Engineering and applied science rely on computational experiments to rigorously study physical systems. The mathematical models used to probe these systems are highly complex, and sampling-intensive studies often require prohibitively many simulations for acceptable accuracy. Surrogate models provide a means of circumventing the high computational expense of sampling such complex models. In particular, polynomial chaos expansions (PCEs) have been successfully used for uncertainty quantification studies of deterministic models where the dominant source of uncertainty is parametric. We discuss an extension to conventional PCE surrogate modeling to enable surrogate construction for stochastic computational models that have intrinsic noise in addition to parametric uncertainty. We develop a PCE surrogate on a joint space of intrinsic and parametric uncertainty, enabled by Rosenblatt transformations, which are evaluated via kernel density estimation of the associated conditional cumulative distributions. Furthermore, we extend the construction to random field data via the Karhunen-Loève expansion. We then take advantage of closed-form solutions for computing PCE Sobol indices to perform a global sensitivity analysis of the model which quantifies the intrinsic noise contribution to the overall model output variance. Additionally, the resulting joint PCE is generative in the sense that it allows generating random realizations at any input parameter setting that are statistically approximately equivalent to realizations from the underlying stochastic model. The method is demonstrated on a chemical catalysis example model and a synthetic example controlled by a parameter that enables a switch from unimodal to bimodal response distributions.
Understanding and accurately characterizing energy dissipation mechanisms in civil structures during earthquakes is an important element of seismic assessment and design. The most commonly used model is attributed to Rayleigh. This paper proposes a systematic approach to quantify the uncertainty associated with Rayleigh's damping model. Bayesian calibration with embedded model error is employed to treat the coefficients of the Rayleigh model as random variables using modal damping ratios. Through a numerical example, we illustrate how this approach works and how the calibrated model can address modeling uncertainty associated with the Rayleigh damping model.
Bayesian inference with a simple Gaussian error model is used to efficiently compute prediction variances for energies, forces, and stresses in the linear SNAP interatomic potential. The prediction variance is shown to have a strong correlation with the absolute error over approximately 24 orders of magnitude. Using this prediction variance, an active learning algorithm is constructed to iteratively train a potential by selecting the structures with the most uncertain properties from a pool of candidate structures. The relative importance of the energy, force, and stress errors in the objective function is shown to have a strong impact upon the trajectory of their respective net error metrics when running the active learning algorithm. Batched training of different batch sizes is also tested against singular structure updates, and it is found that batches can be used to significantly reduce the number of retraining steps required with only minor impact on the active learning trajectory.
A new strategy is presented for computing anharmonic partition functions for the motion of adsorbates relative to a catalytic surface. Importance sampling is compared with conventional Monte Carlo. The importance sampling is significantly more efficient. This new approach is applied to CH3* on Ni(111) as a test case. The motion of methyl relative to the nickel surface is found to be anharmonic, with significantly higher entropy compared to the standard harmonic oscillator model. The new method is freely available as part of the Minima-Preserving Neural Network within the AdTherm package.
The ShakeAlert Earthquake Early Warning (EEW) system aims to issue an advance warning to residents on the West Coast of the United States seconds before the ground shaking arrives, if the expected ground shaking exceeds a certain threshold. However, residents in tall buildings may experience much greater motion due to the dynamic response of the buildings. Therefore, there is an ongoing effort to extend ShakeAlert to include the contribution of building response to provide a more accurate estimation of the expected shaking intensity for tall buildings. Currently, the supposedly ideal solution of analyzing detailed finite element models of buildings under predicted ground-motion time histories is not theoretically or practically feasible. The authors have recently investigated existing simple methods to estimate peak floor acceleration (PFA) and determined these simple formulas are not practically suitable. Instead, this article explores another approach by extending the Pacific Earthquake Engineering Research Center (PEER) performance-based earthquake engineering (PBEE) to EEW, considering that every component involved in building response prediction is uncertain in the EEW scenario. While this idea is not new and has been proposed by other researchers, it has two shortcomings: (1) the simple beam model used for response prediction is prone to modeling uncertainty, which has not been quantified, and (2) the ground motions used for probabilistic demand models are not suitable for EEW applications. In this article, we address these two issues by incorporating modeling errors into the parameters of the beam model and using a new set of ground motions, respectively. We demonstrate how this approach could practically work using data from a 52-story building in downtown Los Angeles. Using the criteria and thresholds employed by previous researchers, we show that if peak ground acceleration (PGA) is accurately estimated, this approach can predict the expected level of human comfort in tall buildings.
Ground heat flux (G0) is a key component of the land-surface energy balance of high-latitude regions. Despite its crucial role in controlling permafrost degradation due to global warming, G0 is sparsely measured and not well represented in the outputs of global scale model simulation. In this study, an analytical heat transfer model is tested to reconstruct G0 across seasons using soil temperature series from field measurements, Global Climate Model, and climate reanalysis outputs. The probability density functions of ground heat flux and of model parameters are inferred using available G0 data (measured or modeled) for snow-free period as a reference. When observed G0 is not available, a numerical model is applied using estimates of surface heat flux (dependent on parameters) as the top boundary condition. These estimates (and thus the corresponding parameters) are verified by comparing the distributions of simulated and measured soil temperature at several depths. Aided by state-of-the-art uncertainty quantification methods, the developed G0 reconstruction approach provides novel means for assessing the probabilistic structure of the ground heat flux for regional permafrost change studies.
