Discrete Space Research Articles

Parameter estimation for Gibbs distributions

A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We consider sampling problems coming from Gibbs distributions , which are families of probability distributions over a discrete space \(\Omega\) with probability mass function of the form \(\mu^{\Omega}_{\beta}(\omega)\propto e^{\beta H(\omega)}\) for \(\beta\) in an interval \([\beta_{\min},\beta_{\max}]\) and \(H(\omega)\in\{0\}\cup[1,n]\) . Two important parameters are the partition function , which is the normalization factor \(Z(\beta)=\sum_{\omega\in\Omega}e^{\beta H(\omega)}\) , and the vector of preimage counts \(c_{x}=|H^{-1}(x)|\) . We develop black-box sampling algorithms to estimate the counts using roughly \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions, where \(q=\log\frac{Z(\beta_{\max})}{Z(\beta_{\min})}\) (ignoring some second-order terms and parameters). We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we estimate all values of the partition function using \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions. This improves over a prior algorithm of Huber (2015) which computes a single point estimate \(Z(\beta_{\max})\) and which uses a slightly larger amount of samples. We show matching lower bounds, demonstrating this complexity is optimal as a function of \(n\) and \(q\) up to logarithmic terms.

Positive Fitted Finite Volume Method for Semilinear Parabolic Systems on Unbounded Domain

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented.

Adaptive unified gas-kinetic scheme for diatomic gases with rotational and vibrational nonequilibrium

Multiscale nonequilibrium physics at large variations of local Knudsen number are encountered in applications of aerospace engineering and micro-electro-mechanical systems, such as high-speed flying vehicles and low pressure of the encapsulation. An accurate description of flow physics in all flow regimes within a single computation requires a genuinely multiscale method. The adaptive unified gas-kinetic scheme (AUGKS) is developed for such multiscale flow simulation. The AUGKS applies discretized velocity space to accurately capture the non-equilibrium physics in the multiscale UGKS, and adaptively employs continuous distribution functions following Chapman–Enskog expansion to efficiently recover near-equilibrium flow region in GKS. The UGKS and GKS are dynamically connected at the cell interface through the fluxes from the discretized and continuous gas distribution functions, which avoids any buffer zone between them. In this study, the AUGKS with rotation and vibration non-equilibrium is developed based on a multiple temperature relaxation model. The real gas effect in different flow regimes has been properly captured. To capture aerodynamic heating accurately, the heat flux modifications from the rotation and vibration modes are also included in the current scheme. Unstructured discrete particle velocity space is adopted to further improve the computational performance of the AUGKS. Numerical tests, including Sod tube, normal shock structure, high-speed flow around the two-dimensional cylinder and three-dimensional sphere and space vehicles, and an unsteady nozzle plume flow from the continuum flow to the background vacuum, have been conducted to validate the current scheme. In comparison with the original UGKS, the current scheme speeds up the computation, reduces the memory requirement, and maintains the equivalent accuracy for multiscale flow simulation, which provides an effective tool for nonequilibrium flow simulations, especially for the flows at low and medium speed.

Explainable time series anomaly detection using masked latent generative modeling

We present a novel time series anomaly detection method that achieves excellent detection accuracy while offering a superior level of explainability. Our proposed method, TimeVQVAE-AD, leverages masked generative modeling adapted from the cutting-edge time series generation method known as TimeVQVAE. The prior model is trained on the discrete latent space of a time–frequency domain. Notably, the dimensional semantics of the time–frequency domain are preserved in the latent space, enabling us to compute anomaly scores across different frequency bands, which provides a better insight into the detected anomalies. Additionally, the generative nature of the prior model allows for sampling likely normal states for detected anomalies, enhancing the explainability of the detected anomalies through counterfactuals. Our experimental evaluation on the UCR Time Series Anomaly archive demonstrates that TimeVQVAE-AD significantly surpasses the existing methods in terms of detection accuracy and explainability. We provide our implementation on GitHub: https://github.com/ML4ITS/TimeVQVAE-AnomalyDetection.

On the convergence of a linearly implicit finite element method for the nonlinear Schrödinger equation

AbstractWe consider a model initial‐ and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second‐order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal‐order error estimates in the and the norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a Courant‐Friedrichs‐Lewy (CFL) condition between the space mesh size and the time step sizes.

