We use functional, Fréchet, derivatives to quantify how thermodynamic outputs of a molecular dynamics (MD) simulation depend on the potential used to compute atomic interactions. Our approach quantifies the sensitivity of the quantities of interest with respect to the input functions as opposed to its parameters as is done in typical uncertainty quantification methods. We show that the functional sensitivity of the average potential energy and pressure in isothermal, isochoric MD simulations using Lennard-Jones two-body interactions can be used to accurately predict those properties for other interatomic potentials (with different functional forms) without re-running the simulations. This is demonstrated under three different thermodynamic conditions, namely a crystal at room temperature, a liquid at ambient pressure, and a high pressure liquid. The method provides accurate predictions as long as the change in potential can be reasonably described to first order and does not significantly affect the region in phase space explored by the simulation. The functional uncertainty quantification approach can be used to estimate the uncertainties associated with constitutive models used in the simulation and to correct predictions if a more accurate representation becomes available.
The study of complex systems is often based on computationally intensive, high-fidelity, simulations. To build confidence in the prediction accuracy of such simulations, the impact of uncertainties in model inputs on the quantities of interest must be measured. This, however, requires a computational budget that is a possibly large multiple of the cost of a single simulation, and thus may become infeasible for expensive simulation models featuring a large number of uncertain inputs and highly nonlinear behavior. Therefore, this work explores multi-fidelity strategies to accelerate the estimati…
The goals of this chapter are twofold. First, we wish to introduce molecular dynamics (MD) and uncertainty quantification (UQ) in a common setting in order to demonstrate how the latter can increase confidence in the former. In some cases, this discussion culminates in our providing practical, mathematical tools that can be used to answer the question, "is this simulation reliable?" However, many questions remain unanswered. Thus, a second goal of this work is to highlight open problems where progress would aid the larger community.
We present a "module-based hybrid" Uncertainty Quantification (UQ) framework for general nonlinear multi-physics simulation. The proposed methodology, introduced in [\hyperlink{ref1}{1}], supports the independent development of each \emph{stochastic} linear or nonlinear physics module equipped with the most suitable probabilistic UQ method: non-intrusive, semi-intrusive or intrusive; and provides a generic framework to couple these stochastic simulation components. Moreover, the methodology is illustrated using a common "global" uncertainty representation scheme based on generalized polynomial…
We propose a multi-fidelity neural network surrogate sampling method for the uncertainty quantification of physical/biological systems described by ordinary or partial differential equations. We first generate a set of low/high-fidelity data by low/high-fidelity computational models, e.g. using coarser/finer discretizations of the governing differential equations. We then construct a two-level neural network, where a large set of low-fidelity data are utilized in order to accelerate the construction of a high-fidelity surrogate model with a small set of high-fidelity data. We then embed the co…
Full Waveform Inversion (FWI) plays a vital role in reconstructing geophysical structures. The Uncertainty Quantification regarding the inversion results is equally important but has been missing out in most of the current geophysical inversions. Mathematically, uncertainty quantification is involved with the inverse Hessian (or the posterior covariance matrix), which is prohibitive in computation and storage for practical geophysical FWI problems. L-BFGS populates as the most efficient Gauss-Newton method; however, in this study, we empower it with the new possibility of accessing the inverse…
We present a scale-bridging approach based on a multi-fidelity (MF) machine-learning (ML) framework leveraging Gaussian processes (GP) to fuse atomistic computational model predictions across multiple levels of fidelity. Through the posterior variance of the MFGP, our framework naturally enables uncertainty quantification, providing estimates of confidence in the predictions. We used Density Functional Theory as high-fidelity prediction, while a ML interatomic potential is used as the low-fidelity prediction. Practical materials design efficiency is demonstrated by reproducing the ternary comp…
Uncertainty estimations are presented of the response of a multiscale in-stent restenosis model, as obtained by both non-intrusive and semi-intrusive uncertainty quantification. The in-stent restenosis model is a fully coupled multiscale simulation of post-stenting tissue growth, in which the most costly submodel is the blood flow simulation. Surrogate modeling for non-intrusive uncertainty quantification takes the whole model as a black-box and maps directly from the three uncertain inputs to the quantity of interest, the neointimal area. The corresponding uncertain estimates matched the resu…
In computational fluid dynamics simulations of industrial flows, models based on the Reynolds-averaged Navier--Stokes (RANS) equations are expected to play an important role in decades to come. However, model uncertainties are still a major obstacle for the predictive capability of RANS simulations. This review examines both the parametric and structural uncertainties in turbulence models. We review recent literature on data-free (uncertainty propagation) and data-driven (statistical inference) approaches for quantifying and reducing model uncertainties in RANS simulations. Moreover, the funda…
