Physical Equations Research Articles

Neuroevolution-Based Network Architecture Evolution in Semiconductor Manufacturing.

Promoted model architectures or algorithms are crucial for intelligent manufacturing since developing them takes a lot of trial and error to embed the domain knowledge into the models correctly. Especially in semiconductor manufacturing, the whole processes depend on complicated physical equations and sophisticated fine-tuning. Therefore, we use a neuroevolution-based model to search the optimized architecture automatically. The collector current value at a particular bias of the silicon-germanium (SiGe) heterojunction bipolar transistor, generated by technology computer-aided design (TCAD), is used as the target dataset with six process parameters as the inputs. The processes include oxidation, dry and wet etching, implantation, annealing, diffusion, and chemical-mechanical polishing. Our work can build a suitable model network with a fast turnaround time, and practical physical constraints are fused in it without domain knowledge extraction. Take the case with 3840 data and one output as an instance. The mean square errors of the train set and validation set, as well as the mean absolute percentage error of the test set, are 1.317 × 10-6, 7.215 × 10-7, and 0.216 while using multilayer perceptron (MLP) and they are 3.285 × 10-7, 1.661 × 10-7, and 0.097 while using NE. The consequences show that the work in this vein is promising. According to the trend plot and results, the ability to extract physic is much better than the traditional (MLP) model.

Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2

This paper brings together methods to solve and/or characterize solutions of some problems of mathematical physics equations involving p-Laplacian and p-pseudo-Laplacian. Using the widely debated results of surjectivity or variational approaches, one may obtain or characterize weak solutions for Dirichlet or Newmann problems for these important operators. The relevance of these operators and the possibility to be involved in the modeling of an important class of real phenomena is once again revealed by their applications. The use of certain variational methods facilitates the complete solution of the problem using appropriate numerical methods and computational algorithms. Some theoretical results are involved to complete the solutions for a sequence of models issued from real phenomena drawing.

Exact traveling wave solutions for the (2+1)-dimensional double sine-Gordon equation using direct integral method

In this paper, the direct integral method and complete discrimination system for quartic polynomial in standard form are applied to the (2+1)-dimensional double sine-Gordon equation. All possible exact traveling wave solutions of different types are successfully obtained from nine subcases. These exact solutions include kink solutions, and periodic solutions. Different from traditional methods (the Jacobi elliptic function method, the Exp-function method, extended Tanh method and the algebraic mapping method, etc.), the direct integral method described in this work is very natural and simple, and gives a new idea to obtain exact solutions for the mathematical physics equations considered.

Identify hidden spreaders of pandemic over contact tracing networks

The COVID-19 infection cases have surged globally, causing devastations to both the society and economy. A key factor contributing to the sustained spreading is the presence of a large number of asymptomatic or hidden spreaders, who mix among the susceptible population without being detected or quarantined. Due to the continuous emergence of new virus variants, even if vaccines have been widely used, the detection of asymptomatic infected persons is still important in the epidemic control. Based on the unique characteristics of COVID-19 spreading dynamics, here we propose a theoretical framework capturing the transition probabilities among different infectious states in a network, and extend it to an efficient algorithm to identify asymptotic individuals. We find that using pure physical spreading equations, the hidden spreaders of COVID-19 can be identified with remarkable accuracy, even with incomplete information of the contract-tracing networks. Furthermore, our framework can be useful for other epidemic diseases that also feature asymptomatic spreading.

Quantum and Component Analysis of P3a and P3b from Auditory Single Trial ERPs Differentiates Borderline Personality Disorder from Schizophrenia

Traditional approaches to EEG modelling use the methods of classical physics to reconstruct scalp potentials in terms of explicit physical models of cortical neuron ensembles. The principal difficulty with such approaches is that the multiplicity of cellular processes, with an intricate array of deterministic and random influencing factors, prevents the creation of consistent biophysical parameter sets. An original, empirically testable solution has been achieved in our previous studies by a radical departure from the deterministic equations of classical physics to the probabilistic reasoning of quantum mechanics. This crucial step relocates the models of elementary bioelectric sources of EEG signals from the cellular to the molecular level where ions are considered as elementary sources of electricity. The rationale is that, despite dramatic differences in cellular machineries, statistical factors governed by the rules of the central limit theorem produce the EEG waveform as a statistical aggregate of the synchronized activity of multiple microscale sources. Based on these innovations, we introduce a method of comprehensive computerized analysis of event related potentials directly from single trial recordings. This method provides a universal model of single trial ERP components in both frequency and time domains. For the first time, this tool provides effective quantification of all significant cognitive components in single trial ERPs and represents a viable alternative to the traditional method of averaging. We demonstrate the clinical significance of the additional information provided by the new method, using ERP data from patients with borderline personality disorder and schizophrenia. Referring to the P300 as an important objective marker of psychiatric disorders, we show that the new method reliably identifies P3a and P3b as the major components of the P3. The diagnostic significance of differentiating the P3a and P3b components of P3 is that it provides an objective electrophysiological measure that distinguishes borderline personality disorder from schizophrenia.