Discrete Space Deep Reinforcement Learning Algorithm Based on Support Vector Machine Recursive Feature Elimination

Algorithms for training agents with experience replay have advanced in several domains, primarily because prioritized experience replay (PER) developed from the double deep Q-network (DDQN) in deep reinforcement learning (DRL) has become a standard. PER-based algorithms have achieved significant success in the image and video domains. However, the exceptional results observed in images and videos are not as effective in many domains with simple action spaces and relatively small states, particularly in discrete action spaces with sparse rewards. Moreover, most advanced techniques may improve sampling efficiency using deep learning algorithms rather than reinforcement learning. However, there is growing evidence that deep learning algorithms cannot generalize during training. Therefore, this study proposes an algorithm suitable for discrete action space environments that uses the sample efficiency of PER based on DDQN but incorporates support vector machine recursive feature elimination (SVM-RFE) without enhancing the sampling efficiency through deep learning algorithms. The proposed algorithm exhibited considerable performance improvements in classical OpenAI Gym environments that did not use images or videos as inputs. In particular, simple discrete space environments with reflection symmetry, such as Cart–Pole, exhibited a faster and more stable learning process. These results suggest that the application of SVM-RFE, which leverages the orthogonality of support vector machines (SVMs) across learning patterns, can be appropriate when the data in the reinforcement learning environment demonstrate symmetry.

Random pore model insights into structural and operational parameters for hydrogen-based iron oxide reduction

The hydrogen-based direct reduction of iron oxide (DRI) provides a sustainable approach to cleaner ironmaking. In this research, for analyzing hydrogen-based DRI in pellet scale, the random pore model (RPM), the most sophisticated non-catalytic gas-solid model, is applied to determine the effects of structural parameters, including pore-size parameter (ψ), Thiele modulus (ϕ), and product layer resistance (β), along with operational parameters such as pressure, temperature, and gas composition. The PDE equations of RPM were solved numerically using the method of lines together with a finite-difference discretization in space. The effects of parameters ψ and ϕ were investigated, revealing that the reaction rate increases with larger pores and decreases with higher values of ϕ over length and time. Kinetics resemble the unreacted shrinking core model (USCM) for small particles, while pore structure dominates for large particles. Parameter determination is influenced by ϕ changes, shifting control from chemical kinetics to diffusion, with operating conditions impacting reaction rate enhancement. Investigations revealed that as ϕ increases from 0.454 to 1.454, a 6 % conversion gradient forms from the pellet center to the surface. Additionally, at 700°C, reducing the temperature by 100°C slows conversion by 15.45 minutes while increasing it accelerates conversion by 4.55 minutes.

The [formula omitted]-Adic Schrödinger equation and the two-slit experiment in quantum mechanics

p-Adic quantum mechanics is constructed from the Dirac–von Neumann axioms identifying quantum states with square-integrable functions on the N-dimensional p-adic space, QpN. This choice is equivalent to the hypothesis of the discreteness of the space. The time is assumed to be a real variable. The p-adic quantum mechanics is motivated by the question: what happens with the standard quantum mechanics if the space has a discrete nature? The time evolution of a quantum state is controlled by a nonlocal Schrödinger equation obtained from a p-adic heat equation by a temporal Wick rotation. This p-adic heat equation describes a particle performing a random motion in QpN. The Hamiltonian is a nonlocal operator; thus, the Schrödinger equation describes the evolution of a quantum state under nonlocal interactions. In this framework, the Schrödinger equation admits complex-valued plane wave solutions, which we interpret as p-adic de Broglie waves. These mathematical waves have all wavelength p−1. In the p-adic framework, the double-slit experiment cannot be explained using the interference of the de Broglie waves. The wavefunctions can be represented as convergent series in the de Broglie waves, but the p-adic de Broglie waves are just mathematical objects. Only the square of the modulus of a wave function has a physical meaning as a time-dependent probability density. These probability densities exhibit interference patterns similar to the ones produced by ‘quantum waves’. In the p-adic framework, in the double-slit experiment, each particle goes through one slit only. The p-adic quantum mechanics is an analog (or model) of the standard one under the hypothesis of the existence of a Planck length. The precise connection between these two theories is an open problem. Finally, we propose the conjecture that the classical de Broglie wave-particle duality is a manifestation of the discreteness of space–time.

