How well do we know our models?

Quantitative predictions of the physical state of the Earth’s subsurface are routinely based on numerical solutions of complex coupled partial differential equations together with estimates of the uncertainties in the material parameters. The resulting high-dimensional problems are computationally prohibitive even for state-of-the-art solver solutions. In this study, we introduce a hybrid physics-based machine learning technique, the non-intrusive reduced basis method, to construct reliable, scalable, and interpretable surrogate models. Our approach, to combine physical process models with data-driven machine learning techniques, allows us to overcome limitations specific to each individual component, and it enables us to carry out probabilistic analyses, such as global sensitivity studies and uncertainty quantification for real-case non-linearly coupled physical problems. It additionally provides orders of magnitude computational gain, while maintaining an accuracy higher than measurement errors. Although in this study we use a thermo-hydro-mechanical reservoir application to illustrate these features, all the theory described is equally valid and applicable to a wider range of geoscientific applications.

The degree to which the interannual variability of vegetation phenology affects hydrological fluxes over land is investigated through a series of simulations with the Mosaic land surface model, run both offline and coupled to the NASA Seasonal-to-Interannual Prediction Project (NSIPP) atmospheric general circulation model (GCM). Over a 9-yr period, from 1982 to 1990, interannual variations of global biophysical land surface parameters (i.e., vegetation density and greenness fraction) are derived from Normalized Difference Vegetation Index (NDVI) data collected by the Advanced Very High Resolution Radiometers (AVHRRs). First the sensitivity of evapotranspiration to interannual variations in vegetation properties is evaluated through offline simulations that ignore feedbacks between the land surface and the atmospheric models, and interannual precipitation variations. Evapotranspiration is shown to be highly sensitive to variations in vegetation over wet continental surfaces that are not densely ve...

Global sensitivity analysis to optimize basin-scale conductive model calibration – A case study from the Upper Rhine Graben

We consider parameter identification problems in parametrized partial differential equations (PDEs). These lead to nonlinear ill-posed inverse problems. One way of solving them is using iterative regularization methods, which typically require numerous amounts of forward solutions during the solution process. In this article we consider the nonlinear Landweber method and couple it with the reduced basis method as a model order reduction technique in order to reduce the overall computational time. In particular, we consider PDEs with a high-dimensional parameter space, which are known to pose difficulties in the context of reduced basis methods. We present a new method that is able to handle such high-dimensional parameter spaces by combining the nonlinear Landweber method with adaptive online reduced basis updates. It is then applied to the inverse problem of reconstructing the conductivity in the stationary heat equation.

Surrogate SDOF models for probabilistic performance assessment of multistory buildings: Methodology and application for steel special moment frames

The reduced basis method is a model-order reduction technique that converts the original system equations into smaller ones using a reduced basis set. This basis set is usually derived from a set of solutions of the system equation with different values of control parameters. If the system model changes affinely with the parameter values, a reduced-order model can be derived. In this abstract, we apply the reduced basis method to method of moments for quasi-periodic array modeling. With this method, a new basis set for a single element is constructed through an offline process. Because of the similarities among elements, the number of basis functions for each element can be much less than directing modeling from geometrical mesh. Numerical example shows that both the computational and memory efficiency is improved compared with direct modeling using method of moments.