In this report we present our findings and outcomes of the NNRDS (analysis of Neural Networks as Random Dynamical Systems) project. The work is largely motivated by the analogy of a large class of neural networks (NNs) with a discretized ordinary differential equation (ODE) schemes. Namely, residual NNs, or ResNets, can be viewed as a discretization of neural ODEs (NODEs) where the NN depth plays the role of the time evolution. We employ several legacy tools from ODE theory, such as stiffness, nonlocality, autonomicity, to enable regularization of ResNets thus improving their generalization capabilities. Furthermore, armed with NN analysis tools borrowed from the ODE theory, we are able to efficiently augment NN predictions with uncertainty overcoming wellknown dimensionality challenges and adding a degree of trust towards NN predictions. Finally, we have developed a Python library QUiNN (Quantification of Uncertainties in Neural Networks) that incorporates improved-architecture ResNets, besides classical feed-forward NNs, and contains wrappers to PyTorch NN models enabling several major classes of uncertainty quantification methods for NNs. Besides synthetic problems, we demonstrate the methods on datasets from climate modeling and materials science.
Neural ordinary differential equations (NODEs) have recently regained popularity as large-depth limits of a large class of neural networks. In particular, residual neural networks (ResNets) are equivalent to an explicit Euler discretization of an underlying NODE, where the transition from one layer to the next is one time step of the discretization. The relationship between continuous and discrete neural networks has been of particular interest. Notably, analysis from the ordinary differential equation viewpoint can potentially lead to new insights for understanding the behavior of neural networks in general. In this work, we take inspiration from differential equations to define the concept of stiffness for a ResNet via the interpretation of a ResNet as the discretization of a NODE. Here, we then examine the effects of stiffness on the ability of a ResNet to generalize, via computational studies on example problems coming from climate and chemistry models. We find that penalizing stiffness does have a unique regularizing effect, but we see no benefit to penalizing stiffness over L2 regularization (penalization of network parameter norms) in terms of predictive performance.
The use of simple models for response prediction of building structures is preferred in earthquake engineering for risk evaluations at regional scales, as they make computational studies more feasible. The primary impediment in their gainful use presently is the lack of viable methods for quantifying (and reducing upon) the modeling errors/uncertainties they bear. This study presents a Bayesian calibration method wherein the modeling error is embedded into the parameters of the model. The method is specifically described for coupled shear-flexural beam models here, but it can be applied to any parametric surrogate model. The major benefit the method offers is the ability to consider the modeling uncertainty in the forward prediction of any degree-of-freedom or composite response regardless of the data used in calibration. The method is extensively verified using two synthetic examples. In the first example, the beam model is calibrated to represent a similar beam model but with enforced modeling errors. In the second example, the beam model is used to represent the detailed finite element model of a 52-story building. Both examples show the capability of the proposed solution to provide realistic uncertainty estimation around the mean prediction.
Runoff is a critical component of the terrestrial water cycle, and Earth system models (ESMs) are essential tools to study its spatiotemporal variability. Runoff schemes in ESMs typically include many parameters so that model calibration is necessary to improve the accuracy of simulated runoff. However, runoff calibration at a global scale is challenging because of the high computational cost and the lack of reliable observational datasets. In this study, we calibrated 11 runoff relevant parameters in the Energy Exascale Earth System Model (E3SM) Land Model (ELM) using a surrogate-assisted Bayesian framework. First, the polynomial chaos expansion machinery with Bayesian compressed sensing is used to construct computationally inexpensive surrogate models for ELM-simulated runoff at 0.5 × 0.5 for 1991-2010. The error metric between the ELM simulations and the benchmark data is selected to construct the surrogates, which facilitates efficient calibration and avoids the more conventional, but challenging, construction of high-dimensional surrogates for the ELM simulated runoff. Second, the Sobol' index sensitivity analysis is performed using the surrogate models to identify the most sensitive parameters, and our results show that, in most regions, ELM-simulated runoff is strongly sensitive to 3 of the 11 uncertain parameters. Third, a Bayesian method is used to infer the optimal values of the most sensitive parameters using an observation-based global runoff dataset as the benchmark. Our results show that model performance is significantly improved with the inferred parameter values. Although the parametric uncertainty of simulated runoff is reduced after the parameter inference, it remains comparable to the multimodel ensemble uncertainty represented by the global hydrological models in ISMIP2a. Additionally, the annual global runoff trend during the simulation period is not well constrained by the inferred parameter values, suggesting the importance of including parametric uncertainty in future runoff projections.
