A neural network solution of first-passage problems

This paper proposes a novel method for solving the first-passage time probability problem of nonlinear stochastic dynamic systems.The safe domain boundary is exactly imposed into the radial basis function neural network(RBF-NN)architecture such that the solution is an admissible function of the boundary-value problem.In this way,the neural network solution can automatically satisfy the safe domain boundaries and no longer requires adding the corresponding loss terms,thus efficiently handling structure failure problems defined by various safe domain boundaries.The effectiveness of the proposed method is demonstrated through three nonlinear stochastic examples defined by different safe domains,and the results are validated against the extensive Monte Carlo simulations(MCSs).

段家河流域泥石流发育的渗流(Percolation)模型

在观察资料的基础上,提出了泥石流发育的渗流(Percolation)模型。由类比得到泥石流爆发的临界值P_c=d/(Z(d-1)),其中d是欧氏空间维数,z是配位数。根据广义Fibonacci数列,得到临界状态时的分形维数D_t(p,q)=In N/2ln,其中p、q是与人类活动、环境因素有关的参数.用以上结果对段家河流域的泥石流的发生作了理论分析。

Percolation on networks with dependence links

As a classical model of statistical physics, the percolation theory provides a powerful approach to analyze the network structure and dynamics. Recently, to model the relations among interacting agents beyond the connection of the networked system, the concept of dependence link is proposed to represent the dependence relationship of agents. These studies suggest that the percolation properties of these networks differ greatly from those of the ordinary networks. In particular, unlike the well known continuous transition on the ordinary networks, the percolation transitions on these networks are discontinuous. Moreover, these networks are more fragile for a broader degree distribution, which is opposite to the famous results for the ordinary networks. In this article, we give a summary of the theoretical approaches to study the percolation process on networks with inter- and inner-dependence links, and review the recent advances in this field, focusing on the topology and robustness of such networks.

Research on the nonlinear spherical percolation model with quadratic pressure gradient and its percolation characteristics

For bottom water reservoir and the reservoir with a thick oil formation, there exists partial penetration completion well and when the well products the oil flow in the porous media takes on spherical percolation. The nonlinear spheri-cal flow equation with the quadratic gradient term is deduced in detail based on the mass conservation principle, and then it is found that the linear percolation is the approximation and simplification of nonlinear percolation. The nonlinear spherical percolation physical and mathematical model under different external boundaries is established, considering the ef-fect of wellbore storage. By variable substitu-tion, the flow equation is linearized, then the Laplace space analytic solution under different external boundaries is obtained and the real space solution is also gotten by use of the nu-merical inversion, so the pressure and the pressure derivative bi-logarithmic nonlinear spherical percolation type curves are drawn up at last. The characteristics of the nonlinear spherical percolation are analyzed, and it is found that the new nonlinear percolation type curves are evidently different from linear per-colation type curves in shape and characteris-tics, the pressure curve and pressure derivative curve of nonlinear percolation deviate from those of linear percolation. The theoretical off-set of the pressure and the pressure derivative between the linear and the nonlinear solution are analyzed, and it is also found that the in-fluence of the quadratic pressure gradient is very distinct, especially for the low permeabil-ity and heavy oil reservoirs. The influence of the non-linear term upon the spreading of pressure is very distinct on the process of percolation, and the nonlinear percolation law stands for the actual oil percolation law in res-ervoir, therefore the research on nonlinear per-colation theory should be strengthened and reinforced.

Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional frac- tional percolation equation （FPE） with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seep- age flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.

Percolation Model of Graphite-modified Asphalt Concrete

The addition of graphite powder in conventional asphalt mixture can produced asp halt concrete with excellent electrical performance. Percolation theor y was employed to discuss the relation between the conductivity and graphite con tent of graphite-modified asphalt concrete. It was found that the results of pe rcolation model are consistent with experimental values. The percolation thresho ld of graphite-modified asphalt concrete is 10.94% graphite content account for the total volume of the binder phase consisting of asphalt and graphite. The cr itical exponent is 3.16, beyond the range of 1.6-2.1 for the standard lattice c ontinuous percolation problem. Its reason is that the tunnel conduction mec hanism originates near the critical percent content, which causes this system to be not universal. Tunnel mechanism is demonstrated by the nonlinear voltage-cu rrent characteristic near percolation threshold.The percolation model is able to well predict the formation and development of conductive network in graphite- modified asphalt concrete.

