Uncertainty quantification of inverse analysis for geomaterials using probabilistic programming

Uncertainty is an essentially challenging for safe construction and long-term stability of geotechnical engineering.The inverse analysis is commonly utilized to determine the physico-mechanical parameters.However,conventional inverse analysis cannot deal with uncertainty in geotechnical and geological systems.In this study,a framework was developed to evaluate and quantify uncertainty in inverse analysis based on the reduced-order model(ROM)and probabilistic programming.The ROM was utilized to capture the mechanical and deformation properties of surrounding rock mass in geomechanical problems.Probabilistic programming was employed to evaluate uncertainty during construction in geotechnical engineering.A circular tunnel was then used to illustrate the proposed framework using analytical and numerical solution.The results show that the geomechanical parameters and associated uncertainty can be properly obtained and the proposed framework can capture the mechanical behaviors under uncertainty.Then,a slope case was employed to demonstrate the performance of the developed framework.The results prove that the proposed framework provides a scientific,feasible,and effective tool to characterize the properties and physical mechanism of geomaterials under uncertainty in geotechnical engineering problems.

High-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer based on probability density evolution method

This paper proposed an efficient research method for high-dimensional uncertainty quantification of projectile motion in the barrel of a truck-mounted howitzer.Firstly,the dynamic model of projectile motion is established considering the flexible deformation of the barrel and the interaction between the projectile and the barrel.Subsequently,the accuracy of the dynamic model is verified based on the external ballistic projectile attitude test platform.Furthermore,the probability density evolution method(PDEM)is developed to high-dimensional uncertainty quantification of projectile motion.The engineering example highlights the results of the proposed method are consistent with the results obtained by the Monte Carlo Simulation(MCS).Finally,the influence of parameter uncertainty on the projectile disturbance at muzzle under different working conditions is analyzed.The results show that the disturbance of the pitch angular,pitch angular velocity and pitch angular of velocity decreases with the increase of launching angle,and the random parameter ranges of both the projectile and coupling model have similar influence on the disturbance of projectile angular motion at muzzle.

Uncertainty quantification of mechanism motion based on coupled mechanism—motor dynamic model for ammunition delivery system

In this paper,a dynamic modeling method of motor driven electromechanical system is presented,and the uncertainty quantification of mechanism motion is investigated based on this method.The main contribution is to propose a novel mechanism-motor coupling dynamic modeling method,in which the relationship between mechanism motion and motor rotation is established according to the geometric coordination of the system.The advantages of this include establishing intuitive coupling between the mechanism and motor,facilitating the discussion for the influence of both mechanical and electrical parameters on the mechanism,and enabling dynamic simulation with controller to take the randomness of the electric load into account.Dynamic simulation considering feedback control of ammunition delivery system is carried out,and the feasibility of the model is verified experimentally.Based on probability density evolution theory,we comprehensively discuss the effects of system parameters on mechanism motion from the perspective of uncertainty quantization.Our work can not only provide guidance for engineering design of ammunition delivery mechanism,but also provide theoretical support for modeling and uncertainty quantification research of mechatronics system.

Uncertainty quantification of predicting stable structures for high-entropy alloys using Bayesian neural networks

High entropy alloys(HEAs)have excellent application prospects in catalysis because of their rich components and configuration space.In this work,we develop a Bayesian neural network(BNN)based on energies calculated with density functional theory to search the configuration space of the CoNiRhRu HEA system.The BNN model was developed by considering six independent features of Co-Ni,Co-Rh,CoRu,Ni-Rh,Ni-Ru,and Rh-Ru in different shells and energies of structures as the labels.The root mean squared error of the energy predicted by BNN is 1.37 me V/atom.Moreover,the influence of feature periodicity on the energy of HEA in theoretical calculations is discussed.We found that when the neural network is optimized to a certain extent,only using the accuracy indicator of root mean square error to evaluate model performance is no longer accurate in some scenarios.More importantly,we reveal the importance of uncertainty quantification for neural networks to predict new structures of HEAs with proper confidence based on BNN.