A Bayesian inference strategy has been used to estimate uncertain inputs to global impurity transport code (GITR) modeling predictions of tungsten erosion and migration in the linear plasma device, PISCES-A. This allows quantification of GITR output uncertainty based on the uncertainties in measured PISCES-A plasma electron density and temperature profiles (n e, T e) used as inputs to GITR. The technique has been applied for comparison to dedicated experiments performed for high (4 × 1022 m-2 s-1) and low (5 × 1021 m-2 s-1) flux 250 eV He-plasma exposed tungsten (W) targets designed to assess the net and gross erosion of tungsten, and corresponding W impurity transport. The W target design and orientation, impurity collector, and diagnostics, have been designed to eliminate complexities associated with tokamak divertor plasma exposures (inclined target, mixed plasma species, re-erosion, etc) to benchmark results against the trace impurity transport model simulated by GITR. The simulated results of the erosion, migration, and re-deposition of W during the experiment from the GITR code coupled to materials response models are presented. Specifically, the modeled and experimental W I emission spectroscopy data for a 429.4 nm line and net erosion through the target and collector mass difference measurements are compared. The methodology provides predictions of observable quantities of interest with quantified uncertainty, allowing estimation of moments, together with the sensitivities to plasma temperature and density.
The UQ Toolkit (UQTk) is a collection of libraries and tools for the quantification of uncertainty in numerical model predictions. Version 3.1.2 offers intrusive and non-intrusive methods for propagating input uncertainties through computational models, tools for sensitivity analysis, methods for sparse surrogate construction, and Bayesian inference tools for inferring parameters from experimental data. This manual discusses the download and installation process for UQTk, provides pointers to the UQ methods used in the toolkit, and describes some of the examples provided with the toolkit.
Flooding impacts are on the rise globally, and concentrated in urban areas. Currently, there are no operational systems to forecast flooding at spatial resolutions that can facilitate emergency preparedness and response actions mitigating flood impacts. We present a framework for real-time flood modeling and uncertainty quantification that combines the physics of fluid motion with advances in probabilistic methods. The framework overcomes the prohibitive computational demands of high-fidelity modeling in real-time by using a probabilistic learning method relying on surrogate models that are trained prior to a flood event. This shifts the overwhelming burden of computation to the trivial problem of data storage, and enables forecasting of both flood hazard and its uncertainty at scales that are vital for time-critical decision-making before and during extreme events. The framework has the potential to improve flood prediction and analysis and can be extended to other hazard assessments requiring intense high-fidelity computations in real-time.
A new method for computing anharmonic thermophysical properties for adsorbates on metal surfaces is presented. Classical Monte Carlo phase space integration is performed to calculate the partition function for the motion of a hydrogen atom on Cu(111). A minima-preserving neural network potential energy surface is used within the integration routine. Two different sampling schema for generating the training data are presented, and two different density functionals are used. The results are benchmarked against direct state counting results by using discrete variable representation. The phase space integration results are in excellent quantitative agreement with the benchmark results. Additionally, both the discrete variable representation and the phase space integration results confirm that the motion of H on Cu(111) is highly anharmonic. The results were applied to calculate the free energy of dissociative adsorption of H2 and the resulting Langmuir isotherms at 400, 800, and 1200 K in a partial pressure range of 0-1 bar. It shows that the anharmonic effects lead to significantly higher predicted surface site fractions of hydrogen.
We present a new geodesic-based method for geometry optimization in a basis set of redundant internal coordinates. Our method updates the molecular geometry by following the geodesic generated by a displacement vector on the internal coordinate manifold, which dramatically reduces the number of steps required to converge to a minimum. Our method can be implemented in any existing optimization code, requiring only implementation of derivatives of the Wilson B-matrix and the ability to numerically solve an ordinary differential equation.
Kreitz, Bjarne; Sargsyan, Khachik; Mazeau, Emily J.; Blondal, Katrin; West, Richard H.; Wehinger, Gregor D.; Turek, Thomas; Goldsmith, C.F.
Automatic mechanism generation is used to determine mechanisms for the CO2 hydrogenation on Ni(111) in a two-stage process while considering the correlated uncertainty in DFT-based energetic parameters systematically. In a coarse stage, all the possible chemistry is explored with gas-phase products down to the ppb level, while a refined stage discovers the core methanation submechanism. Five thousand unique mechanisms were generated, which contain minor perturbations in all parameters. Global uncertainty assessment, global sensitivity analysis, and degree of rate control analysis are performed to study the effect of this parametric uncertainty on the microkinetic model predictions. Comparison of the model predictions with experimental data on a Ni/SiO2 catalyst find a feasible set of microkinetic mechanisms within the correlated uncertainty space that are in quantitative agreement with the measured data, without relying on explicit parameter optimization. Global uncertainty and sensitivity analyses provide tools to determine the pathways and key factors that control the methanation activity within the parameter space. Together, these methods reveal that the degree of rate control approach can be misleading if parametric uncertainty is not considered. The procedure of considering uncertainties in the automated mechanism generation is not unique to CO2 methanation and can be easily extended to other challenging heterogeneously catalyzed reactions.
