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Random Forest and Order Parameters:A Combined Framework for Scenario Recognition for Power Systems with Renewable Penetration 认领 引用
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作者 Xiaolong Xiao Xiaoxing Lu +3 位作者 Ziran Guo Jian Liu Shenglong Wu Ye Cai 《Energy Engineering》 EI 2025年第8期3117-3132,共16页
With the popularization of microgrid construction and the connection of renewable energy sources to the power system,the problem of source and load uncertainty faced by the coordinated operation of multi-microgrid is ... With the popularization of microgrid construction and the connection of renewable energy sources to the power system,the problem of source and load uncertainty faced by the coordinated operation of multi-microgrid is becoming increasingly prominent,and the accuracy of typical scenario predictions is low.In order to improve the accuracy of scenario prediction under source and load uncertainty,this paper proposes a typical scenario identification model based on random forests and order parameters.Firstly,a method for ordinal parameter identification and quantification is provided for the coordinated operating mode of multi-microgrids,taking into account source-load uncertainty.Secondly,the dynamic change characteristics of the order parameters of the daily load curve,wind and solar curve,and load curve of typical scenarios are statistically analyzed to identify the key order parameters that have the most significant impact on the uncertainty of the load.Then,the order parameters and seasonal distribution are used as features to train a random forest classification model to achieve efficient scenario prediction.Finally,the simulation of actual data from a provincial distribution network shows that the proposed method can accurately classify typical scenarios with an accuracy rate of 92.7%.Additionally,sensitivity analysis is conducted to assess how changes in uncertainty levels affect the importance of each order parameter,allowing for adaptive uncertainty mitigation strategies. 展开更多
关键词 Source load uncertainty scenario prediction order parameters random forest
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Temperature dependence of ratio between dielectric anisotropy and order parameter in fluorinated nematic liquid crystals 认领 引用 被引量：5
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作者 马恒 孙瑞芝 +1 位作者 李振新 刘玉芳 《Chinese Physics B》 SCIE EI CAS 2008年第1期255-258,共4页
Temperature dependence of ratio between dielectric anisotropy and order parameter of fluorinated nematic liquid crystal is investigated by using a semi-empirical molecular orbital package that can accurately calculate... Temperature dependence of ratio between dielectric anisotropy and order parameter of fluorinated nematic liquid crystal is investigated by using a semi-empirical molecular orbital package that can accurately calculate an angle between molecular dipole moment and long axis. We optimize the molecular conformations with three semi-empirical Hamiltonians AM1, PM3 and PM5, and then make a comparison between computational results and experimental measurements. It is shown that the results obtained from AM1 method are in good agreement with the measurements. The present study offers an applicable method to predict the dielectric properties of liquid crystal material. 展开更多
关键词 dielectric anisotropy order parameter fluorine nematic liquid crystal
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Order parameters and synchronization of FitzHugh-Nagumo small-world networks 认领 引用 被引量：1
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作者 李延龙 马军 +1 位作者 张巍 刘延君 《Chinese Physics B》 SCIE EI CAS 2009年第10期4598-4602,共5页
This paper numerically investigates the order parameter and synchronisation in the small world connected FitzHugh-Nagumo excitable systems. The simulations show that the order parameter continuously decreases with inc... This paper numerically investigates the order parameter and synchronisation in the small world connected FitzHugh-Nagumo excitable systems. The simulations show that the order parameter continuously decreases with increasing D, the quality of the synchronisation worsens for large noise intensity. As the coupling intensity goes up, the quality of the synchronisation worsens, and it finds that the larger rewiring probability becomes the larger order parameter. It obtains the complete phase diagram for a wide range of values of noise intensity D and control parameter g. 展开更多
关键词 order parameter small world network noise
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A novel approach to topological defects in a vector order parameter system 认领 引用 被引量：1
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作者 任继荣 戎树军 朱涛 《Chinese Physics B》 SCIE EI CAS 2009年第7期2901-2904,共4页
Based on Duan＇s topological current theory,we propose a novel approach to study the topological properties of topological defects in a two-dimensional complex vector order parameter system.This method shows explicitl... Based on Duan＇s topological current theory,we propose a novel approach to study the topological properties of topological defects in a two-dimensional complex vector order parameter system.This method shows explicitly the fine topological structure of defects.The branch processes of defects in the vector order parameter system have also been investigated with this method. 展开更多
