Discrete Space Research Articles

Rediscovery of numerical Lüscher's formula from the neural network* *Partly supported by the National Natural Science Foundation of China (12221005, 11935017, 12070131001, 12175239), (CRC 110 by DFG and NNSFC) and the National Key R&D Program of China (2020YFA0406400)

We present that by predicting the spectrum in discrete space from the phase shift in continuous space, the neural network can remarkably reproduce the numerical Lüscher's formula to a high precision. The model-independent property of the Lüscher's formula is naturally realized by the generalizability of the neural network. This exhibits the great potential of the neural network to extract model-independent relation between model-dependent quantities, and this data-driven approach could greatly facilitate the discovery of the physical principles underneath the intricate data.

On the stability and convergence of discontinuous Galerkin schemes for incompressible flows

Abstract The property that the velocity $\textbf{u}$ belongs to $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretization methods are uniformly stable in the $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$-norm. In this work, we establish that this is indeed the case for discontinuous Galerkin (DG) discretizations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretization is equivalent to the RadauIIA Implicit Runge–Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo–Nirenberg-type inequalities on DG spaces, which might be of independent interest.

Assessing and Improving the Robustness of Bayesian Evidential Learning in One Dimension for Inverting Time-Domain Electromagnetic Data: Introducing a New Threshold Procedure

Understanding the subsurface is of prime importance for many geological and hydrogeological applications. Geophysical methods offer an economical alternative for investigating the subsurface compared to costly borehole investigations. However, geophysical results are commonly obtained through deterministic inversion of data whose solution is non-unique. Alternatively, stochastic inversions investigate the full uncertainty range of the obtained models, yet are computationally more expensive. In this research, we investigate the robustness of the recently introduced Bayesian evidential learning in one dimension (BEL1D) for the stochastic inversion of time-domain electromagnetic data (TDEM). First, we analyse the impact of the accuracy of the numerical forward solver on the posterior distribution, and derive a compromise between accuracy and computational time. We also introduce a threshold-rejection method based on the data misfit after the first iteration, circumventing the need for further BEL1D iterations. Moreover, we analyse the impact of the prior-model space on the results. We apply the new BEL1D with a threshold approach on field data collected in the Luy River catchment (Vietnam) to delineate saltwater intrusions. Our results show that the proper selection of time and space discretization is essential for limiting the computational cost while maintaining the accuracy of the posterior estimation. The selection of the prior distribution has a direct impact on fitting the observed data and is crucial for a realistic uncertainty quantification. The application of BEL1D for stochastic TDEM inversion is an efficient approach, as it allows us to estimate the uncertainty at a limited cost.

AoI minimization of ambient backscatter-assisted EH-CRN with cooperative spectrum sensing

To evaluate the timeliness and freshness of information from energy-constrained devices, we minimize the weighted average age of information (AoI) of the ambient backscatter-assisted energy harvesting cognitive radio network (AB-EH-CRN), where secondary transmitters (STs) follow the non-orthogonal multiple access (NOMA) strategy to transmit status updates. The secondary receiver (SR) schedules the selected actions for STs to perform spectrum access, and receives status updates from them. We formulate the AoI minimization problem where the action selection and transmission power of STs are considered as variables. To exploit spectrum resources in the AB-EH-CRN with unknown signal-to-noise ratios (SNRs) at the STs, we propose the deep neural network (DNN)-based cooperative spectrum sensing (CSS), and adopt an energy threshold approach for STs to determine the energy allocation between spectrum sensing (SS) and packet transmission. Moreover, we address the AoI minimization problem by the proposed hybrid action and energy penalty deep deterministic policy gradient (HAEP-DDPG) algorithm, which adapts to the hybrid continuous and discrete action spaces. Simulation results validate the advantage of the proposed HAEP-DDPG algorithm compared with comparison schemes in terms of AoI.

Well-posedness and Tikhonov regularization of an inverse source problem for a parabolic equation with an integral constraint

Abstract We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d \mathbb{R}^{d} , d ≥ 1 d\geq 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.