Computational intelligence approach for uncertainty quantification using evidence theory

As an alternative or complementary approach to the classical probability theory,the ability of the evidence theory in uncertainty quantification（UQ） analyses is subject of intense research in recent years.Two state-of-the-art numerical methods,the vertex method and the sampling method,are commonly used to calculate the resulting uncertainty based on the evidence theory.The vertex method is very effective for the monotonous system,but not for the non-monotonous one due to its high computational errors.The sampling method is applicable for both systems.But it always requires a high computational cost in UQ analyses,which makes it inefficient in most complex engineering systems.In this work,a computational intelligence approach is developed to reduce the computational cost and improve the practical utility of the evidence theory in UQ analyses.The method is demonstrated on two challenging problems proposed by Sandia National Laboratory.Simulation results show that the computational efficiency of the proposed method outperforms both the vertex method and the sampling method without decreasing the degree of accuracy.Especially,when the numbers of uncertain parameters and focal elements are large,and the system model is non-monotonic,the computational cost is five times less than that of the sampling method.

Analysis of actuator delay and its effect on uncertainty quantification for real-time hybrid simulation

Uncertainties in structure properties can result in different responses in hybrid simulations. Quantification of the effect of these tmcertainties would enable researchers to estimate the variances of structural responses observed from experiments. This poses challenges for real-time hybrid simulation （RTHS） due to the existence of actuator delay. Polynomial chaos expansion （PCE） projects the model outputs on a basis of orthogonal stochastic polynomials to account for influences of model uncertainties. In this paper, PCE is utilized to evaluate effect of actuator delay on the maximum displacement from real-time hybrid simulation of a single degree of freedom （SDOF） structure when accounting for uncertainties in structural properties. The PCE is first applied for RTHS without delay to determine the order of PCE, the number of sample points as well as the method for coefficients calculation. The PCE is then applied to RTHS with actuator delay. The mean, variance and Sobol indices are compared and discussed to evaluate the effects of actuator delay on uncertainty quantification for RTHS. Results show that the mean and the variance of the maximum displacement increase linearly and exponentially with respect to actuator delay, respectively. Sensitivity analysis through Sobol indices also indicates the influence of the single random variable decreases while the coupling effect increases with the increase of actuator delay.

Uncertainty Quantification of Numerical Simulation of Flows around a Cylinder Using Non-intrusive Polynomial Chaos

The uncertainty quantification of flows around a cylinder is studied by the non-intrusive polynomial chaos method. Based on the validation with benchmark results, discussions are mainly focused on the statistic properties of the peak lift and drag coefficients and base pressure drop over the cylinder with the uncertainties of viscosity coefficient and inflow boundary velocity. As for the numerical results of flows around a cylinder, influence of the inflow boundary velocity uncertainty is larger than that of viscosity. The results indeed demonstrate that a five-order degree of polynomial chaos expansion is enough to represent the solution of flow in this study.

Residual Convolution Long Short-Term Memory Network for Machines Remaining Useful Life Prediction and Uncertainty Quantification

Recently,deep learning(DL)has been widely used in the field of remaining useful life(RUL)prediction.Among various DL technologies,recurrent neural network(RNN)and its variant,e.g.,long short-term memory(LSTM)network,have gained extensive attention for their ability to capture temporal dependence.Although existing RNN-based methods have demonstrated their RUL prediction effectiveness,they still suffer from the following two limitations:1)it is difficult for the RNN to directly extract degradation features from original monitoring data and 2)most RNN-based prognostics methods are unable to quantify RUL uncertainty.To address the aforementioned limitations,this paper proposes a new prognostics method named residual convolution LSTM(RC-LSTM)network.In the RC-LSTM,a new ResNet-based convolution LSTM(Res-ConvLSTM)layer is stacked with a convolution LSTM(ConvLSTM)layer to extract degradation representations from monitoring data.Then,under the assumption that the RUL follows a normal distribution,an appropriate output layer is constructed to quantify the uncertainty of prediction results.Finally,the effectiveness and superiority of the RC-LSTM are verified using monitoring data from accelerated bearing degradation tests.

