Operational optimization of copper flotation process based on the weighted Gaussian process regression and index-oriented adaptive differential evolution algorithm

Concentrate copper grade(CCG)is one of the important production indicators of copper flotation processes,and keeping the CCG at the set value is of great significance to the economic benefit of copper flotation industrial processes.This paper addresses the fluctuation problem of CCG through an operational optimization method.Firstly,a density-based affinity propagationalgorithm is proposed so that more ideal working condition categories can be obtained for the complex raw ore properties.Next,a Bayesian network(BN)is applied to explore the relationship between the operational variables and the CCG.Based on the analysis results of BN,a weighted Gaussian process regression model is constructed to predict the CCG that a higher prediction accuracy can be obtained.To ensure the predicted CCG is close to the set value with a smaller magnitude of the operation adjustments and a smaller uncertainty of the prediction results,an index-oriented adaptive differential evolution(IOADE)algorithm is proposed,and the convergence performance of IOADE is superior to the traditional differential evolution and adaptive differential evolution methods.Finally,the effectiveness and feasibility of the proposed methods are verified by the experiments on a copper flotation industrial process.

Reliable calculations of nuclear binding energies by the Gaussian process of machine learning

Reliable calculations of nuclear binding energies are crucial for advancing the research of nuclear physics. Machine learning provides an innovative approach to exploring complex physical problems. In this study, the nuclear binding energies are modeled directly using a machine-learning method called the Gaussian process. First, the binding energies for 2238 nuclei with Z > 20 and N > 20 are calculated using the Gaussian process in a physically motivated feature space, yielding an average deviation of 0.046 MeV and a standard deviation of 0.066 MeV. The results show the good learning ability of the Gaussian process in the studies of binding energies. Then, the predictive power of the Gaussian process is studied by calculating the binding energies for 108 nuclei newly included in AME2020. The theoretical results are in good agreement with the experimental data, reflecting the good predictive power of the Gaussian process. Moreover, the α-decay energies for 1169 nuclei with 50 ≤ Z ≤ 110 are derived from the theoretical binding energies calculated using the Gaussian process. The average deviation and the standard deviation are, respectively, 0.047 MeV and 0.070 MeV. Noticeably, the calculated α-decay energies for the two new isotopes ^ (204 )Ac(Huang et al. Phys Lett B 834, 137484(2022)) and ^ (207) Th(Yang et al. Phys Rev C 105, L051302(2022)) agree well with the latest experimental data. These results demonstrate that the Gaussian process is reliable for the calculations of nuclear binding energies. Finally, the α-decay properties of some unknown actinide nuclei are predicted using the Gaussian process. The predicted results can be useful guides for future research on binding energies and α-decay properties.

Optimization of Generator Based on Gaussian Process Regression Model with Conditional Likelihood Lower Bound Search

The noise that comes from finite element simulation often causes the model to fall into the local optimal solution and over fitting during optimization of generator.Thus,this paper proposes a Gaussian Process Regression(GPR)model based on Conditional Likelihood Lower Bound Search(CLLBS)to optimize the design of the generator,which can filter the noise in the data and search for global optimization by combining the Conditional Likelihood Lower Bound Search method.Taking the efficiency optimization of 15 kW Permanent Magnet Synchronous Motor as an example.Firstly,this method uses the elementary effect analysis to choose the sensitive variables,combining the evolutionary algorithm to design the super Latin cube sampling plan;Then the generator-converter system is simulated by establishing a co-simulation platform to obtain data.A Gaussian process regression model combing the method of the conditional likelihood lower bound search is established,which combined the chi-square test to optimize the accuracy of the model globally.Secondly,after the model reaches the accuracy,the Pareto frontier is obtained through the NSGA-II algorithm by considering the maximum output torque as a constraint.Last,the constrained optimization is transformed into an unconstrained optimizing problem by introducing maximum constrained improvement expectation(CEI)optimization method based on the re-interpolation model,which cross-validated the optimization results of the Gaussian process regression model.The above method increase the efficiency of generator by 0.76%and 0.5%respectively;And this method can be used for rapid modeling and multi-objective optimization of generator systems.

State of health prediction for lithium-ion batteries based on ensemble Gaussian process regression

The performance of lithium-ion batteries(LIBs)gradually declines over time,making it critical to predict the battery’s state of health(SOH)in real-time.This paper presents a model that incorporates health indicators and ensemble Gaussian process regression(EGPR)to predict the SOH of LIBs.Firstly,the degradation process of an LIB is analyzed through indirect health indicators(HIs)derived from voltage and temperature during discharge.Next,the parameters in the EGPR model are optimized using the gannet optimization algorithm(GOA),and the EGPR is employed to estimate the SOH of LIBs.Finally,the proposed model is tested under various experimental scenarios and compared with other machine learning models.The effectiveness of EGPR model is demonstrated using the National Aeronautics and Space Administration(NASA)LIB.The root mean square error(RMSE)is maintained within 0.20%,and the mean absolute error(MAE)is below 0.16%,illustrating the proposed approach’s excellent predictive accuracy and wide applicability.