Multiaxial fatigue life prediction using physics-informed neural networks with sensitive features

Deep learning is a widely used tool for multiaxial fatigue life prediction. However, neural network still needs to solve various problems in the solution dominated by physical equations. This work proposes a physics-informed neural network framework incorporating sensitive features and life prediction models. The framework enhances the training process of the neural network under the constraint of life prediction model. Sensitive features are extracted to select input features with higher importance to the model output. The results show that the filtered sensitive features improve the predictive performance of the neural network. Introducing the Smith-Watson-Topper model as a physical loss degrades the predictive performance of the neural network. On the contrary, introducing Fatemi-Socie and Shang-Wang model as physical loss improves the predictive performance of neural network.

Solitary and Periodic Wave Solutions of the Space-Time Fractional Extended Kawahara Equation

The extended Kawahara (Gardner Kawahara) equation is the improved form of the Korteweg–de Vries (KdV) equation, which is one of the most significant nonlinear evolution equations in mathematical physics. In that research, the analytical solutions of the conformable fractional extended Kawahara equation were acquired by utilizing the Jacobi elliptic function expansion method. The given expansion method was applied to different fractional forms of the extended Kawahara equation, such as the fraction that occurs in time, space, or both time and space by suitably changing the variables. In addition, various types of fractional problems are exhibited to expose the realistic application of the given method, and some of the obtained solutions were illustrated in two- or three-dimensional graphics as proof of the visualization.

A Space-Time Partial Differential Equation Based Physics-Guided Neural Network for Sea Surface Temperature Prediction

Sea surface temperature (SST) prediction has attracted increasing attention, due to its crucial role in understanding the Earth’s climate and ocean system. Existing SST prediction methods are typically based on either physics-based numerical methods or data-driven methods. Physics-based numerical methods rely on marine physics equations and have stable and explicable outputs, while data-driven methods are flexible in adapting to data and are capable of detecting unexpected patterns. We believe that these two types of method are complementary to each other, and their combination can potentially achieve better performances. In this paper, a space-time partial differential equation (PDE) is employed to form a novel physics-based deep learning framework, named the space-time PDE-guided neural network (STPDE-Net), to predict daily SST. Comprehensive experiments for SST prediction were conducted, and the results proved that our method could outperform the traditional finite-difference forecast method and several state-of-the-art deep learning and physics-guided deep learning methods.

A stepwise physics‐informed neural network for solving large deformation problems of hypoelastic materials

AbstractPhysics‐informed neural network (PINN) has been widely concerned for its higher computational accuracy compared with conventional neural network. The merit of PINN mainly comes from its ability to embed known physical laws or equations into data‐based neural networks. However, when dealing with the rate‐dependent nonlinear problems, such as elasto‐plasticity with loading and unloading and hypoelastic large deformation, the conventional PINN cannot obtain satisfactory results. In this article, a stepwise physics‐informed neural network (sPINN) is proposed to solve large deformation problems of hypoelastic materials. The whole process of sPINN can be divided into a series of time steps. In each time step, the rate constitutive equation expressed by Hughes‐Winget algorithm and momentum governing equation are incorporated into the loss function as physical constraints. The displacement and stress fields can be resolved by completing the training process of each time step. Three numerical examples are designed to validate the proposed method by comparing with the solutions of FEM. The results show that sPINN can accurately resolve the displacement and stress fields in path‐dependent large deformation problems. Furthermore, the performance of the sPINN on small data sets are also discussed, which illustrates that sPINN is more capable of predicting the global solution on small data sets as compared with conventional artificial neural network.

Algebra discourses in mathematics and physics textbooks: comparing the use of algebraic symbols

It is agreed that algebra has an important role in physics, particularly through handling symbols. A lot of previous research has focused on how mathematics is used in physics from perspectives where mathematics is taken for granted, and not addressing potential differences of mathematics in the physics classroom and in the mathematics classroom. Studies addressing differences between both subjects have been based on researchers’ own experiences of mathematics in both subjects. Thus, more focused empirical research is needed. The purpose of this study is to clarify similarities and differences between mathematics and physics concerning the use of algebraic symbols. Analyses were based on comparisons between upper secondary textbooks in mathematics and in physics from a discourse perspective. Statistical methods were used to decide if there were any significant differences between the subjects. Results showed an overlap in the algebra discourse in both subjects, but also several differences concerning core aspects of algebra. For example, a higher number of different algebraic symbols in equations in physics than in mathematics, and algebraic symbols are more seldom referred to by words in mathematics than in physics. This can make it difficult for students to identify similarities in the algebraic discourses in both subjects.