Feedback minimization of first-passage failure of quasi integrable Hamiltonian systems

Abstract A nonlinear stochastic optimal control strategy for minimizing the first-passage failure of quasi integrable Hamiltonian systems （multi-degree-of-freedom integrable Hamiltonian systems subject to light dampings and weakly random excitations） is proposed. The equations of motion for a controlled quasi integrable Hamiltonian system are reduced to a set of averaged It6 stochastic differential equations by using the stochastic averaging method. Then, the dynamical programming equations and their associated boundary and final time conditions for the control problems of maximization of reliability and mean first-passage time are formulated. The optimal control law is derived from the dynamical programming equations and the control constraints. The final dynamical programming equations for these control problems are determined and their relationships to the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are separately established. The conditional reliability function and the mean first-passage time of the controlled system are obtained by solving the final dynamical programming equations or their equivalent Kolmogorov and Pontryagin equations. An example is presented to illustrate the application and effectiveness of the proposed control strategy.

A Novel Method for Preparing Polyurethane Based Conductive Composites with Low Percolation Threshold

A novel method for preparing conductive carbon black fllled polymer composites with low percolation threshold from polyurethane emulsion are reported in this paper. The experimental results indicate that with a rise in carbon black concentration the insulator-conductor transition in the emulsion blended composites occurs at 0.8-1.4vol%. In contrast, the solution blended composites exhibit drastic increase in conductivity at conducting filler fraction as high as 12.3-13.3vol%. It is demonstrated that the composites microstructure rather than chemical structure of the matrix polymer predominantly determines the electrical conduction performance of the composites.

Research on percolation model and criticality of seismicity

Making use of modern nonlinear physics theory and earthquake focus theory, combined with seismicity characteristics, the percolation model of earthquake activity is given in this paper. We take the seismogenic process of alarge earthquake as a phase transition process of percolation and apply the renormalization method to phase transition of percolation. The critical property of the system, which is like percolation probability exponential andcorrelative length exponential, etc, can be calculated under the fixed point as which in the renormalization transformation infinite correlative length in percolation phase transition is taken. The percolation phase transition process of two large earthquakes, which are Haicheng and Tangshan event occurred in 1975 and 1976 respectively, hasbeen discussed by means of seismicity data before and after two shocks.

Analysis on nonlinear effect of unsteady percolation in the inhomogeneous shale gas reservoir

The nonlinear effects of unsteady multi-scale shale gas percolation,such as desorption,slippage,diffusion,pressure-dependent viscosity,and compressibility,are investigated by numerical simulation.A new general mathematical model of the problem is built,in which the Gaussian distribution is used to describe the inhomogeneous intrinsic permeability.Based on the Boltzmann transformation,an efficient semi-analytical method is proposed.The problem is then converted into a nonlinear equation in an integral form for the pressure field,and a related explicit iteration scheme is constructed by numerical discretization.The validation examples show that the proposed method has good convergence,and the simulation results also agree well with the results obtained from both numerical and actual data of two vertical fractured test wells in the literature.Desorption,slippage,and diffusion have significant influence on shale gas flows.The accuracy of the usual technique that the product of viscosity and compressibility is approximated as its value at the average formation pressure is examined.

Stochastic averaging method for estimating first-passage statistics of stochastically excited Duffing-Rayleigh-Mathieu system

The first-passage statistics of Duffing-Rayleigh- Mathieu system under wide-band colored noise excitations is studied by using stochastic averaging method. The motion equation of the original system is transformed into two time homogeneous diffusion Markovian processes of amplitude and phase after stochastic averaging. The diffusion process method for first-passage problem is used and the corresponding backward Kolmogorov equation and Pontryagin equation are constructed and solved to yield the conditional reliability function and mean first-passage time with suitable initial and boundary conditions. The analytical results are confirmed by Monte Carlo simulation.

