. For the complementary Lov\'asz theta function we show that its value on the Mycielskian of a graph is completely determined by its value on the original graph, a phenomenon similar to the one discovered for the fractional chromatic number by Larsen, Propp and Ullman. We also consider the possibility of generalizing our results on the Sperner capacity of directed graphs and on the generalized Mycielsky construction. Possible connections with what Zuiddam calls the asymptotic spectrum of graphs are discussed as well.

We propose reinforcement learning to control the dynamical self-assembly of the dodecagonal quasicrystal (DDQC) from patchy particles. The patchy particles have anisotropic interactions with other particles and form DDQC. However, their structures at steady states are significantly influenced by the kinetic pathways of their structural formation. We estimate the best policy of temperature control trained by the Q-learning method and demonstrate that we can generate DDQC with few defects using the estimated policy. The temperature schedule obtained by reinforcement learning can reproduce the desired structure more efficiently than the conventional pre-fixed temperature schedule, such as annealing. To clarify the success of the learning, we also analyse a simple model describing the kinetics of structural changes through the motion in a triple-well potential. We have found that reinforcement learning autonomously discovers the critical temperature at which structural fluctuations enhance the chance of forming a globally stable state. The estimated policy guides the system toward the critical temperature to assist the formation of DDQC.

In the study of the brain, there is a hypothesis that sparse coding is realized in information representation of external stimuli, which is experimentally confirmed for visual stimulus recently. However, unlike the specific functional region in the brain, sparse coding in information processing in the whole brain has not been clarified sufficiently. In this study, we investigate the validity of sparse coding in the whole human brain by applying various matrix factorization methods to functional magnetic resonance imaging data of neural activities in the whole human brain. The result suggests sparse coding hypothesis in information representation in the whole human brain, because extracted features from sparse MF method, SparsePCA or MOD under high sparsity setting, or approximate sparse MF method, FastICA, can classify external visual stimuli more accurately than non-sparse MF method or sparse MF method under low sparsity setting.

We study computational aspects of repulsive Gibbs point processes, which are probabilistic models of interacting particles in a finite-volume region of space. We introduce an approach for reducing a Gibbs point process to the hard-core model, a well-studied discrete spin system. Given an instance of such a point process, our reduction generates a random graph drawn from a natural geometric model. We show that the partition function of a hard-core model on graphs generated by the geometric model concentrates around the partition function of the Gibbs point process. Our reduction allows us to use a broad range of algorithms developed for the hard-core model to sample from the Gibbs point process and approximate its partition function. This is, to the extend of our knowledge, the first approach that deals with pair potentials of unbounded range. We compare the resulting algorithms with recently established results and study further properties of the random geometric graphs with respect to the hard-core model.

Image segmentation, real-value prediction, and cross-modal translation are critical challenges in medical imaging. In this study, we propose a versatile multi-task neural network framework, based on an enhanced Transformer U-Net architecture, capable of simultaneously, selectively, and adaptively addressing these medical image tasks. Validation is performed on a public repository of human brain MR and CT images. We decompose the traditional problem of synthesizing CT images into distinct subtasks, which include skull segmentation, Hounsfield unit (HU) value prediction, and image sequential reconstruction. To enhance the framework's versatility in handling multi-modal data, we expand the model with multiple image channels. Comparisons between synthesized CT images derived from T1-weighted and T2-Flair images were conducted, evaluating the model's capability to integrate multi-modal information from both morphological and pixel value perspectives.

The symmetry of complex networks is a global property that has recently gained attention since MacArthur et al. 2008 showed that many real-world networks contain a considerable number of symmetries. These authors work with a very strict symmetry definition based on the network's automorphism. The potential problem with this approach is that even a slight change in the graph's structure can remove or create some symmetry. Recently, Liu 2020 proposed to use an approximate automorphism instead of strict automorphism. This method can discover symmetries in the network while accepting some minor imperfections in their structure. The proposed numerical method, however, exhibits some performance problems and has some limitations while it assumes the absence of fixed points. In this work, we exploit alternative approaches recently developed for treating the Graph Matching Problem and propose a method, which we will refer to as Quadratic Symmetry Approximator (QSA), to address the aforementioned shortcomings. To test our method, we propose a set of random graph models suitable for assessing a wide family of approximate symmetry algorithms. The performance of our method is also demonstrated on brain networks.

The global minimum point of an optimization problem is of interest in engineering fields and it is difficult to be found, especially for a nonconvex large-scale optimization problem. In this article, we consider a new memetic algorithm for this problem. That is to say, we use the continuation Newton method with the deflation technique to find multiple stationary points of the objective function and use those found stationary points as the initial seeds of the evolutionary algorithm, other than the random initial seeds of the known evolutionary algorithms. Meanwhile, in order to retain the usability of the derivative-free method and the fast convergence of the gradient-based method, we use the automatic differentiation technique to compute the gradient and replace the Hessian matrix with its finite difference approximation. According to our numerical experiments, this new algorithm works well for unconstrained optimization problems and finds their global minima efficiently, in comparison to the other representative global optimization methods such as the multi-start methods (the built-in subroutine GlobalSearch.m of MATLAB R2021b, GLODS and VRBBO), the branch-and-bound method (Couenne, a state-of-the-art open-source solver for mixed integer nonlinear programming problems), and the derivative-free algorithms (CMA-ES and MCS).

Rule-based reasoning is an essential part of human intelligence prominently formalized in artificial intelligence research via logic programs. Describing complex objects as the composition of elementary ones is a common strategy in computer science and science in general. The author has recently introduced the sequential composition of logic programs in the context of logic-based analogical reasoning and learning in logic programming. Motivated by these applications, in this paper we construct a qualitative and algebraic notion of syntactic logic program similarity from sequential decompositions of programs. We then show how similarity can be used to answer queries across different domains via a one-step reduction. In a broader sense, this paper is a further step towards an algebraic theory of logic programming.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.