Comprehensive analysis of uncertainty quantification for the^(58)Ni(n,p)^(58)Co reaction cross section

In this study,we measured the^(58)Ni(n,p)^(58)Co reaction cross section with neutron energies of 1.06,1.86,and 2.85 MeV.The cross section was measured using neutron activation techniques andγ-ray spectroscopy,and it was compared with cross section data available in the EXFOR.Furthermore,we calculated the covariance matrix of the measured cross section for the aforementioned nuclear reaction.The uncertainties of the theoretical calculation for^(58)Ni(n,p)^(58)Co reaction cross section were calculated via Monte Carlo method.In this study,we used uncertainties in the optical model and level density parameters to calculate uncertainties in the theoretical cross sections.The theoretical calculations were performed by using TALYS-1.96.In this study,we aim to analyze the effect of uncertainties of the nuclear model input as well as different experimental variables used to obtain the values of reaction cross section.

Compressor geometric uncertainty quantification under conditions from near choke to near stall

Geometric and working condition uncertainties are inevitable in a compressor,deviating the compressor performance from the design value.It’s necessary to explore the influence of geometric uncertainty on performance deviation under different working conditions.In this paper,the geometric uncertainty influences at near stall,peak efficiency,and near choke conditions under design speed and low speed are investigated.Firstly,manufacturing geometric uncertainties are analyzed.Next,correlation models between geometry and performance under different working conditions are constructed based on a neural network.Then the Shapley additive explanations(SHAP)method is introduced to explain the output of the neural network.Results show that under real manufacturing uncertainty,the efficiency deviation range is small under the near stall and peak efficiency conditions.However,under the near choke conditions,efficiency is highly sensitive to flow capacity changes caused by geometric uncertainty,leading to a significant increase in the efficiency deviation amplitude,up to a magnitude of-3.6%.Moreover,the tip leading-edge radius and tip thickness are two main factors affecting efficiency deviation.Therefore,to reduce efficiency uncertainty,a compressor should be avoided working near the choke condition,and the tolerances of the tip leading-edge radius and tip thickness should be strictly controlled.

A surrogate model for uncertainty quantification and global sensitivity analysis of nonlinear large-scale dome structures

Full-scale dome structures intrinsically have numerous sources of irreducible aleatoric uncertainties.A large-scale numerical simulation of the dome structure is required to quantify the effects of these sources on the dynamic performance of the structure using the finite element method(FEM).To reduce the heavy computational burden,a surrogate model of a dome structure was constructed to solve this problem.The dynamic global sensitivity of elastic and elastoplastic structures was analyzed in the uncertainty quantification framework using fully quantitative variance-and distribution-based methods through the surrogate model.The model considered the predominant sources of uncertainty that have a significant influence on the performance of the dome structure.The effects of the variables on the structural performance indicators were quantified using the sensitivity index values of the different performance states.Finally,the effects of the sample size and correlation function on the accuracy of the surrogate model as well as the effects of the surrogate accuracy and failure probability on the sensitivity index values are discussed.The results show that surrogate modeling has high computational efficiency and acceptable accuracy in the uncertainty quantification of large-scale structures subjected to earthquakes in comparison to the conventional FEM.

Self-Supervised Monocular Depth Estimation by Digging into Uncertainty Quantification

Based on well-designed network architectures and objective functions,self-supervised monocular depth estimation has made great progress.However,lacking a specific mechanism to make the network learn more about the regions containing moving objects or occlusion scenarios,existing depth estimation methods likely produce poor results for them.Therefore,we propose an uncertainty quantification method to improve the performance of existing depth estimation networks without changing their architectures.Our uncertainty quantification method consists of uncertainty measurement,the learning guidance by uncertainty,and the ultimate adaptive determination.Firstly,with Snapshot and Siam learning strategies,we measure the uncertainty degree by calculating the variance of pre-converged epochs or twins during training.Secondly,we use the uncertainty to guide the network to strengthen learning about those regions with more uncertainty.Finally,we use the uncertainty to adaptively produce the final depth estimation results with a balance of accuracy and robustness.To demonstrate the effectiveness of our uncertainty quantification method,we apply it to two state-of-the-art models,Monodepth2 and Hints.Experimental results show that our method has improved the depth estimation performance in seven evaluation metrics compared with two baseline models and exceeded the existing uncertainty method.