Continuum percolation of porous media via random packing of overlapping cube-like particles

The pore configuration in porous medium is assumed to be the randomly distributed cube-like particles which can overlap each other in the periodic cubic domain, and the impact of particle characteristics on the percolation property of these cube-like particle packing systems is analyzed.Firstly, by combining the percolation models and finite-size scaling analysis, three numerical parameters(i.e., percolation transition width △L, local percolation threshold ψ_c(L), and correlation length exponent v) for the cube-like particle systems with shape parameter s in[1.0, +∞] are derived successively. Then, based on the relation between the percolation thresholdψ_c in infinite space and the local percolation threshold ψ_c(L), the corresponding ψ_c with s in[1.0, +∞] are further determined. It is shown from the study that the characteristics of cube-like particles have significant influence on the global percolation threshold ψ_c of the particle packing systems. As the parameter s increases from 1.0 to +∞, the percolation threshold ψ_c will go down persistently. When the surface of cube-like particles is cubical and spherical, respectively, the minimum and maximum thresholds ψ_c,min and ψ_c,max are obtained.

Carbon materials with quasi-graphene layers:The dielectric,percolation properties and the electronic transport mechanism

We investigate the dielectric properties of multi-walled carbon nanotubes（MWCNTs） and graphite filling in SiO2 with the filling concentration of 2-20 wt.% in the frequency range of 10 ^2-10^ 7 Hz.MWCNTs and graphite have general electrical properties and percolation phenomena owing to their quasi-structure made up of graphene layers.Both permittivity ε and conductivity σ exhibit jumps around the percolation threshold.Variations of dielectric properties of the composites are in agreement with the percolation theory.All the percolation phenomena are determined by hopping and migrating electrons,which are attributed to the special electronic transport mechanism of the fillers in the composites.However,the twin-percolation phenomenon exists when the concentration of MWCNTs is between 5-10 wt.% and 15-20 wt.% in the MWCNTs/SiO2 composites,while in the graphite/SiO2 composites,there is only one percolation phenomenon in the graphite concentration of 10-15 wt.%.The unique twin-percolation phenomenon of MWCNTs/SiO2 is described and attributed to the electronic transfer mechanism,especially the network effect of MWCNTs in the composites.The network formation plays an essential role in determining the second percolation threshold of MWCNTs/SiO2.

Exact scaling for the mean first-passage time of random walks on a generalized Koch network with a trap

In this paper, we study the scaling for the mean first-passage time （MFPT） of the random walks on a generalized Koch network with a trap. Through the network construction, where the initial state is transformed from a triangle to a polygon, we obtain the exact scaling for the MFPT. We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order. In addition, we determine the exponents of scaling efficiency characterizing the random walks. Our results are the generalizations of those derived for the Koch network, which shed light on the analysis of random walks over various fractal networks.

Research on Percolation Network and Physical Properties of Graphite/Antimony Composites

The microstructure, friction and wear behaviour of graphite preform and graphite/antimony composites are analysed based on the percolation theory of hydrodynamics to investigate the relationship between the percolation net- work and physical properties of graphite/antimony composites. The result shows that there are two important factors to enhance friction and wear behaviour of graphite/antimony composites at high temperature: 1) the formation of the pore network in the preform, which is called the first percolation and 2) the optimization of infiltration method in the process of infiltrating antimony, which is called the second percolation. By adding some pyrolysate and controlling the roasting process, perfect net pores and sub-micron percolation microstructure may be formed in the graphite preform. By con- trolling the infiltration process, the saturation of molten antimony infiltrating into perfect pores can be optimized.

