Showing papers on "Piecewise linear function published in 2022"

Piecewise-Linear Simplification for Adaptive Synaptic Neuron Model

Abstract: Adaptive synaptic neuron model involves complex activation functions. These nonlinearities lead to complicated hardware implementations, which greatly hinder neuron-based applications. To effectively solve this issue, a piecewise-linear (PWL) activation function with simplified circuit implementation is presented for the adaptive synaptic neuron model in this brief. With this neuron model, the stability evolution mechanism of the equilibrium state is analyzed and the parameter- and initial condition-related neuron dynamics are numerically explored. Afterwards, an analog circuit is designed and manually made using commercially available components. The phase trajectories captured by the hardware experiments verify the feasibility of the PWL activation function. Thus, such a PWL simplification shows superiority in emulating neuron dynamics.

Thermal Analysis of a Novel Linear Oscillating Machine Based on Direct Oil-Cooling Windings

Abstract: High-frequency electromagnetic linear machine is one major component for linear eletro-hydraulic actuator that in turn is extrimely important for the flight control of more-electrical or all-electrical aircrafts. The employment of high-frequency linear machine helps to achieve high reliability and dynamic performance of the actuators. However, the capability of heat dissipation of the linear machine unavoidably constrains the input current, which in turn depresses its output performance. Therefore, this paper proposes a novel electromagnetic linear machine based on the self-powered direct oil-cooling windings. Specifically, the windings are immersed into the oil directly, which helps to decrease the temperature of the linear machine significantly and thus increase its output performance. In addition, the oil is pushed by the motion of the mover itself. No additional pump is required any more. Thus, the system can be simplified greatly. The design concept of high-frequency electromagnetic linear machine along with its operating principle is presented. A high-precision three-dimensional (3D) piecewise modeling method is proposed to simultaneously consider axial, radial and circumferential equivalent thermal circuits. Numerical simulation is conduced to validated the analytical model. A research prototype of the linear oscillating machine based on direct oil-cooling windings is developed and empolyed for experimental study, and the experimental result verifies the mathematical model of the thermal field well.

Periodic Motion and Frequency Energy Plots of Dynamical Systems Coupled with Piecewise Nonlinear Energy Sink

Abstract: The frequency-energy plots (FEPs) of two-degree-of-freedom linear structures attached to a piecewise nonlinear energy sink (PNES) are generated here and thoroughly investigated. This study provides the FEP analysis of such systems for further understanding to nonlinear targeted energy transfer (TET) by the PNES. The attached PNES to the considered linear dynamical systems incorporates a symmetrical clearance zone of zero stiffness content where the boundaries of the zone are coupled with the linear structure by linear stiffness elements. In addition, linear viscous damping is selected to be continuous during the PNES mass oscillation. The underlying nonlinear dynamical behaviour of the considered structure-PNES systems is investigated by generating the fundamental backbone curves of the FEP and the bifurcated subharmonic resonance branches using numerical continuation methods. Accordingly, interesting dynamical behaviour of the nonlinear normal modes (NNMs) of the structure-PNES system on different backbones and subharmonic resonance branches has been observed. In addition, the imposed wavelet transform frequency spectrums on the FEPs have revealed that the TET takes place in multiple resonance captures where it is dominated by the nonlinear action of the PNES.

Piecewise linear neural networks and deep learning

Abstract: As a powerful modelling method, piecewise linear neural networks (PWLNNs) have proven successful in various fields, most recently in deep learning. To apply PWLNN methods, both the representation and the learning have long been studied. In 1977, the canonical representation pioneered the works of shallow PWLNNs learned by incremental designs, but the applications to large-scale data were prohibited. In 2010, rectified linear units (ReLU) advocated the prevalence of PWLNNs in deep learning. Ever since, PWLNNs have been successfully applied to many tasks and achieved excellent performance. In this Primer, we systematically introduce the methodology of PWLNNs by grouping the works into shallow and deep networks. First, different PWLNN representation models are constructed with elaborated examples. With PWLNNs, the evolution of learning algorithms for data is presented and fundamental theoretical analysis follows up for in-depth understandings. Then, representative applications are introduced together with discussions and outlooks. Piecewise linear neural networks (PWLNNs) are a powerful modelling method, particularly in deep learning. In this Primer, Tao et al. introduce the methodology and theoretical analysis of PWLNNs and some of their applications.

