Discrete Space Research Articles

A hybrid heuristic-reinforcement learning-based real-time control model for residential behind-the-meter PV-battery systems

The behind-the-meter (BTM) energy management problem has recently attracted a lot of attention due to the increase in the number of residential photovoltaic (PV)-battery energy storage systems (BESSs). In this work, the use of deep reinforcement learning (DRL) combined with a novel heuristic model for real-time control of home batteries is investigated. The control problem is formulated as a finite Markov Decision Process with discrete time steps, where a proximal policy optimization (PPO) algorithm is employed to train the DRL agent with discrete action space. The agent is trained using real-world measured data to learn the policy for sequential charge/discharge tasks, aiming to minimize daily electricity costs. The battery power is calculated using an innovative heuristic model considering the agent's decision and the battery's available capacity, ensuring demand-supply balance through PV self-consumption and load demand shifting. The performance of the model is evaluated by comparing it to four RL agents and two benchmark models based on rule-based and scenario-based stochastic optimization strategies. The results confirm that the presented model outperforms its counterparts, offering €80.38 savings on electricity bills over 46 days of the test data set. This figure exceeds the savings of the rule-based and stochastic models by €15.64 and €19.38, respectively.

Study of the convergence of the Meshless Lattice Boltzmann Method in Taylor–Green, annular channel and a porous medium flows

The Meshless Lattice Boltzmann Method (MLBM) is a numerical tool that relieves the standard Lattice Boltzmann Method (LBM) from regular lattices and, at the same time, decouples space and velocity discretizations. In this study, we investigate the numerical convergence of MLBM in three benchmark tests: the Taylor–Green vortex, annular (bent) channel and a porous medium flow. We compare our MLBM results to LBM and to the analytical solution of the Navier–Stokes equation. We investigate the method’s convergence in terms of the discretization parameter, the interpolation order, and the LBM streaming distance refinement. We observe that MLBM outperforms LBM in terms of the error value for the same number of nodes discretizing the domain. We find that LBM errors at a given streaming distance δx and timestep length δt are the asymptotic lower bounds of MLBM errors with the same streaming distance and timestep length. Finally, we suggest an expression for the MLBM error that consists of the LBM error and other terms related to the semi-Lagrangian nature of the discussed method itself.

Differentiate data by higher-order structures

The comparison of two objects is a topic that exists widely in different research areas. In this study, the network distance is combined with the k-nearest neighbour network sequence to construct the topological structural distance (TSD). This indicator is used to quantitatively describe the difference between two sets of points in any metric structure. Several datasets were tested to demonstrate the effectiveness of this indicator. Calculations show that TSD can compare the intrinsic structure of low-dimensional and high-dimensional datasets and is not limited by dimension. In addition, TSD can also be used to characterize sets of points in discrete space, such as comparing the structure of different networks. The examples in this study show that TSD is a universal method for analyzing the intrinsic structure of data.

Dual-modality learning and transformer-based approach for high-quality vector font generation

Vector fonts, serving as the fundamental format of fonts, play a significant role in modern media. Described by a set of mathematical equations, vector fonts enable style modifications by adjusting drawing parameters, making them favored by font artists and designers. Due to the non-structural nature of vector font data, the task of vector font generation resembles sequence generation. Existing methods, limited in handling long sequences, are only capable of synthesizing simple character vector fonts. In this paper, we propose a dual-modal learning strategy to convert raster glyph images to vector glyph images in an end-to-end manner. Specifically, by employing vector quantization, we comprehensively utilize the dual-modal information of vector fonts. We quantize image and sequence modal features using a shared codebook, mapping them to the same discrete space and aligning them. Through aligned features, we reconstruct raster glyph images and vector glyph images. During the transformation process of vector glyph data, we redesign the Transformer module for vector glyph data, leveraging multiple stacked sliding window attention mechanisms to model local and global information. By integrating reversible residuals with attention and feedforward layers within the Transformer module, we enhance the model’s capability and stability in handling long sequence data without sacrificing accuracy. Finally, we perform cross-modal model distillation to obtain the model’s backbone network. We further refine the backbone network using a differentiable rasterizer to minimize error accumulation during sequence generation. Qualitative and quantitative results demonstrate that our method achieves high-quality synthesis results in complex Chinese character glyphs. The synthesized vector fonts can be easily converted into TrueType fonts for practical use, holding valuable applications for anyone interested in personalized vector font styles.