Lithium-ion batteries (LIBs) have found widespread application in energy storage due to their high energy density, low self-discharge and low maintenance characteristics. However, the stability and longevity of the batteries under extreme operating conditions is yet to be fully understood. Numerous experimental and simulation studies have been performed to elucidate the effect of temperature, current, depth of discharge, and the number of cycles [1,2,3] on the performance of LIBs. These studies show that the solid-electrolyte interface (SEI) layer and gas formation accelerate at high C-rates due to increased electrolyte decomposition [4,5]. Consequently, the cell's internal resistance increases, resulting in ever-increasing irreversible heat generation inside the cell [6]. This heat generation leads to high internal temperature and a corresponding reduction in cell capacity. The data of cell temperature for a wide range of ambient temperatures and cycling rates is limited because cell performance is typically reported at room temperature. The design of battery packs for electric vehicles(EVs) is also based on test data taken at room conditions. Given that the cell's capacity, specific energy, maximum power output is bound to depend on C-rate and ambient temperature, it is necessary to test and design battery packs based on these parameters.The thermal characteristics of a lithium-ion cell can be predicted using a physics-based thermal model or data-driven methods (DDM) relying on empirical data. Experimental determination of internal cell parameters required in physics-based thermal models is challenging and time-consuming for commercial cells. On the other hand, DDM requires only cell cycling data at specific operating conditions, thereby eliminating the need for internal parameter estimation. This model predicts the experimental input and output data pattern and fits the response surface to estimate the unknown output. In this study, surrogate modelling has been employed to estimate the surface temperature, capacity, energy, and average power of commercial lithium-ion cells for different discharge currents and ambient temperatures. Surrogate modelling, a popular data analysis and reduced-order modelling technique, aims to find a global minimum of a particular objective function using a few objective function evaluations [7]. The surrogate-based model divides experimental data into training and test data sets. The training data set is employed to train the algorithm. After that, the testing data set is used to validate the model's accuracy. Experiments were performed to develop a surrogate model, with the number of experiments decided using the design of experiments[8] for a current range of 0.5C to 3C-rate and ambient temperature range of 0°C to 45°C . The cycling of cells was performed using a high current battery cycler (Arbin), and the ambient temperature was maintained using a thermal chamber (Cincinnati). The surface temperature of commercial 18650 (NMC811) lithium-ion cells was recorded using T-type thermocouples and a National Instruments DAQ module. A polynomial response surface was fitted using surrogate modelling on experimentally obtained data. The response surface shown in figure 1. contains nine data points for the preliminary study, sub-divided into a set of seven training and two testing data. The prediction error sum of squares (PRESS) is currently bounded within 10% due to the limited training data set availability and is expected to be within a band of 1% after adding data from ongoing experiments. The estimation of temperature, capacity, and specific energy can be used to optimize and develop an improved thermal management system for electric vehicles that works under various operating conditions. References Waldmann, T., Wilka, M., Kasper, M., Fleischhammer, M., & Wohlfahrt-Mehrens, M. (2014). Journal of Power Sources, 262, 129-135.Guan, T., Sun, S., Yu, F., Gao, Y., Fan, P., Zuo, P., Du,C., & Yin, G. (2018). Electrochimica Acta, 279, 204-212.Simolka, M., Heger, J. F., Traub, N., Kaess, H., & Friedrich, K. A. (2020). Journal of The Electrochemical Society, 167(11), 110502.Xu, B., Diao, W., Wen, G., Choe, S. Y., Kim, J., & Pecht, M. (2021). Journal of Power Sources, 510, 230390.Rashid, M., & Gupta, A. (2017). Electrochimica Acta, 231, 171-184.Leng, F., Tan, C. M., & Pecht, M. (2015). Scientific reports, 5(1), 1-12.Queipo, N. V., Haftka, R. T., Shyy, W., Goel, T., Vaidyanathan, R., & Tucker, P. K. (2005). Progress in aerospace sciences, 41(1), 1-28.Li, W., Xiao, M., Peng, X., Garg, A., & Gao, L. (2019). Applied Thermal Engineering, 147, 90-100. Figure 1