Analytic Solution to the Piecewise Linear Interface Construction Problem and Its Application in Curvature Calculation for Volume-of-Fluid Simulation Codes

Abstract: The plane-cube intersection problem has been around in literature since 1984 and iterative solutions to it have been used as part of piecewise linear interface construction (PLIC) in computational fluid dynamics simulation codes ever since. In many cases, PLIC is the bottleneck of these simulations regarding compute time, so a faster, analytic solution to the plane-cube intersection would greatly reduce compute time for such simulations. We derive an analytic solution for all intersection cases and compare it to the one previous solution from Scardovelli and Zaleski (Ruben Scardovelli and Stephane Zaleski. "Analytical relations connecting linear interfaces and volume fractions in rectangular grids". In: Journal of Computational Physics 164.1 (2000), pp. 228-237.), which we further improve to include edge cases and micro-optimize to reduce arithmetic operations and branching. We then extend our comparison regarding compute time and accuracy to include two different iterative solutions as well. We find that the best choice depends on the employed hardware platform: on the CPU, Newton-Raphson is fastest with vectorization while analytic solutions perform better without. The reason for this is that vectorization instruction sets do not include trigonometric functions as used in the analytic solutions. On the GPU, the fastest method is our optimized version of the analytic SZ solution. We finally provide details on one of the applications of PLIC: curvature calculation for the Volume-of-Fluid model used for free surface fluid simulations in combination with the lattice Boltzmann method.

Iterative learning feedback control for linear parabolic distributed parameter systems with multiple collocated piecewise observation

Abstract: This paper presents a novel iterative learning feedback control method for linear parabolic distributed parameter systems with multiple collocated piecewise observation. Multiple actuators and sensors distributed at the same position of the spatial domain are utilized to perform collocated piecewise control and measurement operations. The advantage of the proposed method is that it combines the iterative learning algorithm and feedback technique. Not only can it use the iterative learning algorithm to track the desired output trajectory, but also the feedback control approach can be utilized to achieve real-time online update. By utilizing integration by parts, triangle inequality, mean value theorem for integrals and Gronwall lemma, two sufficient conditions based on the inequality constraints for the convergence analysis of the tracking error system are presented. Some simulation experiments are provided to prove the effectiveness of the proposed method.

A Comparison of Two Mixed-Integer Linear Programs for Piecewise Linear Function Fitting

Abstract: The problem of fitting continuous piecewise linear (PWL) functions to discrete data has applications in pattern recognition and engineering, amongst many other fields. To find an optimal PWL function, the positioning of the breakpoints connecting adjacent linear segments must not be constrained and should be allowed to be placed freely. Although the univariate PWL fitting problem has often been approached from a global optimisation perspective, recently, two mixed-integer linear programming approaches have been presented that solve for optimal PWL functions. In this paper, we compare the two approaches: the first was presented by Rebennack and Krasko [Rebennack S, Krasko V (2020) Piecewise linear function fitting via mixed-integer linear programming. INFORMS J. Comput. 32(2):507–530] and the second by Kong and Maravelias [Kong L, Maravelias CT (2020) On the derivation of continuous piecewise linear approximating functions. INFORMS J. Comput. 32(3):531–546]. Both formulations are similar in that they use binary variables and logical implications modelled by big-[Formula: see text] constructs to ensure the continuity of the PWL function, yet the former model uses fewer binary variables. We present experimental results comparing the time taken to find optimal PWL functions with differing numbers of breakpoints across 10 data sets for three different objective functions. Although neither of the two formulations is superior on all data sets, the presented computational results suggest that the formulation presented by Rebennack and Krasko is faster. This might be explained by the fact that it contains fewer complicating binary variables and sparser constraints. Summary of Contribution: This paper presents a comparison of the mixed-integer linear programming models presented in two recent studies published in the INFORMS Journal on Computing. Because of the similarity of the formulations of the two models, it is not clear which one is preferable. We present a detailed comparison of the two formulations, including a series of comparative experimental results across 10 data sets that appeared across both papers. We hope that our results will allow readers to take an objective view as to which implementation they should use.