On improving the efficiency of ADER methods

The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE context, we investigate several strategies to improve this approach. Our initial emphasis is on the order of accuracy of the method in connection with the polynomial discretization of the weak formulation. We demonstrate that precise choices lead to higher-order convergences in comparison to the existing literature. Then, we put ADER methods into a Deferred Correction (DeC) formalism. This allows to determine the optimal number of iterations, which is equal to the formal order of accuracy of the method, and to introduce efficient p-adaptive modifications. These are defined by matching the order of accuracy achieved and the degree of the polynomial reconstruction at each iteration. We provide analytical and numerical results, including the stability analysis of the new modified methods, the investigation of the computational efficiency, an application to adaptivity and an application to hyperbolic PDEs with a Spectral Difference (SD) space discretization.

A cVEM-DG space-time method for the dissipative wave equation

A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time.

Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications

Getting the Lay of the Land in Discrete Space: A Survey of Metric Dimension and Its Applications

Mathematical Model for Excited State Fluid Dynamics

The rational mechanic’s research method is synthesized with the help of mathematical means such as generalized function analysis and tensor analysis. The basic definition of the fluctuation velocity generation is based on phenomenological physics in this article. The basic control equations of general excited state fluid dynamics applicable to the flowfield are obtained based on the basic principles of quantum mechanical superposition states. The simplified basic equations of the excited State are finally obtained through time and space discretization. The basic theory of excited state fluid dynamics is established, providing new ideas for the innovation and application of flow control, fluid mechanical engineering design, and other aspects of research work.

Hierarchical Free Gait Motion Planning for Hexapod Robots Using Deep Reinforcement Learning

This paper addresses the problem of legged locomotion in unstructured environments, and a novel Hierarchical multi-contact motion planning method for hexapod robots is proposed by combining Free Gait motion planning and Deep Reinforcement Learning (HFG-DRL). We structurally decompose the complex free gait multi-contact motion planning task into path planning in discrete state space and gait planning in continuous state space. Firstly, the Soft Deep Q-Network (SDQN) is used to obtain the global prior path information in the Path Planner (PP). Secondly, a Free Gait Planner (FGP) is proposed to obtain the gait sequence. Finally, based on the PP and the FGP, the Center-of-Mass (CoM) sequence is generated by the trained optimal policy using the designed Deep Reinforcement Learning (DRL) algorithm. Experimental results in different environments demonstrate the feasibility, effectiveness, and advancement of the proposed method. Videos are shown at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">http://www.hexapod.cn/hfg-drl.html</uri> .

A nonconforming immersed virtual element method for elliptic interface problems

This paper presents the lowest-order nonconforming immersed virtual element method for solving elliptic interface problems on unfitted polygonal meshes. The local discrete space on each interface mesh element consists of the solutions of local interface problems with Neumann boundary conditions, and the elliptic projection is modified so that its range is the space of broken linear polynomials satisfying the interface conditions. We derive optimal error estimates in the broken H1-norm and L2-norm, under the piecewise H2-regulartiy assumption. In our scheme, the mesh assumptions for error analysis allow small cut elements. Several numerical experiments are provided to confirm the theoretical results.

A deep learning method for the dynamics of classic and conservative Allen-Cahn equations based on fully-discrete operators

The Allen-Cahn equation is a well-known stiff semilinear parabolic equation used to describe the process of phase separation and transition in phase field modeling of multi-component physical systems, while the conservative Allen-Cahn equation is a modified version of the classic Allen-Cahn equation that can additionally conserve the mass. As neural networks and deep learning techniques have achieved significant successes in recent years in scientific and engineering applications, there has been growing interest in developing deep learning algorithms for numerical solutions of partial differential equations. In this paper, we propose a novel deep learning method for predicting the dynamics of the classic and conservative Allen-Cahn equations. Specifically, we design two special convolutional neural network models, one for each of the two equations, to learn the fully-discrete operators between two adjacent time steps. The loss functions of the two models are defined using the residual of the fully-discrete systems, which result from applying the central finite difference discretization in space and the Crank–Nicolson approximation in time. This approach enables us to train the models without requiring any ground-truth data. Moreover, we introduce an effective training strategy that automatically generates useful samples along the time evolution to facilitate training of the models. Finally, we conduct extensive experiments in two and three dimensions to demonstrate outstanding performance of our proposed method, including its dynamics prediction and generalization ability under different scenarios.