Radial basis function-differential quadrature-based physics-informed neural network for steady incompressible flows

In this work, a radial basis function differential quadrature-based physics-informed neural network (RBFDQ-PINN) is proposed to simulate steady incompressible flows. The conventional physics-informed neural network (PINN) makes use of the physical equation as a constraint to ensure that the solution satisfies the physical law and the automatic differentiation (AD) method to calculate derivatives at collocation points. Although the AD-PINN is expedient in evaluating derivatives at arbitrary points, it is time-consuming with higher-order derivatives and may lead to nonphysical solutions with sparse samples. Alternatively, the finite difference (FD) method can facilitate the calculation of derivatives, but the FD-PINN will increase the computational cost when handling random point distributions, especially with higher-order discretization schemes. To address these issues, the radial basis function differential quadrature (RBFDQ) method is incorporated into the PINN to replace the AD method for the calculation of derivatives. The RBFDQ method equips with high efficiency in the calculation of high-order derivatives as compared with the AD method and great flexibility in the distribution of mesh points as compared with the FD method. As a result, the proposed RBFDQ-PINN is not only more efficient and accurate but also applicable to irregular geometries. To demonstrate its effectiveness, the RBFDQ-PINN is tested in sample problems such as the lid-driven cavity flow, the channel flow over a backward-facing step, and the flow around a circular cylinder. Numerical results reveal that the RBFDQ-PINN achieves satisfactory accuracy without any labeled collocation points, whereas the AD-PINN struggles to solve some cases, especially for high Reynolds number flows.

Pi Calculation Reinvented a Trigonometry

In this study, we present a novel method for determining the mathematical value of pi by utilizing the relationship between pi and the tangent function in trigonometry. By examining the relationship between the circumference of a circle and a right-angled triangle, we can derive a new equation for pi, that can be used to simplify the mathematical and physical equations that involve pi.

SOLUTION OF THE LAPLACE EQUATION BY THE METHOD OF SEPARATION OF VARIABLES FOR A LENGTH HOLLOW CYLINDER

The physical model of many electrical objects is a hollow cylinder of finite length. As a basis for constructing analytical models of electrical machines, linear equations of mathematical physics are used. They are the solution of the Laplace three-dimensional partial differential equation, which is widely used in analytical calculations.
 
 The purpose of the study is to solve the Laplace three-dimensional differential equation for a hollow cylinder of finite length, which can be adapted for the electromagnetic calculation of electromechanical devices with cylindrical active parts.
 
 Materials and methods. To solve the Laplace equation in a cylindrical coordinate system, the Fourier variable separation method was used. To obtain a non-trivial solution of the equation, eigenfunctions of the Sturm–Liouville problem were used. Dirichlet, Neumann boundary value problems of the mixed type for hollow cylinders are adapted to the electromagnetic calculation of electromechanical devices having cylindrical active parts.
 
 The results of the study. The Laplace equation given in a cylindrical coordinate system is considered, on the basis of which the Sturm–Liouville equation with zero initial values is compiled to find eigenfunctions. The complete solution of the Lapalace equation with given boundary conditions is obtained as the sum of the solutions of two separate Dirichlet problems with different boundary conditions.
 
 Findings. The obtained analytical expression can be used as a mathematical basis for constructing three-dimensional analytical models of electrical machines with cylindrical active parts and carrying out electromagnetic calculations of the corresponding electromechanical devices.

Structural schemes for one dimension stationary equations

In this paper, we propose a new paradigm for finite differences numerical methods, based on compact schemes to provide high order accurate approximations of a smooth solution. The method involves its derivatives approximations at the grid points and the construction of structural equations deriving from the kernels of a matrix that gathers the variables belonging to a small stencil. Numerical schemes involve combinations of physical equations and the structural relations. We have analysed the spectral resolution of the most common structural equations and performed numerical tests to address both the stability and accuracy issues for popular linear and non-linear problems. Several benchmarks are presented that ensure that the developed technology can cope with several problems that may involve non-linearity.

