We develop adaptive replicated designs for Gaussian process metamodels of stochastic experiments. Adaptive batching is a natural extension of sequential design heuristics with the benefit of replication growing as response features are learned, inputs concentrate, and the metamodeling overhead rises. Motivated by the problem of learning the level set of the mean simulator response we develop four novel schemes: Multi-Level Batching (MLB), Ratchet Batching (RB), Adaptive Batched Stepwise Uncertainty Reduction (ABSUR), Adaptive Design with Stepwise Allocation (ADSA) and Deterministic Design with Stepwise Allocation (DDSA). Our algorithms simultaneously (MLB, RB and ABSUR) or sequentially (ADSA and DDSA) determine the sequential design inputs and the respective number of replicates. Illustrations using synthetic examples and an application in quantitative finance (Bermudan option pricing via Regression Monte Carlo) show that adaptive batching brings significant computational speed-ups with minimal loss of modeling fidelity.
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Parallel surrogate optimization algorithms have proven to be efficient methods for solving expensive noisy optimization problems. In this work we develop a new parallel surrogate optimization algorithm (ProSRS), using a novel tree-based "zoom strategy" to improve the efficiency of the algorithm. We prove that if ProSRS is run for sufficiently long, with probability converging to one there will be at least one point among all the evaluations that will be arbitrarily close to the global minimum. We compare our algorithm to several state-of-the-art Bayesian optimization algorithms on a suite of s…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所）；Aalto University（阿尔托大学）；University of Edinburgh（爱丁堡大学）
We consider Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained. This occurs for example when complex simulator-based statistical models are fitted to data, and synthetic likelihood (SL) method is used to form the noisy log-likelihood estimates using computationally costly forward simulations. We frame the inference task as a sequential Bayesian experimental design problem, where the log-likelihood function is modelled with a hierarchical Gaussian process (GP) surrogate model, which is used to efficiently select additional log-likelihood evaluation…
Applications of ML-Based Surrogates in Bayesian Approaches to Inverse Problems
Izmir University of Economics（伊兹密尔经济大学）；Carnegie Melon University（卡内基梅隆大学）；University of Warwick（华威大学）；Case Western Reserve University（凯斯西储大学）；University of California, Los Angeles（加利福尼亚大学洛杉矶分校）；Advanced Micro Devices, Inc.（超威半导体公司）
Lawrence Berkeley National Laboratory（劳伦斯伯克利国家实验室）
We present an adaptive approach to the construction of Gaussian process surrogates for Bayesian inference with expensive-to-evaluate forward models. Our method relies on the fully Bayesian approach to training Gaussian process models and utilizes the expected improvement idea from Bayesian global optimization. We adaptively construct training designs by maximizing the expected improvement in fit of the Gaussian process model to the noisy observational data. Numerical experiments on model problems with synthetic data demonstrate the effectiveness of the obtained adaptive designs compared to the…
This paper analyzes DONE, an online optimization algorithm that iteratively minimizes an unknown function based on costly and noisy measurements. The algorithm maintains a surrogate of the unknown function in the form of a random Fourier expansion (RFE). The surrogate is updated whenever a new measurement is available, and then used to determine the next measurement point. The algorithm is comparable to Bayesian optimization algorithms, but its computational complexity per iteration does not depend on the number of measurements. We derive several theoretical results that provide insight on how…
Department of Basic Neuroscience, University of Geneva（日内瓦大学基础神经科学系）
Variational Bayesian Monte Carlo (VBMC) is a recently introduced framework that uses Gaussian process surrogates to perform approximate Bayesian inference in models with black-box, non-cheap likelihoods. In this work, we extend VBMC to deal with noisy log-likelihood evaluations, such as those arising from simulation-based models. We introduce new `global' acquisition functions, such as expected information gain (EIG) and variational interquantile range (VIQR), which are robust to noise and can be efficiently evaluated within the VBMC setting. In a novel, challenging, noisy-inference benchmark …
When a black-box optimization objective can only be evaluated with costly or noisy measurements, most standard optimization algorithms are unsuited to find the optimal solution. Specialized algorithms that deal with exactly this situation make use of surrogate models. These models are usually continuous and smooth, which is beneficial for continuous optimization problems, but not necessarily for combinatorial problems. However, by choosing the basis functions of the surrogate model in a certain way, we show that it can be guaranteed that the optimal solution of the surrogate model is integer. …
SURGE: Approximation and Training Free Particle Filter for Diffusion Surrogate
Department of Mechanical Engineering, Northwestern University, Evanston, IL, United States；Institute for Computational \& Mathematical Engineering, Stanford University, Stanford, CA, United States；Department of Industrial Engineering \& Management Sciences, Northwestern University, Evanston, IL, United States
A Simple Approximate Bayesian Inference Neural Surrogate for Stochastic Petri Net Models