关键词 topological defects order parameter topological current branch processes
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A tentative model for the explanation of B?th law using the order parameter of seismicity in natural time 认领 引用 被引量：1
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作者 Konstantina A. Papadopoulou Efthimios S. Skordas Nicholas V. Sarlis 《Earthquake Science》 CSCD 2016年第6期311-319,共9页
Using the order parameter of seismicity defined in natural time, we suggest a simple model for the expla- nation of Bath law, according to which a mainshock differs in magnitude from its largest aftershock by approxim... Using the order parameter of seismicity defined in natural time, we suggest a simple model for the expla- nation of Bath law, according to which a mainshock differs in magnitude from its largest aftershock by approximately 1.2 regardless of the mainshock magnitude. In addition, the validity of Bath law is studied in the Global Centroid Moment Tensor catalogue by using two different aftershock definitions. It is found that the mean of this difference, when considering all the pairs mainshock-largest aftershock, does not markedly differ from 1.2 and the corresponding distributions do not depend on the mainshock＇s magnitude threshold in a statistically significant manner. Finally, the analysis of the cumulative distribution functions provides evidence in favour of the proposed model. 展开更多
关键词 Bath law . Aftershocks .Order parameter .Natural time
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Spatial distribution order parameter prediction of collective system using graph network 认领 引用
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作者 赵慧敏 王瑞 +1 位作者 赵偲 郑文 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第5期566-572,共7页
In the past few decades, the study of collective motion phase transition process has made great progress. It is also important for the description of the spatial distribution of particles. In this work, we propose a n... In the past few decades, the study of collective motion phase transition process has made great progress. It is also important for the description of the spatial distribution of particles. In this work, we propose a new order parameter φ to quantify the degree of order in the spatial distribution of particles. The results show that the spatial distribution order parameter can effectively describe the transition from a disorderly moving phase to a phase with a coherent motion of the particle distribution and the same conclusion could be obtained for systems with different sizes. Furthermore, we develop a powerful molecular dynamic graph network(MDGNet) model to realize the long-term prediction of the self-propelled collective system solely from the initial particle positions and movement angles. Employing this model, we successfully predict the order parameters of the specified time step. And the model can also be applied to analyze other types of complex systems with local interactions. 展开更多
关键词 order parameter graph network collective system active matter
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The gas discharge visualization(GDV)order parameter model based on the principle of mastering both permanence and change 认领 引用
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作者 XIN Yu ZHANG Lei +3 位作者 ZHAO Qiancheng SHE Yurong SHE Zhensu SONG Shuna 《Digital Chinese Medicine》 CAS CSCD 2024年第3期231-240,共10页
Objective To investigate the human body’s complex system,and classify and characterize the human body’s health states with“a comprehensive integrated method from qualitative to quantitative”.Methods This paper int... Objective To investigate the human body’s complex system,and classify and characterize the human body’s health states with“a comprehensive integrated method from qualitative to quantitative”.Methods This paper introduces the concept of“order parameters”and proposes a method for establishing an order parameter model of gas discharge visualization(GDV)based on the principle of“mastering both permanence and change(MBPC)”.The method involved the fol-lowing three steps.First,average luminous intensity(I)and average area(S)of the GDV im-ages were calculated to construct the phase space,and the score of the health questionnaire was calculated as the health deviation index(H).Second,the k-means++clustering method was employed to identify subclasses with the same health characteristics based on the data samples,and to statistically determine the symptom-specific frequencies of the subclasses.Third,the distance(d)between each sample and the“ideal health state”,which determined in the phase space of each subclass,was calculated as an order parameter describing the health imbalance,and a linear mapping was established between the d and the H.Further,the health implications of GDV signals were explored by analyzing subclass symptom profiles.We also compare the mean square error(MSE)with classification methods based on age,gen-der,and body mass index(BMI)indices to verify that the phase space