A massively parallel method to build large transition rate matrices from temperature accelerated molecular dynamics trajectories is presented. Bayesian Markov model analysis is used to estimate the expected residence time in the known state space, providing crucial uncertainty quantification for higher scale simulation schemes such as kinetic Monte Carlo or cluster dynamics. The estimators are additionally used to optimize where exploration is performed and the degree of temperature ac- celeration on the fly, giving an autonomous, optimal procedure to explore the state space of complex systems…
Parameterization of interatomic forcefields is a necessary first step in performing molecular dynamics simulations. This is a non-trivial global optimization problem involving quantification of multiple empirical variables against one or more properties. We present EZFF, a lightweight Python library for parameterization of several types of interatomic forcefields implemented in several molecular dynamics engines against multiple objectives using genetic-algorithm-based global optimization methods. The EZFF scheme provides unique functionality such as the parameterization of hybrid forcefields …
The goal of inverse (quantum) approaches is to devise methods and approaches capable of efficiently searching chemical space in such a way that the design of novel materials and compounds with specific properties is as direct and efficient as possible. Here, we review the current state of the field with a focus on the most recent developments. We discuss the importance of heuristic rules and human intuition for rational compound design. Moreover, we elaborate on options for reliable uncertainty quantification for computational results, which is crucial for a truly predictive application of any…
In computational materials science, mechanical properties are typically extracted from simulations by means of analysis routines that seek to mimic their experimental counterparts. However, simulated data often exhibit uncertainties that can propagate into final predictions in unexpected ways. Thus, modelers require data analysis tools that (i) address the problems posed by simulated data, and (ii) facilitate uncertainty quantification. In this manuscript, we discuss three case studies in materials modeling where careful data analysis can be leveraged to address specific instances of these iss…
College of Engineering, Peking University（北京大学工学院）；School of Environmental Science and Engineering, Southern University of Science and Technology（南方科技大学环境科学与工程学院）；Peng Cheng Laboratory（鹏城实验室）
Subsurface flow problems usually involve some degree of uncertainty. Consequently, uncertainty quantification is commonly necessary for subsurface flow prediction. In this work, we propose a methodology for efficient uncertainty quantification for dynamic subsurface flow with a surrogate constructed by the Theory-guided Neural Network (TgNN). The TgNN here is specially designed for problems with stochastic parameters. In the TgNN, stochastic parameters, time and location comprise the input of the neural network, while the quantity of interest is the output. The neural network is trained with a…
Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学研究所）；Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Max Planck Institute for Plasma Physics（马克斯·普朗克等离子体物理研究所）；Technical University of Munich（慕尼黑工业大学）
Multi-fidelity Monte Carlo methods leverage low-fidelity and surrogate models for variance reduction to make tractable uncertainty quantification even when numerically simulating the physical systems of interest with high-fidelity models is computationally expensive. This work proposes a context-aware multi-fidelity Monte Carlo method that optimally balances the costs of training low-fidelity models with the costs of Monte Carlo sampling. It generalizes the previously developed context-aware bi-fidelity Monte Carlo method to hierarchies of multiple models and to more general types of low-fidel…
We propose a numerical method for discovering unknown parameterized dynamical systems by using observational data of the state variables. Our method is built upon and extends the recent work of discovering unknown dynamical systems, in particular those using deep neural network (DNN). We propose a DNN structure, largely based upon the residual network (ResNet), to not only learn the unknown form of the governing equation but also take into account the random effect embedded in the system, which is generated by the random parameters. Once the DNN model is successfully constructed, it is able to…
In the context of Monte Carlo (MC) simulation of particle transport Uncertainty Quantification (UQ) addresses the issue of predicting non statistical errors affecting the physical results, i.e. errors deriving mainly from uncertainties in the physics data and/or in the model they embed. In the case of a single uncertainty a simple analytical relation exists among its the Probability Density Function (PDF) and the corresponding PDF for the output of the simulation: this allows a complete statistical analysis of the results of the simulation. We examine the extension of this result to the multi-…
We are interested in the development of surrogate models for uncertainty quantification and propagation in problems governed by stochastic PDEs using a deep convolutional encoder-decoder network in a similar fashion to approaches considered in deep learning for image-to-image regression tasks. Since normal neural networks are data intensive and cannot provide predictive uncertainty, we propose a Bayesian approach to convolutional neural nets. A recently introduced variational gradient descent algorithm based on Stein's method is scaled to deep convolutional networks to perform approximate Baye…
We propose an efficient surrogate modeling technique for uncertainty quantification. The method is based on a well-known dimension-adaptive collocation scheme. We improve the scheme by enhancing sparse polynomial surrogates with conformal maps and adjoint error correction. The methodology is applied to Maxwell's source problem with random input data. This setting comprises many applications of current interest from computational nanoplasmonics, such as grating couplers or optical waveguides. Using a non-trivial benchmark model we show the benefits and drawbacks of using enhanced surrogate mode…