Fast Remaining Capacity Estimation for Lithium-ion Batteries Based on Short-time Pulse Test and Gaussian Process Regression

It remains challenging to effectively estimate the remaining capacity of the secondary lithium-ion batteries that have been widely adopted for consumer electronics,energy storage,and electric vehicles.Herein,by integrating regular real-time current short pulse tests with data-driven Gaussian process regression algorithm,an efficient battery estimation has been successfully developed and validated for batteries with capacity ranging from 100%of the state of health(SOH)to below 50%,reaching an average accuracy as high as 95%.Interestingly,the proposed pulse test strategy for battery capacity measurement could reduce test time by more than 80%compared with regular long charge/discharge tests.The short-term features of the current pulse test were selected for an optimal training process.Data at different voltage stages and state of charge(SOC)are collected and explored to find the most suitable estimation model.In particular,we explore the validity of five different machine-learning methods for estimating capacity driven by pulse features,whereas Gaussian process regression with Matern kernel performs the best,providing guidance for future exploration.The new strategy of combining short pulse tests with machine-learning algorithms could further open window for efficiently forecasting lithium-ion battery remaining capacity.

Multi-output Gaussian Process Regression Model with Combined Kernel Function for Polyester Esterification Processes

In polyester fiber industrial processes,the prediction of key performance indicators is vital for product quality.The esterification process is an indispensable step in the polyester polymerization process.It has the characteristics of strong coupling,nonlinearity and complex mechanism.To solve these problems,we put forward a multi-output Gaussian process regression(MGPR)model based on the combined kernel function for the polyester esterification process.Since the seasonal and trend decomposition using loess(STL)can extract the periodic and trend characteristics of time series,a combined kernel function based on the STL and the kernel function analysis is constructed for the MGPR.The effectiveness of the proposed model is verified by the actual polyester esterification process data collected from fiber production.

Quality prediction of batch process using the global-local discriminant analysis based Gaussian process regression model

The conventional single model strategy may be ill- suited due to the multiplicity of operation phases and system uncertainty. A novel global-local discriminant analysis （GLDA） based Gaussian process regression （GPR） approach is developed for the quality prediction of nonlinear and multiphase batch processes. After the collected data is preprocessed through batchwise unfolding, the hidden Markov model （HMM） is applied to identify different operation phases. A GLDA algorithm is also presented to extract the appropriate process variables highly correlated with the quality variables, decreasing the complexity of modeling. Besides, the multiple local GPR models are built in the reduced- dimensional space for all the identified operation phases. Furthermore, the HMM-based state estimation is used to classify each measurement sample of a test batch into a corresponding phase with the maximal likelihood estimation. Therefore, the local GPR model with respect to specific phase is selected for online prediction. The effectiveness of the proposed prediction approach is demonstrated through the multiphase penicillin fermentation process. The comparison results show that the proposed GLDA-GPR approach is superior to the regular GPR model and the GPR based on HMM （HMM-GPR） model.

Multiple Model Soft Sensor Based on Affinity Propagation, Gaussian Process and Bayesian Committee Machine

Presented is a multiple model soft sensing method based on Affinity Propagation （AP）, Gaussian process （GP） and Bayesian committee machine （BCM）. AP clustering arithmetic is used to cluster training samples according to their operating points. Then, the sub-models are estimated by Gaussian Process Regression （GPR）. Finally, in order to get a global probabilistic prediction, Bayesian committee mactnne is used to combine the outputs of the sub-estimators. The proposed method has been applied to predict the light naphtha end point in hydrocracker fractionators. Practical applications indicate that it is useful for the online prediction of quality monitoring in chemical processes.

Novel methodology for casting process optimization using Gaussian process regression and genetic algorithm

High pressure die casting (HPDC) is a versatile material processing method for mass-production of metal parts with complex geometries,and this method has been widely used in manufacturing various products of excellent dimensional accuracy and productivity. In order to ensure the quality of the components,a number of variables need to be properly set. A novel methodology for high pressure die casting process optimization was developed,validated and applied to selection of optimal parameters,which incorporate design of experiment (DOE),Gaussian process (GP) regression technique and genetic algorithms (GA). This new approach was applied to process optimization for cast magnesium alloy notebook shell. After being trained,using data generated by PROCAST (FEM-based simulation software),the GP model approximated well with the simulation by extracting useful information from the simulation results. With the help of MATLAB,the GP/GA based approach has achieved the optimum solution of die casting process condition settings.