Exploring discrete space–time models for information transfer: Analogies from mycelial networks to the cosmic web

Fungal mycelium networks are large scale biological networks along which nutrients, metabolites flow. Recently, we discovered a rich spectrum of electrical activity in mycelium networks, including action-potential spikes and trains of spikes. Reasoning by analogy with animals and plants, where travelling patterns of electrical activity perform integrative and communicative mechanisms, we speculated that waves of electrical activity transfer information in mycelium networks. Using a new discrete space–time model with emergent radial spanning-tree topology, hypothetically comparable mycelial morphology and physically comparable information transfer, we provide physical arguments for the use of such a model, and by considering growing mycelium network by analogy with growing network of matter in the cosmic web, we develop mathematical models and theoretical concepts to characterise the parameters of the information transfer.

A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem

In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests on (1) discrete spaces that are spanned by vectors of polynomials whose components are attached to mesh entities and (2) discrete operators obtained mimicking integration by parts formulas. We provide an in-depth study of the algebraic properties of the local complex, showing that it is exact on mesh elements with trivial topology. The new DDR complex is used to design a numerical scheme for the approximation of biharmonic problems, for which we provide detailed stability and convergence analyses. Numerical experiments complete the theoretical results.

ZeroGrads: Learning Local Surrogates for Non-Differentiable Graphics

Gradient-based optimization is now ubiquitous across graphics, but unfortunately can not be applied to problems with undefined or zero gradients. To circumvent this issue, the loss function can be manually replaced by a "surrogate" that has similar minima but is differentiable. Our proposed framework, ZeroGrads , automates this process by learning a neural approximation of the objective function, which in turn can be used to differentiate through arbitrary black-box graphics pipelines. We train the surrogate on an actively smoothed version of the objective and encourage locality, focusing the surrogate's capacity on what matters at the current training episode. The fitting is performed online, alongside the parameter optimization, and self-supervised, without pre-computed data or pre-trained models. As sampling the objective is expensive (it requires a full rendering or simulator run), we devise an efficient sampling scheme that allows for tractable run-times and competitive performance at little overhead. We demonstrate optimizing diverse non-convex, non-differentiable black-box problems in graphics, such as visibility in rendering, discrete parameter spaces in procedural modelling or optimal control in physics-driven animation. In contrast to other derivative-free algorithms, our approach scales well to higher dimensions, which we demonstrate on problems with up to 35k interlinked variables.

Targeted materials discovery using Bayesian algorithm execution

Rapid discovery and synthesis of future materials requires intelligent data acquisition strategies to navigate large design spaces. A popular strategy is Bayesian optimization, which aims to find candidates that maximize material properties; however, materials design often requires finding specific subsets of the design space which meet more complex or specialized goals. We present a framework that captures experimental goals through straightforward user-defined filtering algorithms. These algorithms are automatically translated into one of three intelligent, parameter-free, sequential data collection strategies (SwitchBAX, InfoBAX, and MeanBAX), bypassing the time-consuming and difficult process of task-specific acquisition function design. Our framework is tailored for typical discrete search spaces involving multiple measured physical properties and short time-horizon decision making. We demonstrate this approach on datasets for TiO2 nanoparticle synthesis and magnetic materials characterization, and show that our methods are significantly more efficient than state-of-the-art approaches. Overall, our framework provides a practical solution for navigating the complexities of materials design, and helps lay groundwork for the accelerated development of advanced materials.

Fast, approximation-free molecular simulation of the SPC/Fw water model using non-reversible Markov chains

In a world made of atoms, computer simulations of molecular systems such as proteins in water play an enormous role in science. Software packages for molecular simulation have been developed for decades. They all discretize Hamilton’s equations of motion and treat long-range potentials through cutoffs or discretization of reciprocal space. This introduces severe approximations and artifacts that must be controlled algorithmically. Here, we bring to fruition a paradigm for molecular simulation that relies on modern concepts in statistics to explore the thermodynamic equilibrium with an exact and efficient non-reversible Markov process. It is free of all discretizations, approximations, and cutoffs. We explicitly demonstrate that this approach reaches a break-even point with traditional molecular simulation performed at high precision, but without any of its approximations. We stress the potential of our paradigm for crucial applications in biophysics and other fields, and as a practical approach to molecular simulation. We set out a strategy to reach our goal of rigorous molecular simulation.

