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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 give a structure preserving spatio-temporal discretization for incompressible magnetohydrodynamics (MHD) on the sphere. Discretization in space is based on the theory of geometric quantization, which yields a spatially discretized analogue of the MHD equations as a finite-dimensional Lie--Poisson system on the dual of the magnetic extension Lie algebra $\mathfrak{f}=\mathfrak{su}(N)\ltimes\mathfrak{su}(N)^{*}$. We also give accompanying structure preserving time discretizations for Lie--Poisson systems on the dual of semidirect product Lie algebras of the form $\mathfrak{f}=\mathfrak{g}\ltimes\mathfrak{g^{*}}$, where $\mathfrak{g}$ is a $J$-quadratic Lie algebra. Critically, the time integration method is free of computationally costly matrix exponentials. We prove that the full method preserves the underlying geometry, namely the Lie--Poisson structure and all the Casimirs. To showcase the method, we apply it to two models for magnetic fluids: incompressible magnetohydrodynamics and Hazeltine's model.

This manuscript is devoted to investigating the conservation laws of incompressible Navier-Stokes equations(NSEs), written in the energy-momentum-angular momentum conserving(EMAC) formulation, after being linearized by the two-level methods. With appropriate correction steps(e.g., Stoke/Newton corrections), we show that the two-level methods, discretized from EMAC NSEs, could preserve momentum, angular momentum, and asymptotically preserve energy. Error estimates and (asymptotic) conservative properties are analyzed and obtained, and numerical experiments are conducted to validate the theoretical results, mainly confirming that the two-level linearized methods indeed possess the property of (almost) retainability on conservation laws. Moreover, experimental error estimates and optimal convergence rates of two newly defined types of pressure approximation in EMAC NSEs are also obtained.

While the semi-blind source separation-based acoustic echo cancellation (SBSS-AEC) has received much research attention due to its promising performance during double-talk compared to the traditional adaptive algorithms, it suffers from system latency and nonlinear distortions. To circumvent these drawbacks, the recently developed ideas on convolutive transfer function (CTF) approximation and nonlinear expansion have been used in the iterative projection (IP)-based semi-blind source separation (SBSS) algorithm. However, because of the introduction of CTF approximation and nonlinear expansion, this algorithm becomes computationally very expensive, which makes it difficult to implement in embedded systems. Thus, we attempt in this paper to improve this IP-based algorithm, thereby developing an element-wise iterative source steering (EISS) algorithm. In comparison with the IP-based SBSS algorithm, the proposed algorithm is computationally much more efficient, especially when the nonlinear expansion order is high and the length of the CTF filter is long. Meanwhile, its AEC performance is as good as that of IP-based SBSS.

Due to their intrinsic capabilities on parallel signal processing, optical neural networks (ONNs) have attracted extensive interests recently as a potential alternative to electronic artificial neural networks (ANNs) with reduced power consumption and low latency. Preliminary confirmation of the parallelism in optical computing has been widely done by applying the technology of wavelength division multiplexing (WDM) in the linear transformation part of neural networks. However, inter-channel crosstalk has obstructed WDM technologies to be deployed in nonlinear activation in ONNs. Here, we propose a universal WDM structure called multiplexed neuron sets (MNS) which apply WDM technologies to optical neurons and enable ONNs to be further compressed. A corresponding back-propagation (BP) training algorithm is proposed to alleviate or even cancel the influence of inter-channel crosstalk on MNS-based WDM-ONNs. For simplicity, semiconductor optical amplifiers (SOAs) are employed as an example of MNS to construct a WDM-ONN trained with the new algorithm. The result shows that the combination of MNS and the corresponding BP training algorithm significantly downsize the system and improve the energy efficiency to tens of times while giving similar performance to traditional ONNs.