An Improved Machine Learning Model for Pure Component Property Estimation

Information on the physicochemical properties of chemical species is an important prerequisite when performing tasks such as process design and product design. However, the lack of extensive data and high experimental costs hinder the development of prediction techniques for these properties. Moreover, accuracy and predictive capabilities still limit the scope and applicability of most property estimation methods. This paper proposes a new Gaussian process-based modeling framework that aims to manage a discrete and high-dimensional input space related to molecular structure representation with the group-contribution approach. A warping function is used to map discrete input into a continuous domain in order to adjust the correlation between different compounds. Prior selection techniques, including prior elicitation and prior predictive checking, are also applied during the building procedure to provide the model with more information from previous research findings. The framework is assessed using datasets of varying sizes for 20 pure component properties. For 18 out of the 20 pure component properties, the new models are found to give improved accuracy and predictive power in comparison with other published models, with and without machine learning.

Evolutionary Multimodal Multiobjective Optimization for Traveling Salesman Problems

Multimodal multiobjective optimization problems (MMOPs) are commonly seen in real-world applications. Many evolutionary algorithms have been proposed to solve continuous MMOPs. However, little effort has been made to solve combinatorial (or discrete) MMOPs. Searching for equivalent Pareto optimal solutions in the discrete decision space is challenging. Moreover, the true Pareto optimal solutions of a combinatorial MMOP are usually difficult to know, which has limited the development of its optimizer. In this paper, we first propose a test problem generator for multimodal multiobjective traveling salesman problems (MMTSPs). It can readily generate MMTSPs with known Pareto optimal solutions. Then we propose a novel evolutionary algorithm to solve MMTSPs. In our proposed algorithm, we develop two new edge assembly crossover operators, which are specialized in searching for superior solutions to MMTSPs. Moreover, the proposed algorithm uses a new environmental selection operator to maintain a good balance between the objective space diversity and decision space diversity. We compare our algorithm with five state-of-the-art designs. Experimental results convincingly show that our algorithm is powerful in solving MMTSPs.

A novel structure learning method of Bayesian networks based on the neighboring complete node ordering search

Learning the optimal structure of a Bayesian network (BN) from observational data has received considerable research attention. In most structure learning methods based on scoring and searching, the discrete space of BNs is used, and the scale of the network grows exponentially with the number of nodes n, that is, O(n!2Cn2). Any existing scoring function in the BN space has multiple peaks. Thus, when using heuristic algorithms, local optimum problems may occur while searching in this space. As the search process involves only a small part of the space, the final learning result is not always ideal. In this study, the scoring and searching task is implemented in the complete node ordering space, and a novel neighbor operation is proposed for improving learning accuracy. This technique reduces the search space to O(n!) and facilitates optimized learning of a structure. Furthermore, techniques for efficiency optimization are proposed, which address the problems of high computational complexity and repetitive calculation of family scores.

Multi-Attribute Decision-Making Method in Preventive Maintenance of Asphalt Pavement Based on Optimized Triangular Fuzzy Number

By choosing the right pavement maintenance plan, we can reduce resource utilization, reduce environmental pollution, and extend road life, which is important for improving social sustainability. At present, the selection of road maintenance programs mostly adopts multiple attribute decision-making (MADA), in particular, the analytic hierarchy process (AHP) is often used. However, this method needs to use expert scoring data, which leads to strong subjectivity and poor reliability. Therefore, it reduces the science of road maintenance scheme selection. In order to reduce the subjectivity of the score and obtain a more suitable road maintenance scheme, this paper applies a multi-criteria decision-making method that characterizes attribute information by triangular fuzzy numbers (TFN) in the discrete decision space. Firstly, we invite experts to score the importance of the selection of pavement preventive maintenance technical solutions with respect to the indicators affecting the selection of solutions. Secondly, the two indicators of similarity and reliability are used to quantitatively evaluate the indicators and programs, respectively. Finally, we compare the weighted programs according to the overall possibility degree of each program. In actual cases, the overall possibility degree of each scheme obtained by this method is 1.0002–0.0477, and the optimal solution is fog sealing technology. The decision-making model applied in this paper can be considered in multiple dimensions, which can scientifically reduce the subjectivity of expert scoring. The best maintenance plan can also be quickly obtained through the simple calculation method in this paper.