possesses the ability to portray the health status of the human body.Results This study preliminarily tested the reliability of the order parameter model on data samples provided by 20 participants.Based on the discovered linear law,the current model can use d calculated by measuring the GDV signal to predict H(R2>0.77).Combined with the symptom profiles of the subclasses,we explain the classification basis of the phase space based on the pattern identification.Compared with common classification methods based on age,gender,BMI,etc.,the MSE of phase space-based classification was reduced by an order of magnitude.Conclusion In this study,the GDV order parameter model based on MBPC can identify sub-classes and characterize individual health levels,and explore the TCM health meanings of the GDV signals by using subjective-objective methods,which holds significance for establishing mathematical models from TCM diagnosis principles to interpret human body signals. 展开更多
关键词 Gas discharge visualization(GDV) Traditional Chinese medicine(TCM) Order parameters Math-physical model Individualized health assessment
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Polarizabilities and Orientational Order Parameter in N-(p-n-Ethoxybenzylidene)- p-n-Alkoxy Anilines, 2O.Om LC Compounds 认领 引用
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作者 Putcha Srinivasa Sastry Chintalapati Srinivas +1 位作者 Pokkunuri Pardhasaradhi Venkata Gopala Krishna Murthy Pisipati 《Journal of Surface Engineered Materials and Advanced Technology》 2016年第3期134-148,共15页
n.m, n.Om, nO.m and nO.Om compounds play an important role in the fundamental and applied aspects. As a part of our systematic studies of the above homologous series regarding the synthesis, characterization and phase... n.m, n.Om, nO.m and nO.Om compounds play an important role in the fundamental and applied aspects. As a part of our systematic studies of the above homologous series regarding the synthesis, characterization and phase transition studies, here the authors report the polarizabilities and orientational order parameter, S in N-(p-n-ethoxybenzylidene)-p-n-alkoxy anilines, 2O.Om with m = 3 to 10 liquid crystalline compounds except with m = 5 which has been already published. The rest of the compounds exhibit the nematic LC phase except with m = 10 which shows the smectic-C phase along the nematic phase which is originally not detected by Godzwon et al. But the author’s observations with TM and differential scanning calorimeter have shown the existence on SmC phase in addition to nematic phase. Further, SmC is clarified using the histogram technique whose details are given below and are published elsewhere. The orientstonal order parameter, S is estimated using different methods and compared with the value obtained from birefringence, δn = (ne – no) where no field model is used. It is observed in these compounds case as in other LC compounds Vuks isotropic model is favored compared to that of anisotropic model proposed by Neugebauer. 展开更多
关键词 Orientstonal Order Parameter S 2O.Om LC Compounds Vuks Isotropic Model Neugebauer Anisotropic Model Nematic Phase
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Shape phase transition in the odd Sm nuclei: effective order parameter and odd–even effect 认领 引用
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作者 张宇 关鑫 +3 位作者 王印 左岩 包丽娜 潘峰 《Chinese Physics C》 SCIE CAS CSCD 2015年第10期48-54,共7页
Some binding energy related quantities serving as effective order parameters have been used to analyze the shape phase transition in the odd Sm nuclei. It is found that the signals of phase transition in the odd Sm nu... Some binding energy related quantities serving as effective order parameters have been used to analyze the shape phase transition in the odd Sm nuclei. It is found that the signals of phase transition in the odd Sm nuclei are greatly enhanced in contrast to the even Sm nuclei. A further analysis shows that the transitional behaviors related to pairing in the Sm nuclei can be well described by the mean field plus pairing interaction model, with a monotonic decrease in the pairing strength G. 展开更多
关键词 shape phase transition effective order parameter odd even effects
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Order parameter analysis of synchronization transitions on star networks 认领 引用 被引量：4
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作者 Hong-Bin Chen Yu-Ting Sun +2 位作者 Jian Gao Can Xu Zhi-Gang Zheng 《Frontiers of physics》 SCIE CSCD 2017年第6期11-20,共10页
The collective behaviors of populations of coupled oscillators have attracted significant attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechan... The collective behaviors of populations of coupled oscillators have attracted significant attention in recent years. In this paper, an order parameter approach is proposed to study the low-dimensional dynamical mechanism of collective synchronizations, by adopting the star-topology of coupled oscil- lators as a prototype system. The order parameter equation of star-linked phase oscillators can be obtained in terms of the Watanabe--Strogatz transformation, Ott--Antonsen ansatz, and the ensem- ble order parameter approach. Different solutions of the order parameter equation correspond to the diverse collective states, and different bifurcations reveal various transitions among these collective states. The properties of various transitions in the star-network model are revealed by using tools of nonlinear dynamics such as time reversibility analysis and linear stability analysis. 展开更多