Cognitive Radio Network (CRN) provides effective capabilities for resource allocation with the valuable spectrum resources in the network. It provides the effective allocation of resources to the unlicensed users or Secondary Users (SUs) to access the spectrum those are unused by the licensed users or Primary Users (Pus). This paper develops an Optimal Relay Selection scheme with the spectrum-sharing scheme in CRN. The proposed Cross-Layer Spider Swarm Shifting is implemented in CRN for the optimal relay selection with Spider Swarm Optimization (SSO). The shortest path is estimated with the data shifting model for the data transmission path in the CRN. This study examines a cognitive relay network (CRN) with interference restrictions imposed by a mobile end user (MU). Half-duplex communication is used in the proposed system model between a single primary user (PU) and a single secondary user (SU). Between the SU source and SU destination, an amplify and forward (AF) relaying mechanism is also used. While other nodes (SU Source, SU relays, and PU) are supposed to be immobile in this scenario, the mobile end user (SU destination) is assumed to travel at high vehicle speeds. The suggested method achieves variety by placing a selection combiner at the SU destination and dynamically selecting the optimal relay for transmission based on the greatest signal-to-noise (SNR) ratio. The performance of the proposed Cross-Layer Spider Swarm Shifting model is compared with the Spectrum Sharing Optimization with QoS Guarantee (SSO-QG). The comparative analysis expressed that the proposed Cross-Layer Spider Swarm Shifting model delay is reduced by 15% compared with SSO-QG. Additionally, the proposed Cross-Layer Spider Swarm Shifting exhibits the improved network performance of ~25% higher throughput compared with SSO-QG.

Stochastic differential equation (SDE in short) solvers find numerous applications across various fields. However, in practical simulations, we usually resort to using Ito-Taylor series-based methods like the Euler-Maruyama method. These methods often suffer from the limitation of fixed time scales and recalculations for different Brownian motions, which lead to computational inefficiency, especially in generative and sampling models. To address these issues, we propose a novel approach: learning a mapping between the solution of SDE and corresponding Brownian motion. This mapping exhibits versatility across different scales and requires minimal paths for training. Specifically, we employ the DeepONet method to learn a nonlinear mapping. And we also assess the efficiency of this method through simulations conducted at varying time scales. Additionally, we evaluate its generalization performance to verify its good versatility in different scenarios.

This work is motivated by the need of efficient numerical simulations of gas flows in the serpentine channels used in proton-exchange membrane fuel cells. In particular, we consider the Poisson problem in a 2D domain composed of several long straight rectangular sections and of several bends corners. In order to speed up the resolution, we propose a 0D model in the rectangular parts of the channel and a Finite Element resolution in the bends. To find a good compromise between precision and time consuming, the challenge is double: how to choose a suitable position of the interface between the 0D and the 2D models and how to control the discretization error in the bends. We shall present an \textit{a posteriori} error estimator based on an equilibrated flux reconstruction in the subdomains where the Finite Element method is applied. The estimates give a global upper bound on the error measured in the energy norm of the difference between the exact and approximate solutions on the whole domain. They are guaranteed, meaning that they feature no undetermined constants. (global) Lower bounds for the error are also derived. An adaptive algorithm is proposed to use smartly the estimator for aforementioned double challenge. A numerical validation of the estimator and the algorithm completes the work. \end{abstract}

A countable structure is indivisible if for every coloring with finite range there is a monochromatic isomorphic subcopy of the structure. Each indivisible structure $\mathcal{S}$ naturally corresponds to an indivisibility problem $\mathsf{Ind}\ \mathcal{S}$, which outputs such a subcopy given a presentation and coloring. We investigate the Weihrauch complexity of the indivisibility problems for two structures: the rational numbers $\mathbb{Q}$ as a linear order, and the equivalence relation $\mathscr{E}$ with countably many equivalence classes each having countably many members. We separate the Weihrauch degrees of both $\mathsf{Ind}\ \mathbb{Q}$ and $\mathsf{Ind}\ \mathscr{E}$ from several benchmark problems, showing in particular that $\mathsf{C}_\mathbb{N} \vert_\mathrm{W} \mathsf{Ind}\ \mathbb{Q}$ and hence $\mathsf{Ind}\ \mathbb{Q}$ is strictly weaker than the problem of finding an interval in which some color is dense for a given coloring of $\mathbb{Q}$; and that the Weihrauch degree of $\mathsf{Ind}\ \mathscr{E}_k$ is strictly between those of $\mathsf{SRT}^2_k$ and $\mathsf{RT}^2_k$, where $\mathsf{Ind}\ \mathcal{S}_k$ is the restriction of $\mathsf{Ind}\ \mathcal{S}$ to $k$-colorings.

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.