Optimal transport with nonlinear mobilities: A deterministic particle approximation result

Abstract We study the discretisation of generalised Wasserstein distances with nonlinear mobilities on the real line via suitable discrete metrics on the cone of N ordered particles, a setting which naturally appears in the framework of deterministic particle approximation of partial differential equations. In particular, we provide a Γ-convergence result for the associated discrete metrics as N → ∞ {N\to\infty} to the continuous one and discuss applications to the approximation of one-dimensional conservation laws (of gradient flow type) via the so-called generalised minimising movements, proving a convergence result of the schemes at any given discrete time step τ > 0 {\tau>0} . This the first work of a series aimed at sheding new lights on the interplay between generalised gradient-flow structures, conservation laws, and Wasserstein distances with nonlinear mobilities.

π-Light: Programmatic Interpretable Reinforcement Learning for Resource-Limited Traffic Signal Control

The recent advancements in Deep Reinforcement Learning (DRL) have significantly enhanced the performance of adaptive Traffic Signal Control (TSC). However, DRL policies are typically represented by neural networks, which are over-parameterized black-box models. As a result, the learned policies often lack interpretability, and cannot be deployed directly in the real-world edge hardware due to resource constraints. In addition, the DRL methods often exhibit limited generalization performance, struggling to generalize the learned policy to other geographical regions. These factors limit the practical application of learning-based approaches. To address these issues, we suggest the use of an inherently interpretable program for representing the control policy. We present a new approach, Programmatic Interpretable reinforcement learning for traffic signal control (π-light), designed to autonomously discover non-differentiable programs. Specifically, we define a Domain Specific Language (DSL) and transformation rules for constructing programs, and utilize Monte Carlo Tree Search (MCTS) to find the optimal program in a discrete space. Extensive experiments demonstrate that our method consistently outperforms baseline approaches. Moreover, π-Light exhibits superior generalization capabilities compared to DRL, enabling training and evaluation across intersections from different cities. Finally, we analyze how the learned program policies can directly deploy on edge devices with extremely limited resources.

G2P-DDM: Generating Sign Pose Sequence from Gloss Sequence with Discrete Diffusion Model

The Sign Language Production (SLP) project aims to automatically translate spoken languages into sign sequences. Our approach focuses on the transformation of sign gloss sequences into their corresponding sign pose sequences (G2P). In this paper, we present a novel solution for this task by converting the continuous pose space generation problem into a discrete sequence generation problem. We introduce the Pose-VQVAE framework, which combines Variational Autoencoders (VAEs) with vector quantization to produce a discrete latent representation for continuous pose sequences. Additionally, we propose the G2P-DDM model, a discrete denoising diffusion architecture for length-varied discrete sequence data, to model the latent prior. To further enhance the quality of pose sequence generation in the discrete space, we present the CodeUnet model to leverage spatial-temporal information. Lastly, we develop a heuristic sequential clustering method to predict variable lengths of pose sequences for corresponding gloss sequences. Our results show that our model outperforms state-of-the-art G2P models on the public SLP evaluation benchmark. For more generated results, please visit our project page: https://slpdiffusier.github.io/g2p-ddm.