Joint asymptotic distribution of exceedances point process and partial sum of stationary Gaussian sequence

Let {Xi}i=1^∞ be a standardized stationary Gaussian sequence with covariance function τ（n） =EX1Xn＋1, Sn =∑i=1^nXi,and X^-n=Sn/n.And let Nn be the point process formed by the exceedances of random level （x/√2 log n＋√2 log n-log（4π log n）/2√log n） √1-τ（n） ＋ X^-n by X1,X2,…, Xn. Under some mild conditions, Nn and Sn are asymptotically independent, and Nn converges weakly to a Poisson process on （0,1].

Multimodal process monitoring based on transition-constrained Gaussian mixture model

Reliable process monitoring is important for ensuring process safety and product quality.A production process is generally characterized bymultiple operation modes,and monitoring thesemultimodal processes is challenging.Most multimodal monitoring methods rely on the assumption that the modes are independent of each other,which may not be appropriate for practical application.This study proposes a transition-constrained Gaussian mixture model method for efficient multimodal process monitoring.This technique can reduce falsely and frequently occurring mode transitions by considering the time series information in the mode identification of historical and online data.This process enables the identified modes to reflect the stability of actual working conditions,improve mode identification accuracy,and enhance monitoring reliability in cases of mode overlap.Case studies on a numerical simulation example and simulation of the penicillin fermentation process are provided to verify the effectiveness of the proposed approach inmultimodal process monitoring with mode overlap.

Gaussian process regression-based quaternion unscented Kalman robust filter for integrated SINS/GNSS

High-precision filtering estimation is one of the key techniques for strapdown inertial navigation system/global navigation satellite system(SINS/GNSS)integrated navigation system,and its estimation plays an important role in the performance evaluation of the navigation system.Traditional filter estimation methods usually assume that the measurement noise conforms to the Gaussian distribution,without considering the influence of the pollution introduced by the GNSS signal,which is susceptible to external interference.To address this problem,a high-precision filter estimation method using Gaussian process regression(GPR)is proposed to enhance the prediction and estimation capability of the unscented quaternion estimator(USQUE)to improve the navigation accuracy.Based on the advantage of the GPR machine learning function,the estimation performance of the sliding window for model training is measured.This method estimates the output of the observation information source through the measurement window and realizes the robust measurement update of the filter.The combination of GPR and the USQUE algorithm establishes a robust mechanism framework,which enhances the robustness and stability of traditional methods.The results of the trajectory simulation experiment and SINS/GNSS car-mounted tests indicate that the strategy has strong robustness and high estimation accuracy,which demonstrates the effectiveness of the proposed method.

LEAST SQUARES TYPE ESTIMATION FOR DISCRETELY OBSERVED NON-ERGODIC GAUSSIAN ORNSTEIN-UHLENBECK PROCESSES

In this article, we consider the drift parameter estimation problem for the nonergodic Ornstein-Uhlenbeck process defined as dXt = OXtdt + dGt, i > 0 with an unknown parameter θ> 0, where G is a Gaussian process. We assume that the process {xt,t≥ 0} is observed at discrete time instants t1=△n,…, tn = n△n, and we construct two least squares type estimators θn and θn for θ on the basis of the discrete observations ,{xti,i= 1,…, n} as →∞. Then, we provide sufficient conditions, based on properties of G, which ensure that θn and θn are strongly consistent and the sequences √n△n(θn-θ) and √n△n(θn-θ) are tight. Our approach offers an elementary proof of [11], which studied the case when G is a fractional Brownian motion with Hurst parameter H∈(1/2, 1). As such, our results extend the recent findings by [11] to the case of general Hurst parameter H∈(0,1). We also apply our approach to study subfractional Ornstein-Uhlenbeck and bifractional Ornstein-Uhlenbeck processes.

PARAMETER ESTIMATION FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A GENERAL GAUSSIAN NOISE

In this paper,we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process(G_(t))t≥0.The second order mixed partial derivative of the covariance function R(t,s)=E[GtGs]can be decomposed into two parts,one of which coincides with that of fractional Brownian motion and the other of which is bounded by(ts)^(β-1)up to a constant factor.This condition is valid for a class of continuous Gaussian processes that fails to be self-similar or to have stationary increments;some examples of this include the subfractional Brownian motion and the bi-fractional Brownian motion.Under this assumption,we study the parameter estimation for a drift parameter in the Ornstein-Uhlenbeck process driven by the Gaussian noise(G_(t))t≥0.For the least squares estimator and the second moment estimator constructed from the continuous observations,we prove the strong consistency and the asympotic normality,and obtain the Berry-Esséen bounds.The proof is based on the inner product's representation of the Hilbert space(h)associated with the Gaussian noise(G_(t))t≥0,and the estimation of the inner product based on the results of the Hilbert space associated with the fractional Brownian motion.