关键词 Kuramoto model synchronization order parameter Ott-Antonsen ansatz star network
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From microscopic theory to macroscopic theory——symmetries and order parameters of rigid molecules 认领 引用 被引量：1
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作者 XU Jie ZHANG PingWen 《Science China Mathematics》 SCIE 2014年第3期443-468,共26页
Density functional theory is used to describe the phase behaviors of rigid molecules. The construc- tion of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C2v... Density functional theory is used to describe the phase behaviors of rigid molecules. The construc- tion of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C2v symmetry. Molecular symmetries lead to the symmetries of the kernel function and the density function, enabling a reduction of configuration space. By approximating the kernel function with a polynomial, the system can be fully characterized by some moments corresponding to the form of the kernel function. The symmetries of the kernel function determine the form of the polynomial, while the coefficients are determined by the tem- perature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Combining the analysis and some simulation results, we propose a minimal set of order parameters for bent-core molecules. 展开更多
关键词 liquid crystals symmetry order parameter bent-core molecules excluded volume
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Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model 认领 引用
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作者 Yan-Wei Dai Sheng-Hao Li Xi-Hao Chen 《Chinese Physics B》 SCIE EI CAS CSCD 2022年第7期163-168,共6页
We investigate quantum phase transitions for q-state quantum Potts models(q=2,3,4)on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with ... We investigate quantum phase transitions for q-state quantum Potts models(q=2,3,4)on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme.We extend the universal order parameter to a two-dimensional lattice system,which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G.The universal order parameter is zero in the symmetric phase,and it ranges from zero to unity in the symmetry-broken phase.The ground-state fidelity per lattice site is computed,and a pinch point is identified on the fidelity surface near the critical point.The results offer another example highlighting the connection between(i)critical points for a quantum many-body system undergoing a quantum phase-transition and(ii)pinch points on a fidelity surface.In addition,we discuss three quantum coherence measures:the quantum Jensen–Shannon divergence,the relative entropy of coherence,and the l1norm of coherence,which are singular at the critical point,thereby identifying quantum phase transitions. 展开更多
关键词 quantum phase transitions universal order parameter fidelity
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SPECTRUM DISTRIBUTION OF THE SEDOND ORDER GENERALIZED DISTRIBUTED PARAMETER SYSTEMS 认领 引用 被引量：2
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作者 葛照强 朱广田 马勇镐 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2003年第10期1224-1232,共9页
Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expr... Spectrum distribution of the second order generalized distributed parameter system was discussed via the functional analysis and operator theory in Hilbert space. The solutions of the problem and the constructive expression of the solutions are given by the generalized inverse one of bounded linear operator. This is theoretically important for studying the stabilization and asymptotic stability of the second order generalized distributed parameter system. 展开更多
关键词 second order generalized distributed parameter system spectrum distribution Hilbert space generalized inverse one of bounded linear operatiors
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CONSTITUTIVE EQUATION WITH ORDERED PARAMETERS FOR VISCOELASTIC MATERIAL CONTAINING DEFECTS 认领 引用 被引量：3
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作者 Yuan Longwei 《Acta Mechanica Solida Sinica》 SCIE 1997年第1期11-27,共17页
Based on the experimental results of local temperature field formed in the evolution period of defects, the defect field is defined by this internal temperature field. The evolutionary processes of statistically corre... Based on the experimental results of local temperature field formed in the evolution period of defects, the defect field is defined by this internal temperature field. The evolutionary processes of statistically correlative meso-scopic defects are analysed with the growth rate and nucleation rate of the meso-defect, and it is deemed that the dynamic failure process of the viscoelastic material with defects can be quantitatively described by the normal method of the procedure of heat wave transmission. The defect field is regarded as a complex system of the union of the real sets and null sets, its main characteristics depending on the stratum hypothesis about the activities of the subsystems of the highest stratum described by ordered parameters. The fluctuation of ordered parameters is demonstrated by means of the projection operator method. The constitutive equation with ordered parameters for the meso-defect evolutionary state of viscoelastic material with defects is deduced from the nonlinear rheological dynamic approach, and its solution is obtained. 展开更多