Modeling weathering processes of spilled oil on the sea surface based on dynamic Bayesian network

Oil spills pose a serious threat to the ecological environment and human health. When the spilled oil enters seawater, oil weathering processes like evaporation, emulsification, biodegradation, photooxidation and dissolution will occur. These processes have high uncertainty and directly affect the spatial distribution and fate of oil spills. This paper presents a stochastic modeling method for weathering process of spilled oil on the sea surface based on Dynamic Bayesian Network (DBN). The physical equations of evaporation, emulsification, biodegradation, photooxidation and dissolution are converted into DBN, and then all the individual DBNs are integrated to obtain the entire DBN of the oil weathering process. The developed DBN could intuitively and graphically express the logical relations and conditional probabilities among physical variables. This method is helpful to find out the influencing mechanism of multi-random variables on the oil weathering process. The proposed method is applied to one type of light oil that is spilled on the sea surface of the Bohai Sea. Based on the developed DBN model, the oil weathering process in the first 18 h are studied by quantitative analysis. Through sensitivity analysis, it shows that the photooxidation and dissolution have greater impact on the heavy oil fraction than biodegradation. For oil-water mixture, it is greatly affected by evaporation in the first few hours, but emulsification becomes the leading weathering process in the 18th hour. The effects of dissolution are low, but higher than biodegradation and photooxidation. The key parameters affecting each oil weathering process are determined. Uncertainty analysis is carried out to reveal the possible upper and lower limit values of some physical quantities caused by the weathering process.

Lump-Type Solutions, Lump Solutions, and Mixed Rogue Waves for Coupled Nonlinear Generalized Zakharov Equations

This article studies diverse forms of lump-type solutions for coupled nonlinear generalized Zakharov equations (CNL-GZEs) in plasma physics through an appropriate transformation approach and bilinear equations. By utilizing the positive quadratic assumption in the bilinear equation, the lump-type solutions are derived. Similarly, by employing a single exponential transformation in the bilinear equation, the lump one-soliton solutions are derived. Furthermore, by choosing the double exponential ansatz in the bilinear equation, the lump two-soliton solutions are found. Interaction behaviors are observed and we also establish a few new solutions in various dimensions (3D and contour). Furthermore, we compute rogue-wave solutions and lump periodic solutions by employing proper hyperbolic and trigonometric functions.

Generalized (G’ / G) – expansion method for the loaded shallow water wave equation

This article is devoted to finding solutions for the traveling wave of the loaded wave equation in shallow water. One of the approaches to finding solutions by the expansion method (G / G) is given, which is one of the most effective ways to obtain solutions. When parameters are taken as special values, solitary waves are also derived from traveling waves. Solutions for the traveling wave are expressed by hyperbolic and trigonometric functions. This method is easy to implement using well-known software packages that allow solving complex nonlinear evolutionary equations of mathematical physics.

Application of the polynomial function method to the variable-coefficient Kadomtsev–Petviashvili equation

In this paper, we research a (2+1)-dimensional variable-coefficient Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics. The lump, lump-soliton and lump-periodic solutions are derived based on the variable-coefficient polynomial function method. The effect of variable coefficients on the amplitude and velocity of solitons is analyzed and shown by some 3D graphs, contour plots and density graphs.

Assessing plate reconstruction models using plate driving force consistency tests

Plate reconstruction models are constructed to fit constraints such as magnetic anomalies, fracture zones, paleomagnetic poles, geological observations and seismic tomography. However, these models do not consider the physical equations of plate driving forces when reconstructing plate motion. This can potentially result in geodynamically-implausible plate motions, which has implications for a range of work based on plate reconstruction models. We present a new algorithm that calculates time-dependent slab pull, ridge push (GPE force) and mantle drag resistance for any topologically closed reconstruction, and evaluates the residuals—or missing components—required for torques to balance given our assumed plate driving force relationships. In all analyzed models, residual torques for the present-day are three orders of magnitude smaller than the typical driving torques for oceanic plates, but can be of the same order of magnitude back in time—particularly from 90 to 50 Ma. Using the Pacific plate as an example, we show how our algorithm can be used to identify areas and times with high residual torques, where either plate reconstructions have a high degree of geodynamic implausibility or our understanding of the underlying geodynamic forces is incomplete. We suggest strategies for plate model improvements and also identify times when other forces such as active mantle flow were likely important contributors. Our algorithm is intended as a tool to help assess and improve plate reconstruction models based on a transparent and expandable set of a priori dynamic constraints.

Study on the Nonlinear Dynamics of the (3+1)-Dimensional Jimbo-Miwa Equation in Plasma Physics

The Jimbo-Miwa equation (JME) that describes certain interesting (3+1)-dimensional waves in plasma physics is studied in this work. The Hirota bilinear equation is developed via the Cole-Hopf transform. Then, the symbolic computation, together with the ansatz function schemes, are utilized to seek exact solutions. Some new solutions, such as the multi-wave complexiton solution (MWCS), multi-wave solution (MWS) and periodic lump solution (PLS), are successfully constructed. Additionally, different types of travelling wave solutions (TWS), including the dark, bright-dark and singular periodic wave solutions, are disclosed by employing the sub-equation method. Finally, the physical characteristics and interaction behaviors of the extracted solutions are depicted graphically by assigning appropriate parameters. The obtained outcomes in this paper are more general and newer. Additionally, they reveal that the used methods are concise, direct, and can be employed to study other partial differential equations (PDEs) in physics.