Denoising Diffusion Models on Model-Based Latent Space

With the recent advancements in the field of diffusion generative models, it has been shown that defining the generative process in the latent space of a powerful pretrained autoencoder can offer substantial advantages. This approach, by abstracting away imperceptible image details and introducing substantial spatial compression, renders the learning of the generative process more manageable while significantly reducing computational and memory demands. In this work, we propose to replace autoencoder coding with a model-based coding scheme based on traditional lossy image compression techniques; this choice not only further diminishes computational expenses but also allows us to probe the boundaries of latent-space image generation. Our objectives culminate in the proposal of a valuable approximation for training continuous diffusion models within a discrete space, accompanied by enhancements to the generative model for categorical values. Beyond the good results obtained for the problem at hand, we believe that the proposed work holds promise for enhancing the adaptability of generative diffusion models across diverse data types beyond the realm of imagery.

Space Discretization in Quantum Physics and Precision Principle

Space Discretization in Quantum Physics and Precision Principle

New options for explicit all Mach number schemes by suitable choice of time integration methods

Many low-Mach or all-Mach number codes are based on space discretizations which in combination with the first order explicit Euler method as time integration would lead to an unstable scheme. In this paper, we investigate how the choice of a suitable explicit time integration method can stabilize these schemes. We restrict ourselves to some old prototypical examples in order to find directions for further research in this field.

Numerical Modeling and Simulation of Fractured-Vuggy Reservoirs Based on Field Outcrops

We propose a novel workflow to investigate the complex flow behaviors and remaining oil distribution related to the oil–gas–water three-phase system based on information from typical outcrops of fractured-vuggy reservoirs. A refined geological model is built to represent the size, geometry, and spatial distribution of the karst caves and fractures extracted from the field outcrop photographs. The combination of the perpendicular bisector (PEBI) grid technique and the control-volume finite difference method is adopted for space discretization. We have validated the numerical model against experimental data. Numerical simulations were performed to explore the impacts of the permeability of karst cave and natural fractures and the position of natural water bodies upon oil production performance. Numerical results indicate that (1) the cave permeability has few impacts on the oil production, yet the fracture permeability plays a significant role in determining the oil recovery; (2) a higher permeability of the fractures will lead to a longer period of time for no-water oil production and, thus, a higher oil recovery; (3) the position of natural water body shows significant impacts on oil recovery, e.g., a short distance between the natural water body and the production well tends to form preferential passages, causing severe reduction of water flooding range; and (4) the distribution of remaining oil is controlled by spatial patterns of the fractured-vuggy system and reservoir development schemes. We found that the remaining oil is mainly distributed along the model boundaries and at the corner of the caves with single or multiple connection/s to fractures.

PS-Insar point cloud densification using Sentinel-1 and TerraSAR-X data

ABSTRACT The persistent scatterer interferometric SAR (PS-InSAR) was presented to overcome the temporal and spatial circumspection in the studied areas. The interferometric phase contains different phase components, with many researchers being particularly interested in the displacement component. In this article, contrary to the conventional use of this method in evaluating the displacement phase, we try to estimate the residual phase component of the topography contribution by changing master image selection criteria and spatial baseline maximization. By calculating the height error of the topography model utilized in PS-InSAR processing, also known as the height phase, it becomes possible to determine the height correction for each permanent scatterer. There is a challenge with generating 3D models of urban buildings due to the absence of acceptable point clouds. To overcome this obstacle, we used shape-completion learning methods. The training process directly transforms from discrete point cloud space to dense space by extracting the general geometric structure of the desired shape while preserving the details. The loss criterion of this network is based on the Chamfer Distance, which measures the distance between the non-dense input and dense output point clouds. We generated 30,000 individual building datasets for network training. We utilized two SAR image datasets consisting of 27 Sentinel-1 and 20 TerraSAR-X images. After evaluating the test dataset of 3D building models and comparing them with aerial LiDAR reference data, we calculated the Root Mean Square Error (RMSE) for the PS height point clouds. For Sentinel-1 VH and VV channels, the RMSE values were 6.369 m and 5.478 m, respectively. As for TerraSAR-X HH channel, the RMSE was 4.317 m.