Percolation analysis of large-scale wireless balloon networks

Recent advancements in wireless technology have tested Wireless Balloon Networks (WBNs) as an ideal solution for the provision of internet facilities in deprived and challenging areas. A few high profile companies, such as Google, Space Data Inc., etc., have already made news by initiating projects based on high-altitude WBNs in order to provide internet facilities in remote areas. Unfortunately, the technical details have mainly been kept confidential so far. In this paper, we attempt to analyze the percolation properties of large-scale WBNs, considering both homogenous and heterogenous wireless nodes. In order to do so, we modeled a WBN as a large-scale random network where the path-loss models of homogenous and heterogenous WBNs were reduced to GDM (Gilbert's Disk Model) and RGDM (Random Gilbert's Disk Model), respectively. The bounds of the critical density regime were derived for both percolation models. Additionally, this paper implemented an experimental test bed for the WBN percolation model. Consequently, the findings of this research may prove crucial in estimating critical network properties.

FIRST-PASSAGE TIME OF QUASI-NON-INTEGRABLE-HAMILTONIAN SYSTEM

Studies on first-passage failure are extended to the multi-degree-of-freedom quasi-non-integrable-Hamiltonian systems under parametric excitations of Gaussian white noises in this paper. By the stochastic averaging method of energy envelope, the system's energy can be modeled as a one-dimensional approximate diffusion process by which the classical Pontryagin equation with suitable boundary conditions is applicable to analyzing the statistical moments of the first-passage time of an arbitrary order. An example is studied in detail and some numerical results are given to illustrate the above procedure.

Transient Properties of a Bistable System with Delay Time Driven by Non-Gaussian and Gaussian Noises:Mean First-Passage Time

The mean first-passage time of a bistable system with time-delayed feedback driven by multiplicative non-Gaussian noise and additive Gaussian white noise is investigated. Firstly, the non-Markov process is reduced to the Markov process through a path-integral approach; Secondly, the approximate Fokker-Planck equation is obtained by applying the unified coloured noise approximation, the small time delay approximation and the Novikov Theorem. The functional analysis and simplification are employed to obtain the approximate expressions of MFPT. The effects of non-Gaussian parameter （measures deviation from Gaussian character） r, the delay time τ, the noise correlation time to, the intensities D and a of noise on the MFPT are discussed. It is found that the escape time could be reduced by increasing the delay time τ, the noise correlation time τ0, or by reducing the intensities D and α. As far as we know, this is the first time to consider the effect of delay time on the mean first-passage time in the stochastic dynamical system.

Stochastic Optimal Control of First-Passage Failure for Rectangular Thin Plate Vibration Model under Gaussian White-Noise Excitations

A rectangular thin plate vibration model subjected to inplane stochastic excitation is simplified to a quasinonintegrable Hamiltonian system with two degrees of freedom. Subsequently a one-dimensional Ito stochastic differential equation for the system is obtained by applying the stochastic averaging method for quasi-nonintegrable Hamiltonian systems. The conditional reliability function and conditional probability density are both gained by solving the backward Kolmogorov equation numerically. Finally, a stochastic optimal control model is proposed and solved. The numerical results show the effectiveness of this method.

Effect of Percolation Pattern on Yields and Accumulation of Copper and Cadmium in the Rice Plants with Soil Dressing Models

A greenhouse experiment was conducted using stratified paddy field models, which were prepared by polluted soil with 12.5 cm soil dressing. The paddy models were assembled with open and closed percolation system to focus the effect of percolation pattern on accumulation of cupper （Cu） and cadmium （Cd） in rice plants in contaminated paddy filed with soil dressing models. Percolation pattern has significant effect on soil environment, especially redox potential that may influence mobilization of Cd, Cu and other elements. In open system percolation models, the plowsole and subsoil were in oxidative condition （600 mV）, whereas reduction condition （-200 mV） was measured in closed system percolation models. Accumulations of Cu and Cd in all parts of rice plants （roots, grains, stems and leaves） were found higher in an open system percolation with the paddy field model comparatively than in a closed system percolation. The soil redox （Eh） condition influenced by the percolation pattern might be one of the main factors for uptake and accumulation of Cu and Cd in rice plants. The plant height and stem number were found lower in open system percolation comparatively to closed system percolation. In the yields section, the average panicle length, number of panicle and rice grain/hill and weight of grain/hill were lower in open system percolation than the closed system percolation.