In this work we describe dimensionality reduction methods to solve parametric problems governed by partial differential equations (PDE). The goal of these methodologies is to elucidate and reduce the dimensionality of the manifold containing the family of solutions of a parametric problem. This leads to a reduced system that can often be solved in real time. These techniques are being successfully applied in many fields in science and engineering, e.g. [1,2] as well as in geodynamics [3,4].The application of these techniques involves two phases: i) creation of a reduced space, often done via a sampling of the parametric space and a singular value decomposition, and ii) use of the reduced space to find a new solution within the family.Here we want to compare methodologies that, in their second step, include or neglect the physics described by the PDEs. We call these, physics-based and physics-agnostic approaches. Reduced Basis methods being an example of the first, and surrogate modelling one of the second.Applications of flow in porous media are used as examples to test the strengths and weaknesses of the different approaches. These methodologies are a potential tool to be used in situations where it is unaffordable to obtain a very large training set (big data).REFERENCES[1] Rocas M., A. García-González, S. Zlotnik, X. Larráyoz and P. Díez. Nonintrusive Uncertainty Quantification for automotive crash problems with VPS/Pamcrash. Finite Elements in Analysis & Design, Vol. 193, doi:10.1016/j.finel.2021.103556, 2021.[2] Muixí A., S. Zlotnik, P. Calvet, M. Espanol, I. Lodoso-Torrecilla, M.P. Ginebra, P. Díez and A. García-González. A multiparametric advection-diffusion reduced-order model for molecular transport in scaffolds for osteoinduction. Biomechanics and Modeling in Mechanobiology, doi:10.1007/s10237-022-01577-2, 2022.[3] Ortega O., S. Zlotnik, J.C. Afonso and P. Díez. Fast Stokes flow simulations for geophysical-geodynamic inverse problems and sensitivity analyses based on reduced order modeling. Journal of Geophysical Research: Solid Earth, Vol. 125, 1–25, doi:10.1029/2019JB018314, 2020.[4] Manassero M., J.C. Afonso, F. Zyserman, A. Jones, S. Zlotnik and I. Fomin. A Reduced Order Approach for Probabilistic Inversions of 3D Magnetotelluric Data II: Joint inver- sion of MT and Surface-Wave Data. Journal of Geophysical Research - Solid Earth, doi:10.1029/2021JB021962, 2021.

Surrogate models have been used as a substitute for traditional engineering simulation to get target simulation results faster. These surrogate models usually need sufficient data to fulfill their optimum performance. However, the acquisition of high quality datasets for cases like welding residual stress is both difficult and costly. This work developed a workflow that can rapidly develop machine learning based surrogate models to predict post-weld residual stresses using finite element (FE) simulation data. The workflow comprises five steps: (i) baseline FE model creation, (ii) FE model processing (iii) scripting (iv) data generation and (v) surrogate model development. The workflow presented in the paper automates all the steps involved in generating, calculating, and extracting results from FE simulations, with the aim of reducing the time and cost required for data acquisition. In addition, to minimize the possibility of FE analyses not completing successfully during the automated data generation process and to ensure that the training data can be obtained without any manual intervention, the FE model was simplified without significant loss of accuracy. A single weld bead-on-plate structure, taken from the NeT project Task Group 1, was used as a case study to validate the developed workflow. The overall aim of this work was to validate the feasibility of developing a surrogate model for specific welding conditions with no experimental dataset and with the aid of simulations, and to investigate the predictive performance of an Artificial Neural Network based surrogate model. Four welding parameters, arc advance speed, net energy input rate, arc travel length and position, were considered as the input for the surrogate model to predict the longitudinal stress along the line running across the sample thickness in the mid-length of the specimen. The relative root mean square error of the predictive surrogate model for the test dataset was 0.0024. Residual stress prediction based on the NeT-TG1 benchmark welding case met the R6 validation criteria and the developed model generates predictions that are in line with other methods used by contributors to the NeT-TG1 round robin.

This paper attempts to develop a new automated multipoint model-order reduction (MOR) technique, based on matching moments of the system input–output function, which would be suited for fast and accurate computation of scattering parameters for electromagnetic (EM) systems over a wide frequency band. To this end, two questions are addressed. Firstly, the cost of the wideband reduced model generation is optimized by automating a greedy multipoint MOR scheme. This is achieved by introducing a new dual local-global model convergence scheme, which applies fast and reliable a posteriori error estimates to check both local model convergence, used to select the number of moments at a single expansion point, and global model convergence, used to optimally select the expansion points. Secondly, the question of optimal convergence measure is addressed by proposing an enhanced a posteriori error estimator particularly suited for scattering parameter computations for lossy EM systems. The effectiveness and efficiency of the proposed automated scheme is verified through numerical simulations using reduced-order models for examples of a bandstop dielectric resonator filter and a dielectric resonator antenna for a wide frequency band, and compared against the results obtained using the full-order model, a reduced model generated with the optimal greedy point selection algorithm, as well as the reduced-order models obtained using the reduced basis method (RBM) and the single-point second-order Arnoldi method for passive order reduction (SAPOR) method.