Scalar curvature for metric spaces: Defining curvature for quantum gravity without coordinates

Geometrical properties of spacetime are difficult to study in nonperturbative approaches to quantum gravity like causal dynamical triangulations (CDT), where one uses simplicial manifolds to define the gravitational path integral, instead of Riemannian manifolds. In particular, in CDT one only relies on two mathematical tools, a distance measure and a volume measure. In this paper, we define a notion of scalar curvature, for metric spaces endowed with a volume measure or a random walk, without assuming nor using notions of tensor calculus. Furthermore, we directly define the Ricci scalar, without the need of defining and computing the Riemann or the Ricci tensor . For this, we make use of quantities, like the surface of a geodesic sphere, or the return probability of scalar diffusion processes, that can be computed in these metric spaces, as in a Riemannian manifold, where they receive scalar curvature contributions. Our definitions recover the classical results of scalar curvature when the sets are Riemannian manifolds. We propose two methods to compute the scalar curvature in these spaces, and we compare their features in natural implementations in discrete spaces. The defined generalized scalar curvatures are easily implemented on discrete spaces, like graphs. We present the results of our definitions on random triangulations of a 2D sphere and plane. Additionally, we show the results of our generalized scalar curvatures on the quantum geometries of 2D CDT, where we find that all our definitions indicate a flat ground state of the gravitational path integral. Published by the American Physical Society 2024

Research on Leak Detection of Low-Pressure Gas Pipelines in Buildings Based on Improved Variational Mode Decomposition and Robust Kalman Filtering.

Aiming at the complex characteristics of negative pressure waves in low-pressure pipelines inside of buildings, we proposed an estimation method of pressure fluctuation trends based on the robust Kalman filter and the improved VMD, which can be used for leakage detection. The reconstructed baseline signal can accurately describe the fluctuation trend of the negative pressure wave after the pressure drop, and quantitatively express the characteristic difference between the leakage condition and the gas usage condition. The robust Kalman filter was used to estimate the pressure fluctuations. The parameters of VMD were adaptively calculated based on the WAA and discrete scale space. The trend components contained in the IMFs were separated by a reconstruction based on the Fourier series. Based on the simulation signal, the method can accurately restore the trend component contained in the complex pressure signal. Based on the actual signals, the accuracy of small leakage detection is 96.7% and the accuracy of large leakage detection is 73.3%.

Influence of control parameters on the stabilization of an Euler-Bernoulli flexible beam

In this work, we numerically study the influence of control parameters on the stabilization of a flexible Euler-Bernoulli beam fixed at one end and subjected at the other end to a force control and a punctual moment control proportional respectively to velocity and rotation velocity. First, we analyze the displacement stabilization and the asymptotic behavior of the beam energy using a stable numerical scheme, resulting from the Crank-Nicholson algorithm for time discretization and the finite element method based on the approximation by Hermite's cubic polynomial functions, for discretization in space. Then, by means of the finite element method, we represent the spectrum of the operator associated with this beam problem and we carry out a qualitative study of thelocus of the eigenvalues according to the positive control parameters. From these studies we conclude that rotation velocity control has more effect on the stabilization of the beam compared to velocity control. Finally, this result is confirmed by a sensitivity study on the control parameters involved in the stabilization of the beam.

Aging suppression in Multistrip Multigap Resistive Plate Chambers for high counting rate experiments

A long term operation of Multi-Strip Multi-Gap Resistive Plate Chambers (MSMGRPC) with gas mixtures based on C2H2F4 and SF6 leads to aging effects observed as depositions on the surface of the resistive electrodes, especially in the region of the spacers used for defining the gas gaps between the resistive electrodes. The observed higher noise rate and dark current has a negative impact on the performance of the chamber and leads to an increase of the data volume in a free running readout mode operation. MSMGRPC prototypes designed with a direct gas flow through the gas gaps and minimization of the number of spacers in the active area were developed as mitigation solution. Three new prototypes of different granularities were assembled using fishing line as spacers and investigated for aging effects. Although a significant reduction in the dark current and dark counting rate was evidenced, noise rate localized around the fishing line spacers, even though reduced, still remains. In this paper, a new generation of direct flow chambers based on discrete spacers is presented. The results of their aging investigations show that, even at lower gas flows, the aging effects become negligible.