Peer Learning: Learning Complex Policies in Groups from Scratch via Action Recommendations

Peer learning is a novel high-level reinforcement learning framework for agents learning in groups. While standard reinforcement learning trains an individual agent in trial-and-error fashion, all on its own, peer learning addresses a related setting in which a group of agents, i.e., peers, learns to master a task simultaneously together from scratch. Peers are allowed to communicate only about their own states and actions recommended by others: "What would you do in my situation?". Our motivation is to study the learning behavior of these agents. We formalize the teacher selection process in the action advice setting as a multi-armed bandit problem and therefore highlight the need for exploration. Eventually, we analyze the learning behavior of the peers and observe their ability to rank the agents' performance within the study group and understand which agents give reliable advice. Further, we compare peer learning with single agent learning and a state-of-the-art action advice baseline. We show that peer learning is able to outperform single-agent learning and the baseline in several challenging discrete and continuous OpenAI Gym domains. Doing so, we also show that within such a framework complex policies from action recommendations beyond discrete action spaces can evolve.

Iterative Token Evaluation and Refinement for Real-World Super-resolution

Real-world image super-resolution (RWSR) is a long-standing problem as low-quality (LQ) images often have complex and unidentified degradations. Existing methods such as Generative Adversarial Networks (GANs) or continuous diffusion models present their own issues including GANs being difficult to train while continuous diffusion models requiring numerous inference steps. In this paper, we propose an Iterative Token Evaluation and Refinement (ITER) framework for RWSR, which utilizes a discrete diffusion model operating in the discrete token representation space, i.e., indexes of features extracted from a VQGAN codebook pre-trained with high-quality (HQ) images. We show that ITER is easier to train than GANs and more efficient than continuous diffusion models. Specifically, we divide RWSR into two sub-tasks, i.e., distortion removal and texture generation. Distortion removal involves simple HQ token prediction with LQ images, while texture generation uses a discrete diffusion model to iteratively refine the distortion removal output with a token refinement network. In particular, we propose to include a token evaluation network in the discrete diffusion process. It learns to evaluate which tokens are good restorations and helps to improve the iterative refinement results. Moreover, the evaluation network can first check status of the distortion removal output and then adaptively select total refinement steps needed, thereby maintaining a good balance between distortion removal and texture generation. Extensive experimental results show that ITER is easy to train and performs well within just 8 iterative steps.

Probabilistic Offline Policy Ranking with Approximate Bayesian Computation

In practice, it is essential to compare and rank candidate policies offline before real-world deployment for safety and reliability. Prior work seeks to solve this offline policy ranking (OPR) problem through value-based methods, such as Off-policy evaluation (OPE). However, they fail to analyze special case performance (e.g., worst or best cases), due to the lack of holistic characterization of policies’ performance. It is even more difficult to estimate precise policy values when the reward is not fully accessible under sparse settings. In this paper, we present Probabilistic Offline Policy Ranking (POPR), a framework to address OPR problems by leveraging expert data to characterize the probability of a candidate policy behaving like experts, and approximating its entire performance posterior distribution to help with ranking. POPR does not rely on value estimation, and the derived performance posterior can be used to distinguish candidates in worst-, best-, and average-cases. To estimate the posterior, we propose POPR-EABC, an Energy-based Approximate Bayesian Computation (ABC) method conducting likelihood-free inference. POPR-EABC reduces the heuristic nature of ABC by a smooth energy function, and improves the sampling efficiency by a pseudo-likelihood. We empirically demonstrate that POPR-EABC is adequate for evaluating policies in both discrete and continuous action spaces across various experiment environments, and facilitates probabilistic comparisons of candidate policies before deployment.

Learning Ultrametric Trees for Optimal Transport Regression

Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the space. However, measures supported on trees admit a closed-form optimal transport that can be computed in linear time. In this paper, we aim to find an optimal tree structure for a given discrete metric space so that the tree-Wasserstein distance approximates the optimal transport distance in the original space. One of our key ideas is to cast the problem in ultrametric spaces. This helps us optimize over the space of ultrametric trees --- a mixed-discrete and continuous optimization problem --- via projected gradient decent over the space of ultrametric matrices. During optimization, we project the parameters to the ultrametric space via a hierarchical minimum spanning tree algorithm, equivalent to the closest projection to ultrametrics under the supremum norm. Experimental results on real datasets show that our approach outperforms previous approaches (e.g. Flowtree, Quadtree) in approximating optimal transport distances. Finally, experiments on synthetic data generated on ground truth trees show that our algorithm can accurately uncover the underlying trees.