关键词 rheology of bodies with defects constitutive equation local temperature field heat waves ordered parameter stratum system
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Shape phase transitions in nuclei:Effective order parameters and trajectories 认领 引用 被引量：3
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作者 ZHANG Yu HOU ZhanFeng LIU YuXin 《Science China(Physics,Mechanics & Astronomy)》 SCIE EI CAS 2011年第S1期88-97,共10页
We analyze systematically the effective order parameters in nuclear shape phase transition both in experiments and in the interacting boson model.We find that energy ratios and B(E2)ratios can distinguish the first fr... We analyze systematically the effective order parameters in nuclear shape phase transition both in experiments and in the interacting boson model.We find that energy ratios and B(E2)ratios can distinguish the first from the second-order phase transition in theory above a certain boson number N(about 50),but in experiments,only those quantities,such as E(L1+)/E(02+)and B(E2;(L+2)1→L1)/B(E2;21→01),etc.,of which the monotonous transitional behavior in the second-order phase transition is broken in the first order phase transition independent of N,are qualified as the effective order parameters.By implementing the originally proposed effective order parameters and the new ones,we find that the isotones with neutron number Nn=62 are a trajectory of the second order phase transition.In addition,we predict that the transitional behavior of isomer shifts of Xe,Ba isotopes and Nn=62 isotones is approximately monotonous due to the finiteness of nuclear system. 展开更多
关键词 nuclear shape phase transition interacting boson model effective order parameter finite particle number effect transition trajectory
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Finite-Size Scaling Behaviors of Eigen Microstates near the Critical Point 认领 引用
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作者 Gaoke Hu Jia-Qi Dong +4 位作者 Yongwen Zhang Teng Liu Wen-Long You Maoxin Liu Xiaosong Chen 《Chinese Physics Letters》 SCIE EI CAS CSCD 2026年第2期1-12,共12页
In the eigen microstate approach(EMA),phase emergence is signaled by the condensation of an eigen microstate with a finite eigenvalue.The associated eigenvalue acts as an order parameter,exhibiting finite-size scaling... In the eigen microstate approach(EMA),phase emergence is signaled by the condensation of an eigen microstate with a finite eigenvalue.The associated eigenvalue acts as an order parameter,exhibiting finite-size scaling(FSS)behaviors within the critical regime.The eigen microstate derived from EMA precisely characterizes the spatial structure of the emergent phase,which remains less explored in systems lacking translational invariance.Here,we propose an FSS form for the eigen microstate itself and numerically validate it using the two-dimensional Ising model under various boundary conditions.This result enables the derivation of FSS relations for correlation functions,which we verify through numerical simulations.Our results complete the FSS formalism for the EMA. 展开更多
关键词 condensation eigen microstate finite size scaling eigen microstate approach critical regimethe fss form eigen microstate order parameter eigen microstate approach ema phase emergence
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THEORY OF INTRAMOLECULAR ORIENTATIONAL ORDER OF MESOGENIC UNITS IN MC-LCPS AND ITS APPLICATION 认领 引用
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作者 王晓工 刘德山 周其庠 《Chinese Journal of Polymer Science》 SCIE EI CAS 1993年第4期17-25,共9页
New concepts such as intramolecular orientational order parameter and corresponding model as well as theory were proposed to describe the intramolecular orientation of mesogenic units in the liquid crystalline polymer... New concepts such as intramolecular orientational order parameter and corresponding model as well as theory were proposed to describe the intramolecular orientation of mesogenic units in the liquid crystalline polymer chains. The relationship between the intramolecular orientational order parameter and the molecular geometrical parameters such as the bond angle, the bond rotational angle and the rotational potential energy of chemical bonds was deduced. A significant even-odd oscillation of the intramolecular orientational order parameter of LCPs with different length of flexible spacer was found and rationally related to even-odd zig-zag manner of transition properties The verification and application of the theory are also discussed. The isotropic transition temperature predicted by the theory is shown to be in favourable agreement with the experiments. 展开更多
关键词 Liquid crystal polymer Intramolecular orientation order parameter, Theory.