Gradient-based optimization for quantum architecture search

Quantum Architecture Search (QAS) has shown significant promise in designing quantum circuits for Variational Quantum Algorithms (VQAs). However, existing QAS algorithms primarily explore circuit architectures within a discrete space, which is inherently inefficient. In this paper, we propose a Gradient-based Optimization for Quantum Architecture Search (GQAS), which leverages a circuit encoder, decoder, and predictor. Initially, the encoder embeds circuit architectures into a continuous latent representation. Subsequently, a predictor utilizes this continuous latent representation as input and outputs an estimated performance for the given architecture. The latent representation is then optimized through gradient descent within the continuous latent space based on the predicted performance. The optimized latent representation is finally mapped back to a discrete architecture via the decoder. To enhance the quality of the latent representation, we pre-train the encoder on a substantial dataset of circuit architectures using Self-Supervised Learning (SSL). Our simulation results on the Variational Quantum Eigensolver (VQE) indicate that our method outperforms the current Differentiable Quantum Architecture Search (DQAS).

Intelligent decision-making for a “Three-Variable” frequency-hopping pattern based on OC-CDRL

The frequency hopping pattern of the existing frequency hopping communication system is not designed according to the electromagnetic interference environment, resulting in blind anti-jamming. Therefore, to address this problem, a “three-variable” frequency-hopping pattern is proposed, where the frequency, hopping rate, and instantaneous bandwidth of the frequency-hopping signal vary randomly based on the background electromagnetic interference. The decision-making problem of the “three-variable” frequency-hopping pattern is modeled as a Markov decision process (MDP) by constructing the state-action-reward tuple. The designed frequency varies continuously within a small frequency band selected from a pseudo-random sequence to alleviate the problem of dimension explosion in decision-making. At the same time, discrete values for the hopping rate and instantaneous bandwidth are designed. To solve this MDP problem efficiently, a combined deep reinforcement learning algorithm (OC-CDRL) based on optimistic exploration and conservative estimation is proposed, which combines the features of TD3 and D3QN algorithms and designs the corresponding states, actions, and rewards to deal with continuous and discrete action spaces, respectively. To address the problem that the D3QN algorithm tends to fall into local optimal solutions, an optimistic exploration strategy (OES) for action selection is proposed to improve the degree of exploration. Moreover, the loss function is improved by conservatively estimating state–action pairs outside the experience replay buffer, reducing the overestimation of the optimistic action-value function and increasing the stability and convergence of the algorithm. Comparative simulation results of the algorithms in different electromagnetic interference environments show that the OC-CDRL algorithm effectively avoids most regions with higher interference and has better adaptability and anti-jamming capability.

Microscopic simulation of bicycle traffic flow incorporating cyclists’ heterogeneous dynamics and non-lane-based movement strategies

Cycling as a mode of transport is on an upward trend as a low-emission alternative to driving in urbanized areas nowadays. With the increasing number of cyclists, it is of great importance to assess the capacity of cycling infrastructure in practice. Simulation models are useful tools to investigate bicycle flow performance considering cyclists’ distinct moving behaviors. However, existing bicycle simulation models are restricted by either space discretization, lane-based setup, adaptation from models for car traffic, or complicated calibration requirement in a force-based environment. In addition, cyclists’ decision-making ability in the operational-level cycling behavior are not well-captured in these models. This paper proposes a comprehensible microscopic bicycle simulation model which includes a detailed decision-making process and the ability to simulate continuous-space lateral movement. The model consists of three levels, maneuver decision, movement planning, and physical acceleration. It is able to simulate bicycle flow dynamics in undersaturated traffic conditions on an exclusive bike path. As we do not intend to show the empirical validity of the proposed model, the simulation experiment aims at verifying the model and exploring bicycle flow performance in various scenarios by estimating the fundamental diagrams (FDs). The effect of different path widths on bicycle flow capacity is first explored. Other behavioral factors, including desired speed heterogeneity, overtaking incentive, and safety region size perceived by cyclists, which can potentially influence the shape of the FD are also tested. The model can be further extended to simulate relatively complex cycling behavior with cooperative and anticipative strategies and investigate bicycle flow characteristics in congested traffic conditions.