Longwall mining automation horizon control: Coal seam gradient identification using piecewise linear fitting

Abstract: Horizon control, maintaining the alignment of the shearer’s exploitation gradient with the coal seam gradient, is a key technique in longwall mining automation. To identify the coal seam gradient, a geological model of the coal seam was constructed using in-seam seismic surveying technology. By synthesizing the control resolution of the range arm and the geometric characteristics of the coal seam, a gradient identification method based on piecewise linear representation (PLR) is proposed. To achieve the maximum exploitation rate within the shearer’s capacity, the control resolution of the range arm is selected as the threshold parameter of PLR. The control resolution significantly influenced the number of line segments and the fitting error. With the decrease of the control resolution from 0.01 to 0.02 m, the number of line segments decreased from 65 to 15, which was beneficial to horizon control. However, the average fitting error increased from 0.055 to 0.14 m, which would induce a decrease in the exploitation rate. To avoid significant deviation between the cutting range and the coal seam, the control resolution of the range arm must be lower than 0.02 m. In a field test, the automated horizon control of the longwall face was realized by coal seam gradient identification.

Learning of Continuous and Piecewise-Linear Functions With Hessian Total-Variation Regularization

Abstract: We develop a novel 2D functional learning framework that employs a sparsity-promoting regularization based on second-order derivatives. Motivated by the nature of the regularizer, we restrict the search space to the span of piecewise-linear box splines shifted on a 2D lattice. Our formulation of the infinite-dimensional problem on this search space allows us to recast it exactly as a finite-dimensional one that can be solved using standard methods in convex optimization. Since our search space is composed of continuous and piecewise-linear functions, our work presents itself as an alternative to training networks that deploy rectified linear units, which also construct models in this family. The advantages of our method are fourfold: the ability to enforce sparsity, favoring models with fewer piecewise-linear regions; the use of a rotation, scale and translation-invariant regularization; a single hyperparameter that controls the complexity of the model; and a clear model interpretability that provides a straightforward relation between the parameters and the overall learned function. We validate our framework in various experimental setups and compare it with neural networks.

Passenger-oriented traffic control for rail networks: An optimization model considering crowding effects on passenger choices and train operations

Abstract: In public transport, e.g., railways, crowding is of major influence on passenger satisfaction and also on system performance. We study the passenger-oriented traffic control problem by means of integrated optimization, particularly considering the crowding effects on passenger route choices and on train traffic. The goal is to find the system optimum solution by adapting train schedules and rerouting passengers. A mixed-integer nonlinear programming (MINLP) model is proposed, identifying the train orders and departure and arrival times, as well as finding the best route for passengers, with the objective of minimizing passenger disutility and train delay. In the model, we allow free splits of the passengers in a group onto different routes and reasonable passenger transfers between trains. We value train crowding by using time multiplier, which is defined as a piecewise constant function of the train crowding ratio (also called load factor), indicating that passengers perceive a longer travel time on a more crowded train. Moreover, we assume variations of the minimum train dwell time, caused by the alighting and boarding passengers. The nonlinear terms in the MINLP model are linearized by using an exact reformulation method and three transformation properties, resulting in an equivalent mixed-integer linear programming (MILP) model. In the experiences, we adopt a real-world railway network, i.e., the urban railway network in Zürich city, to examine the proposed approach. The results demonstrate the effectiveness of the model. The results show that, by considering the crowding effects, some passengers are forced to choose the routes that are less crowded but have larger travel/delay times, which leads to the improved passenger comfort and makes the planned train timetable less affected (in terms of delays). We also find that flexibility in train schedules brings more possibilities to serve better the passengers. Moreover, it is observed that if the train dwell time is highly sensitive to the alighting and boarding passengers, then the transport network will become vulnerable and less reliable, which should be avoided in real operations.