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参数化泊松方程的模型降阶预处理 认领 引用
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作者 胡奇晓 徐友才 张世全 《四川大学学报（自然科学版）》 CAS CSCD 北大核心 2026年第3期559-565,共7页
本文针对数值求解参数化泊松方程提出了一种基于模型降阶思想的预处理共轭梯度(Preconditioned Conjugate Gradient,PCG)算法。为加快所得离散线性方程组的求解速度,本文首先基于模型降阶思想构造了一般形式的预处理矩阵,并证明其对称... 本文针对数值求解参数化泊松方程提出了一种基于模型降阶思想的预处理共轭梯度(Preconditioned Conjugate Gradient,PCG)算法。为加快所得离散线性方程组的求解速度,本文首先基于模型降阶思想构造了一般形式的预处理矩阵,并证明其对称正定性。在算法的off-line阶段,本文基于少量参数对应的全模型解数据,利用PCG算法并结合本征正交分解(Proper Orthogonal Decomposition,POD)方法生成了一组动态预处理矩阵。然后,在on-line阶段,本文利用动态预处理矩阵结合PCG算法建立所需算法。为了验证算法的性能,本文分别在单位矩形区域和L形区域上数值求解参数化泊松方程。结果显示,在相同精度条件下,算法的平均计算时间比标准共轭梯度算法快40倍以上。 展开更多
关键词 参数化泊松方程 模型降阶 预处理矩阵 共轭梯度
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面向SCADE高阶语言结构的程序转换和模型检验技术 认领 引用
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作者 王慧杰 陈哲 韦依洋 《计算机科学与探索》 CSCD 北大核心 2026年第7期1954-1962,共9页
同步数据流语言被广泛应用于安全攸关系统的研发,例如Lustre语言和SCADE语言。SCADE语言相较于Lustre语言,增加了多种易用且具有高阶编程概念的语言结构。安全攸关系统的安全性验证需借助模型检验来完成形似化的系统证明。但目前SCADE... 同步数据流语言被广泛应用于安全攸关系统的研发,例如Lustre语言和SCADE语言。SCADE语言相较于Lustre语言,增加了多种易用且具有高阶编程概念的语言结构。安全攸关系统的安全性验证需借助模型检验来完成形似化的系统证明。但目前SCADE程序的验证工具存在的问题有不支持所有的高阶语言结构,能够验证的程序模型局限在可扩展标记结构,导致现阶段对大型复杂程序的验证能力较差。主要研究对含有高阶语言结构(map、fold等12类高阶迭代操作符以及参数化节点)的SCADE程序进行模型检验。提出了从高阶语言结构到Lustre程序的基于语义等价思想的转换算法,针对转换后的程序结合现有工具完成模型检验;提出了时钟数据流等价性的概念评价转换前后程序语义的等价性;在含有高阶语言结构的SCADE测试集上进行实验,通过实验评估了依据转换算法设计的模型检验工具的有效性,可以完成对含有高阶语言结构程序的模型检验。提出和实现的方法提高了对SCADE程序进行模型检验的语言兼容能力。 展开更多
关键词 SCADE 高阶迭代器 参数化节点 程序转换 模型检验
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An optimization-based equilibrium measure describing fixed points of non-equilibrium dynamics:application to the edge of chaos 认领 引用 被引量：1
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作者 Junbin Qiu Haiping Huang 《Communications in Theoretical Physics》 SCIE CAS CSCD 2025年第3期140-156,共17页
Understanding neural dynamics is a central topic in machine learning,non-linear physics,and neuroscience.However,the dynamics are non-linear,stochastic and particularly non-gradient,i.e.,the driving force cannot be wr... Understanding neural dynamics is a central topic in machine learning,non-linear physics,and neuroscience.However,the dynamics are non-linear,stochastic and particularly non-gradient,i.e.,the driving force cannot be written as the gradient of a potential.These features make analytic studies very challenging.The common tool is the path integral approach or dynamical mean-field theory.Still,the drawback is that one has to solve the integro-differential or dynamical mean-field equations,which is computationally expensive and has no closed-form solutions in general.From the associated Fokker-Planck equation,the steady-state solution is generally unknown.Here,we treat searching for the fixed points as an optimization problem,and construct an approximate potential related to the speed of the dynamics,and find that searching for the ground state of this potential is equivalent to running approximate stochastic gradient dynamics or Langevin dynamics.Only in the zero temperature limit,can the distribution of the original fixed points be achieved.The resultant stationary state of the dynamics exactly follows the canonical Boltzmann measure.Within this framework,the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method,which leads naturally to order parameters for the non-equilibrium steady states.Our theory reproduces the well-known result of edge-of-chaos.Furthermore,the order parameters characterizing the continuous transition are derived,and the order parameters are explained as fluctuations and responses of the steady states.Our method thus opens the door to analytically studying the fixed-point landscape of the deterministic or stochastic high dimensional dynamics. 展开更多
关键词 high-dimensional chaos phase transitions neural networks order parameters statistical physics
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