Fast Uncertainty Quantification of Electromechanical Oscillation Frequency on Varying Generator Damping

This letter develops a fast analytical method for uncertainty quantification of electromechanical oscillation frequency due to varying generator dampings. By employing the techniques of matrix determinant reduction, two types of uncertainty analysis are investigated to quantify the impact of the generator damping on electromechanical oscillation frequency, i.e., interval analysis and probabilistic analysis. The proposed analytical frequency estimation formula is verified against conventional methods on two transmission system models. Then, Monte Carlo experiments and interval analysis are respectively conducted to verify the established lower/upper bound formulae and probability distribution formulae. Results demonstrate the accuracy and speed of the proposed method.

Uncertainty quantification in the Permian Basin using conventional and modified bootstrap methodology

Various uncertainty quantification methodologies are presented using a combination of several deter-ministic decline curve analysis models and two bootstrapping algorithms.These probabilistic models are applied to 126 sample wells from the Permian basin.Results are presented for 12-72 months of pro-duction hindcast given an average well production history of 103 months.Based on the coverage rate and the forecast error(with the coverage rate being more significant in our choice of the best probabilistic models)and using up to one-half of the available production history for a group of sample wells from the Permian Basin,we find that the CBM-SEPD combination is the best probabilistic model for the Central Basin Platform,the MBM-Arps combination is the best probabilistic model for the Delaware Basin,the CBM-Arps is the best probabilistic model for the Midland Basin,and the best probabilistic model for the overall Permian Basin is the CBM-Arps when early time data is used as hindcast and CBM-SEPD for when one-quarter to one-half of the data is used as hindcast.When three-quarters or more of the available production history is used for analysis,the MBM-SEPD probabilistic model is the best combination in terms of both coverage rate and forecast error for all the sub-basins in the Permian.The novelty of this work lies in its extension of bootstrapping methods to other decline curve analysis models.This work also offers the engineer guidance on the best choice of probabilistic model whilst attempting to forecast production from the Permian Basin.

Uncertainty-Aware Physical Simulation of Neural Radiance Fields for Fluids

This paper presents a novel framework aimed at quantifying uncertainties associated with the 3D reconstruction of smoke from2Dimages.This approach reconstructs color and density fields from 2D images using Neural Radiance Field(NeRF)and improves image quality using frequency regularization.The NeRF model is obtained via joint training ofmultiple artificial neural networks,whereby the expectation and standard deviation of density fields and RGB values can be evaluated for each pixel.In addition,customized physics-informed neural network(PINN)with residual blocks and two-layer activation functions are utilized to input the density fields of the NeRF into Navier-Stokes equations and convection-diffusion equations to reconstruct the velocity field.The velocity uncertainties are also evaluated through ensemble learning.The effectiveness of the proposed algorithm is demonstrated through numerical examples.The presentmethod is an important step towards downstream tasks such as reliability analysis and robust optimization in engineering design.

Uncertainty-Aware Deep Learning: A Promising Tool for Trustworthy Fault Diagnosis

Recently,intelligent fault diagnosis based on deep learning has been extensively investigated,exhibiting state-of-the-art performance.However,the deep learning model is often not truly trusted by users due to the lack of interpretability of“black box”,which limits its deployment in safety-critical applications.A trusted fault diagnosis system requires that the faults can be accurately diagnosed in most cases,and the human in the deci-sion-making loop can be found to deal with the abnormal situa-tion when the models fail.In this paper,we explore a simplified method for quantifying both aleatoric and epistemic uncertainty in deterministic networks,called SAEU.In SAEU,Multivariate Gaussian distribution is employed in the deep architecture to compensate for the shortcomings of complexity and applicability of Bayesian neural networks.Based on the SAEU,we propose a unified uncertainty-aware deep learning framework(UU-DLF)to realize the grand vision of trustworthy fault diagnosis.Moreover,our UU-DLF effectively embodies the idea of“humans in the loop”,which not only allows for manual intervention in abnor-mal situations of diagnostic models,but also makes correspond-ing improvements on existing models based on traceability analy-sis.Finally,two experiments conducted on the gearbox and aero-engine bevel gears are used to demonstrate the effectiveness of UU-DLF and explore the effective reasons behind.