A point-picking game

In this article we study a version of the point-picking game defined by Berner and Juhász. Given a space X, the closed game CG(X) on X between Player O and Player P is played as follows: Player O chooses a non-empty open set U1⊂X, then Player P chooses a point x1∈U1, then Player O chooses a non-empty open set U2⊂X, then Player P chooses a point x2∈U2, and so on. An infinite sequence w=(U1,x1,U2,x2,…) such that Un is a nonempty open set and xn∈Un for every n∈N is called a play inCG(X). We will say that PlayerPwins w if {xn:n∈N} is closed in X, otherwise, PlayerOwins w. We prove that if Player O does not have a Markov winning strategy in CG(X) then X is selectively closed. We show that if X is the σ-product {f∈{0,1}ω1:f−1(1) is finite }, then Player O has a winning strategy in CG(X), yet X is selectively closed and selectively discrete. We also construct a selectively discrete space with a stationary winning strategy for Player O in CG(X).

Effectiveness of meshless methods for advection dominant groundwater contaminant transport problems

Meshless methods are efficient tools in prediction of fate of groundwater contamination. In these methods, the choice of amount of discretization in space and time affects the accuracy of solution and computational time. Peclet number is a controlling parameter of such numerical transport simulation models and is the measure of space discretization adequacy for a given velocity and dispersion coefficient. Greater value of Peclet number indicates highly advective flow with negligible diffusion. In this study, the effect of Peclet number on accuracy of meshless methods is analysed. Three meshless methods of three different categories of formulation are considered, namely, Radial Point Collocation Method (RPCM), a meshless strong form method, weak form Meshless Petrov Galerkin Method (MLPG) and hybrid RPCM-MLPG Meshless Weak Strong (MWS) form method. These are applied for modelling Groundwater Contaminant Transport (GCT) phenomenon. The three models are demonstrated using hypothetical case studies and the results are verified with analytical solutions. After a successful validation, the models are subjected to changes in Peclet number and the errors with respect to analytical solutions are studied. It is found that MLPG-GCT model is the most stable and MWS-GCT model is more accurate within less computational time for reasonably high Peclet numbers.

Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time

In this paper, we address a class of boundary layer originated singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions having large time delay; for the higher order numerical analysis. The boundary layer originated nonuniform mesh, is generated by adaptive moving mesh approach based on equidistribution principle. The mesh is obtained by solving a nonlinear system; involving the discrete problem and the discrete nonlinear mesh generating functions. We provide the theoretical mesh structure. The difference scheme for Robin type boundary conditions having normalized flux; uses a combination of space and time discretizations so that the order of convergence in space is higher than the usual linear accurate upwind discretization in space. The present theoretical convergence analysis requires less regularity of the solution and proves to be exactly second order parameter uniform accurate in space. Numerical experiments highly validate the theoretical findings and also show that the present approach is better than the well-known upwind discretizations.

Energy-efficient joint resource allocation in 5G HetNet using Multi-Agent Parameterized Deep Reinforcement learning

Small cells are a promising technique to improve the capacity and throughput of future wireless networks. However, user association and power allocation in heterogeneous networks is complicated by the dense deployment of small cells, resulting in non-convex and combinatorial problems. Conventionally, machine learning techniques are applied to the joint optimization problem, which has different action spaces. Gauging the continuous spaces to discrete spaces results in the loss of granularity due to discretization (e.g. potential power values in power allocation). Due to its hybrid action space, it is sub-optimal to solve joint user association (discrete spaces) and power allocation (continuous spaces) problems by applying traditional machine learning approaches. This work proposes a Multi-Agent Parameterized Deep Reinforcement Learning (MA-PDRL) approach to address the joint user association and power allocation problem efficiently. According to simulation results, the proposed multi-agent PDRL performs better in energy efficiency and QoS satisfaction than WMMSE, game theory, Q-learning, and DRL techniques.