Reduced basis methods are projection-based model order reduction techniques for reducing the computational complexity of solving parametrized partial differential equation problems. In this work we discuss the design of pyMOR, a freely available software library of model order reduction algorithms, in particular reduced basis methods, implemented with the Python programming language. As its main design feature, all reduction algorithms in pyMOR are implemented generically via operations on well-defined vector array, operator and discretization interface classes. This allows for an easy integration with existing open-source high-performance partial differential equation solvers without adding any model reduction specific code to these solvers. Besides an in-depth discussion of pyMOR's design philosophy and architecture, we present several benchmark results and numerical examples showing the feasibility of our approach.

We present the first surrogate model for gravitational waveforms from the coalescence of precessing binary black holes. We call this surrogate model NRSur4d2s. Our methodology significantly extends recently introduced reduced-order and surrogate modeling techniques, and is capable of directly modeling numerical relativity waveforms without introducing phenomenological assumptions or approximations to general relativity. Motivated by GW150914, LIGO's first detection of gravitational waves from merging black holes, the model is built from a set of $276$ numerical relativity (NR) simulations with mass ratios $q \leq 2$, dimensionless spin magnitudes up to $0.8$, and the restriction that the initial spin of the smaller black hole lies along the axis of orbital angular momentum. It produces waveforms which begin $\sim 30$ gravitational wave cycles before merger and continue through ringdown, and which contain the effects of precession as well as all $\ell \in \{2, 3\}$ spin-weighted spherical-harmonic modes. We perform cross-validation studies to compare the model to NR waveforms \emph{not} used to build the model, and find a better agreement within the parameter range of the model than other, state-of-the-art precessing waveform models, with typical mismatches of $10^{-3}$. We also construct a frequency domain surrogate model (called NRSur4d2s_FDROM) which can be evaluated in $50\, \mathrm{ms}$ and is suitable for performing parameter estimation studies on gravitational wave detections similar to GW150914.

Among Series-Parallel Hybrid Electric Vehicle (SPHEV) powertrains, the Power-Split architecture with a planetary gear has an exemplary energetic efficiency in mixed driving conditions. Nevertheless, a simple SPHEV architecture can be realized without a planetary gear. It consists of 2 Electric Machines (EM) mounted on the engine shaft and separated by a clutch. With no power-split operation, this architecture allows the vehicle to operate in pure electric, or series hybrid, or parallel hybrid mode. It was proven to be less efficient than a reference Power-Split SPHEV: the Toyota Hybrid System (THS). The aim of this paper is to investigate the potential of efficiency improvement of the simple SPHEV powertrain by topology modification: the addition of gears for the components or a gearbox with few number of ratios. Two new variants of SPHEVs are proposed. The versions of SPHEVs and the reference THS are optimized by a bi-level optimization technique using Genetic Algorithm and Dynamic Programming. Compared to the simple SPHEV, results show an efficiency worsening in one variant and an efficiency improvement in another variant with a fuel consumption comparable to the one of THS. A global sensitivity study is then performed on the worsened variant. The sensitivities of the added gears are determined and an elimination of some is suggested. A new variant with fewer gears is therefore proposed and optimized. The efficiency is improved but remains less than the one of THS.

Global Sensitivity Analysis (GSA) is the study of the influence of any given input on the outputs of a model. In the context of engineering design, GSA has been widely used to understand both individual and collective contributions of design variables on the design objectives. So far, global sensitivity studies have often been limited to design spaces with only quantitative (numerical) design variables. However, many engineering systems also contain, if not only, qualitative (categorical) design variables in addition to quantitative design variables. In this paper, we integrate Latent Variable Gaussian Process (LVGP) with Sobol’ analysis to develop the first metamodel-based mixed-variable GSA method. Through numerical case studies, we validate and demonstrate the effectiveness of our proposed method for mixed-variable problems. Furthermore, while the proposed GSA method is general enough to benefit various engineering design applications, we integrate it with multi-objective Bayesian optimization (BO) to create a sensitivity-aware design framework in accelerating the Pareto front design exploration for metal-organic framework (MOF) materials with many-level combinatorial design spaces. Although MOFs are constructed only from qualitative variables that are notoriously difficult to design, our method can utilize sensitivity analysis to navigate the optimization in the many-level large combinatorial design space, greatly expediting the exploration of novel MOF candidates.

Global Sensitivity Analysis of Arrhythmic Risk Biomarkers in Cardiac Single Cell Models