Advances in the study of uncertainty quantification of large-scale hydrological modeling system

The regional hydrological system is extremely complex because it is affected not only by physical factors but also by human dimensions.And the hydrological models play a very important role in simulating the complex system.However,there have not been effective methods for the model reliability and uncertainty analysis due to its complexity and difficulty.The uncertainties in hydrological modeling come from four important aspects：uncertainties in input data and parameters,uncertainties in model structure,uncertainties in analysis method and the initial and boundary conditions.This paper systematically reviewed the recent advances in the study of the uncertainty analysis approaches in the large-scale complex hydrological model on the basis of uncertainty sources.Also,the shortcomings and insufficiencies in the uncertainty analysis for complex hydrological models are pointed out.And then a new uncertainty quantification platform PSUADE and its uncertainty quantification methods were introduced,which will be a powerful tool and platform for uncertainty analysis of large-scale complex hydrological models.Finally,some future perspectives on uncertainty quantification are put forward.

Nonlinear uncertainty quantification of the impact of geometric variability on compressor performance using an adjoint method

Manufactured blades are inevitably different from their design intent,which leads to a deviation of the performance from the intended value.To quantify the associated performance uncertainty,many approaches have been developed.The traditional Monte Carlo method based on a Computational Fluid Dynamics solver(MC-CFD)for a three-dimensional compressor is prohibitively expensive.Existing alternatives to the MC-CFD,such as surrogate models and secondorder derivatives based on the adjoint method,can greatly reduce the computational cost.Nevertheless,they will encounter’the curse of dimensionality’except for the linear model based on the adjoint gradient(called MC-adj-linear).However,the MC-adj-linear model neglects the nonlinearity of the performance function.In this work,an improved method is proposed to circumvent the lowaccuracy problem of the MC-adj-linear without incurring the high cost of other alternative models.The method is applied to the study of the aerodynamic performance of an annular transonic compressor cascade,subject to prescribed geometric variability with industrial relevance.It is found that the proposed method achieves a significant accuracy improvement over the MC-adj-linear with low computational cost,showing the great potential for fast uncertainty quantification.

A one-dimensional polynomial chaos method in CFD–Based uncertainty quantification for ship hydrodynamic performance

A one-dimensional non-intrusive Polynomial Chaos （PC） method is applied in Uncertainty Quantification （UQ） studies for CFD-based ship performances simulations. The uncertainty properties of Expected Value （EV） and Standard Deviation （SD） are evaluated by solving the PC coefficients from a linear system of algebraic equations. The one-dimensional PC with the Legendre polynomials is applied to： （1） stochastic input domain and （2） Cumulative Distribution Function （CDF） image domain, allowing for more flexibility. The PC method is validated with the Monte-Carlo benchmark results in several high-fidelity, CFD-based, ship UQ problems, evaluating the geometrical, operational and environmental uncertainties for the Delft Catamaran 372. Convergence is studied versus PC order P for both EV and SD, showing that high order PC is not necessary for present applications. Comparison is carried out for PC with/without the least square minimization when solving the PC coefficients. The least square minimization, using larger number of CFD samples, is recommended for current test cases. The study shows the potentials of PC method in Robust Design Optimization （RDO） and Reliability-Based Design Optimization （RBDO） of ship hydrodynamic performances.

Uncertainty quantification and reduction in the characterization of subsurface stratigraphy using limited geotechnical investigation data

Subsurface stratigraphy is critical to the design,construction,and subsequent performance of geotechnical structures.However,in practice it is impossible to identify the stratigraphy of a subsurface geological domain with absolute certainty,due to the limitations imposed by geotechnical investigation techniques and project budgets.This paper presents a subsurface stratigraphic modeling and uncertainty quantification approach,which is established based on an improved and extended geological modeling technique previously established by the author and others,for simulating the stratigraphy of both two-dimensional(2D)and three-dimensional(3D)cases with more complex geological features.Furthermore,this approach provides quantitative evaluation of the amount of stratigraphic uncertainty in the current interpretation and enables the systematic reduction of stratigraphic uncertainty through the investigation of additional targeted borehole drilling locations.Illustrative examples,including artificial cases as well as two real cases from existing geotechnical projects,are presented in this study to demonstrate the use of the proposed analysis approach.