MULTI-SCALE GAUSSIAN PROCESSES MODEL

A novel model named Multi-scale Gaussian Processes (MGP) is proposed. Motivated by the ideas of multi-scale representations in the wavelet theory, in the new model, a Gaussian process is represented at a scale by a linear basis that is composed of a scale function and its different translations. Finally the distribution of the targets of the given samples can be obtained at different scales. Compared with the standard Gaussian Processes (GP) model, the MGP model can control its complexity conveniently just by adjusting the scale pa-rameter. So it can trade-off the generalization ability and the empirical risk rapidly. Experiments verify the fea-sibility of the MGP model, and exhibit that its performance is superior to the GP model if appropriate scales are chosen.

Dynamic soft sensor development based on Gaussian mixture regression for fermentation processes

The dynamic soft sensor based on a single Gaussian process regression(GPR) model has been developed in fermentation processes.However,limitations of single regression models,for multiphase/multimode fermentation processes,may result in large prediction errors and complexity of the soft sensor.Therefore,a dynamic soft sensor based on Gaussian mixture regression(GMR) was proposed to overcome the problems.Two structure parameters,the number of Gaussian components and the order of the model,are crucial to the soft sensor model.To achieve a simple and effective soft sensor,an iterative strategy was proposed to optimize the two structure parameters synchronously.For the aim of comparisons,the proposed dynamic GMR soft sensor and the existing dynamic GPR soft sensor were both investigated to estimate biomass concentration in a Penicillin simulation process and an industrial Erythromycin fermentation process.Results show that the proposed dynamic GMR soft sensor has higher prediction accuracy and is more suitable for dynamic multiphase/multimode fermentation processes.

A genetic Gaussian process regression model based on memetic algorithm

Gaussian process(GP)has fewer parameters,simple model and output of probabilistic sense,when compared with the methods such as support vector machines.Selection of the hyper-parameters is critical to the performance of Gaussian process model.However,the common-used algorithm has the disadvantages of difficult determination of iteration steps,over-dependence of optimization effect on initial values,and easily falling into local optimum.To solve this problem,a method combining the Gaussian process with memetic algorithm was proposed.Based on this method,memetic algorithm was used to search the optimal hyper parameters of Gaussian process regression(GPR)model in the training process and form MA-GPR algorithms,and then the model was used to predict and test the results.When used in the marine long-range precision strike system(LPSS)battle effectiveness evaluation,the proposed MA-GPR model significantly improved the prediction accuracy,compared with the conjugate gradient method and the genetic algorithm optimization process.

Soft sensor modeling based on Gaussian processes

In order to meet the demand of online optimal running, a novel soft sensor modeling approach based on Gaussian processes was proposed. The approach is moderately simple to implement and use without loss of performance. It is trained by optimizing the hyperparameters using the scaled conjugate gradient algorithm with the squared exponential covariance function employed. Experimental simulations show that the soft sensor modeling approach has the advantage via a real-world example in a refinery. Meanwhile, the method opens new possibilities for application of kernel methods to potential fields.

Sparse Additive Gaussian Process with Soft Interactions

This paper presents a novel variable selection method in additive nonparametric regression model. This work is motivated by the need to select the number of nonparametric components and number of variables within each nonparametric component. The proposed method uses a combination of hard and soft shrinkages to separately control the number of additive components and the variables within each component. An efficient algorithm is developed to select the importance of variables and estimate the interaction network. Excellent performance is obtained in simulated and real data examples.

Multi-fidelity Gaussian process based empirical potential development for Si:H nanowires

In material modeling,the calculation speed using the empirical potentials is fast compared to the first principle calculations,but the results are not as accurate as of the first principle calculations.First principle calculations are accurate but slow and very expensive to calculate.In this work,first,the H-H binding energy and H2-H2 interaction energy are calculated using the first principle calculations which can be applied to the Tersoff empirical potential.Second,the H-H parameters are estimated.After fitting H-H parameters,the mechanical properties are obtained.Finally,to integrate both the low-fidelity empirical potential data and the data from the high-fidelity firstprinciple calculations,the multi-fidelity Gaussian process regression is employed to predict the HH binding energy and the H2-H2 interaction energy.Numerical results demonstrate the accuracy of the developed empirical potentials.