Multiagent Gumbel MuZero: Efficient Planning in Combinatorial Action Spaces

AlphaZero and MuZero have achieved state-of-the-art (SOTA) performance in a wide range of domains, including board games and robotics, with discrete and continuous action spaces. However, to obtain an improved policy, they often require an excessively large number of simulations, especially for domains with large action spaces. As the simulation budget decreases, their performance drops significantly. In addition, many important real-world applications have combinatorial (or exponential) action spaces, making it infeasible to search directly over all possible actions. In this paper, we extend AlphaZero and MuZero to learn and plan in more complex multiagent (MA) Markov decision processes, where the action spaces increase exponentially with the number of agents. Our new algorithms, MA Gumbel AlphaZero and MA Gumbel MuZero, respectively without and with model learning, achieve superior performance on cooperative multiagent control problems, while reducing the number of environmental interactions by up to an order of magnitude compared to model-free approaches. In particular, we significantly improve prior performance when planning with much fewer simulation budgets. The code and appendix are available at https://github.com/tjuHaoXiaotian/MA-MuZero.

Optimizing the Finishing Treatment of Potato Harvester Transmission Shafts

The potato harvester’s distinctive specifications include the spatially segregated drive of its units from an external engine, operating under stochastic conditions. This setup complicates the loading of its components, including transmission shafts, and creates the prerequisites for decreased durability, leading to an upsurge in fatigue failures. To counteract this process, transmission shafts undergo a special treatment aimed at mitigating concentrated defects. (Research purpose) The purpose of this research is to formulate an objective function for controlling the finishing treatment of transmission shafts in a potato harvester. This function is designed to maximize potato yield while monitoring concentrated defects. (Materials and methods) The implementation of the method requires equipment capable of being controlled through a technical vision channel. The control scheme employs mathematical tools for describing discrete and continuous random variables, Markov processes with discrete time and continuous state space, the maximum likelihood method, and methods of numerical optimization in multifactor space. (Results and discussion) The research has yielded formulas for calculating the probabilities of successful and unsuccessful outcomes of finishing treatment based on the criterion of concentrated defect presence. The paper provides the results of testing the proposed algorithm for predicting the yield of viable products using the proposed method based on interim measurements of defect characteristics during the treatment process. Additionally, a scheme is proposed for integrating the obtained results into the production process of manufacturing transmission shafts for potato harvesters. (Conclusions) An algorithm has been proposed to control the finishing treatment of the transmission shaft based on a predictive assessment of the probability of manufacturing a product that adheres to the specified technical criteria regarding geometry and surface cleanliness. The predictive probability of achieving a surface with the required cleanliness without exceeding permissible defect limits is considered as the objective function. The solution to the problem is achieved through the selection of appropriate technological parameters.

Variable selection in Bayesian multiple instance regression using shotgun stochastic search

In multiple instance learning (MIL), a bag represents a sample that has a set of instances, each of which is described by a vector of explanatory variables, but the entire bag only has one label/response. Though many methods for MIL have been developed to date, few have paid attention to interpretability of models and results. The proposed Bayesian regression model stands on two levels of hierarchy, which transparently show how explanatory variables explain and instances contribute to bag responses. Moreover, two selection problems are simultaneously addressed; the instance selection to find out the instances in each bag responsible for the bag response, and the variable selection to search for the important covariates. To explore a joint discrete space of indicator variables created for selection of both explanatory variables and instances, the shotgun stochastic search algorithm is modified to fit in the MIL context. Also, the proposed model offers a natural and rigorous way to quantify uncertainty in coefficient estimation and outcome prediction, which many modern MIL applications call for. The simulation study shows the proposed regression model can select variables and instances with high performance (AUC greater than 0.86), thus predicting responses well. The proposed method is applied to the musk data for prediction of binding strengths (labels) between molecules (bags) with different conformations (instances) and target receptors. It outperforms all existing methods, and can identify variables relevant in modeling responses.