Analysis on the Number of Linear Regions of Piecewise Linear Neural Networks

Abstract: Deep neural networks (DNNs) are shown to be excellent solutions to staggering and sophisticated problems in machine learning. A key reason for their success is due to the strong expressive power of function representation. For piecewise linear neural networks (PLNNs), the number of linear regions is a natural measure of their expressive power since it characterizes the number of linear pieces available to model complex patterns. In this article, we theoretically analyze the expressive power of PLNNs by counting and bounding the number of linear regions. We first refine the existing upper and lower bounds on the number of linear regions of PLNNs with rectified linear units (ReLU PLNNs). Next, we extend the analysis to PLNNs with general piecewise linear (PWL) activation functions and derive the exact maximum number of linear regions of single-layer PLNNs. Moreover, the upper and lower bounds on the number of linear regions of multilayer PLNNs are obtained, both of which scale polynomially with the number of neurons at each layer and pieces of PWL activation function but exponentially with the number of layers. This key property enables deep PLNNs with complex activation functions to outperform their shallow counterparts when computing highly complex and structured functions, which, to some extent, explains the performance improvement of deep PLNNs in classification and function fitting.

Piecewise linear neural networks and deep learning

Abstract: As a powerful modelling method, piecewise linear neural networks (PWLNNs) have proven successful in various fields, most recently in deep learning. To apply PWLNN methods, both the representation and the learning have long been studied. In 1977, the canonical representation pioneered the works of shallow PWLNNs learned by incremental designs, but the applications to large-scale data were prohibited. In 2010, rectified linear units (ReLU) advocated the prevalence of PWLNNs in deep learning. Ever since, PWLNNs have been successfully applied to many tasks and achieved excellent performance. In this Primer, we systematically introduce the methodology of PWLNNs by grouping the works into shallow and deep networks. First, different PWLNN representation models are constructed with elaborated examples. With PWLNNs, the evolution of learning algorithms for data is presented and fundamental theoretical analysis follows up for in-depth understandings. Then, representative applications are introduced together with discussions and outlooks. Piecewise linear neural networks (PWLNNs) are a powerful modelling method, particularly in deep learning. In this Primer, Tao et al. introduce the methodology and theoretical analysis of PWLNNs and some of their applications.

Dynamics of leptospirosis disease in context of piecewise classical-global and classical-fractional operators

Abstract: In this paper, we use newly introduced piecewise classical-global and classical-fractional operators to study the dynamics of the Leptospirosis disease model. The existence and uniqueness of the solution to piecewise derivatives are examined for the suggested problem. The piecewise iterative Newton polynomial method is used to obtain an approximate solution to the suggested problem. In addition, a numerical scheme for the piecewise Leptospirosis model with integer and fractional orders is established. The numerical simulation of the piecewise derivable problem under consideration is presented in classical as well as various fractional orders.

Interior Point Methods Can Exploit Structure of Convex Piecewise Linear Functions with Application in Radiation Therapy

Abstract: Auxiliary variables are often used to model a convex piecewise linear function in the framework of linear optimization. This work shows that such variables yield a block diagonal plus low rank structure in the reduced KKT system of the dual problem. We show how the structure can be detected efficiently and derive the linear algebra formulas for an interior point method which exploits such a structure. The structure is detected in 36% of the cases in Netlib. Numerical results on the inverse planning problem in radiation therapy show an order of magnitude speed-up compared to the state-of-the-art interior point solver CPLEX and considerable improvements in dose distribution compared to current algorithms.