School of Computing and Augmented Intelligence, Arizona State University（计算与增强智能学院，亚利桑那州立大学）；School of Mathematical and Statistical Sciences, Arizona State University（数学与统计科学学院，亚利桑那州立大学）；School of Life Sciences, Arizona State University（生命科学学院，亚利桑那州立大学）
Datasets in engineering applications are often limited and contaminated, mainly due to unavoidable measurement noise and signal distortion. Thus, using conventional data-driven approaches to build a reliable discriminative model, and further applying this identified surrogate to uncertainty analysis remains to be very challenging. A deep learning approach is presented to provide predictions based on limited and noisy data. To address noise perturbation, the Bayesian learning method that naturally facilitates an automatic updating mechanism is considered to quantify and propagate model uncertai…
In complex large-scale systems such as climate, important effects are caused by a combination of confounding processes that are not fully observable. The identification of sources from observations of system state is vital for attribution and prediction, which inform critical policy decisions. The difficulty of these types of inverse problems lies in the inability to isolate sources and the cost of simulating computational models. Surrogate models may enable the many-query algorithms required for source identification, but data challenges arise from high dimensionality of the state and source,…
The Gaussian Process (GP)-based surrogate model has the inherent capability of capturing the anomaly arising from limited data, lack of data, missing data, and data inconsistencies (noisy/erroneous data) present in the modeling and simulation component of the digital twin framework, specifically for the accident tolerant fuel (ATF) concepts. However, GP will not be very accurate when we have limited high-fidelity (experimental) data. In addition, it is challenging to apply higher dimensional functions (>20-dimensional function) to approximate predictions with the GP. Furthermore, noisy data or…
SMT 2.0: A Surrogate Modeling Toolbox with a focus on Hierarchical and Mixed Variables Gaussian Processes
ONERA（法国航空航天研究院）；Université de Toulouse（图卢兹大学）；ISAE-SUPAERO（高等航空航天学院）；Polytechnique Montréal（蒙特利尔理工学院）；German Aerospace Center (DLR)（德国航空航天中心）；University of California San Diego（加州大学圣地亚哥分校）；ICA（高等航空航天学院）；INSA（国立应用科学学院）；CNRS（法国国家科学研究中心）；MINES ALBI（阿尔比矿业学校）；UPS（图卢兹第三大学）；University of Michigan（密歇根大学）
Bayesian optimization is an approach to optimizing objective functions that take a long time (minutes or hours) to evaluate. It is best-suited for optimization over continuous domains of less than 20 dimensions, and tolerates stochastic noise in function evaluations. It builds a surrogate for the objective and quantifies the uncertainty in that surrogate using a Bayesian machine learning technique, Gaussian process regression, and then uses an acquisition function defined from this surrogate to decide where to sample. In this tutorial, we describe how Bayesian optimization works, including Gau…
Robust Learning of Diffusion Models with Extremely Noisy Conditions
University of Auckland（奥克兰大学）；Shandong University（山东大学）；University of Melbourne（墨尔本大学）；King Abdullah University of Science and Technology（国王 Abdullah 科学与技术大学）
Rochester Institute of Technology（罗切斯特理工学院）；Dalhousie University（达尔豪斯大学）
Estimation of patient-specific model parameters is important for personalized modeling, although sparse and noisy clinical data can introduce significant uncertainty in the estimated parameter values. This importance source of uncertainty, if left unquantified, will lead to unknown variability in model outputs that hinder their reliable adoptions. Probabilistic estimation model parameters, however, remains an unresolved challenge because standard Markov Chain Monte Carlo sampling requires repeated model simulations that are computationally infeasible. A common solution is to replace the simula…
Heteroscedastic Gaussian process regression, based on the concept of chained Gaussian processes, is used to build surrogates to predict site-specific loads on an offshore wind turbine. Stochasticity in the inflow turbulence and irregular waves results in load responses that are best represented as random variables rather than deterministic values. Moreover, the effect of these stochastic sources on the loads depends strongly on the mean environmental conditions -- for instance, at low mean wind speeds, inflow turbulence produces much less variability in loads than at high wind speeds. Statisti…
Delta smelt is an endangered fish species in the San Francisco estuary that have shown an overall population decline over the past 30 years. Researchers have developed a stochastic, agent-based simulator to virtualize the system, with the goal of understanding the relative contribution of natural and anthropogenic factors suggested as playing a role in their decline. However, the input configuration space is high-dimensional, running the simulator is time-consuming, and its noisy outputs change nonlinearly in both mean and variance. Getting enough runs to effectively learn input--output dynami…
Randomizing the Fourier-transform (FT) phases of temporal-spatial data generates surrogates that approximate examples from the data-generating distribution. We propose such FT surrogates as a novel tool to augment and analyze training of neural networks and explore the approach in the example of sleep-stage classification. By computing FT surrogates of raw EEG, EOG, and EMG signals of under-represented sleep stages, we balanced the CAPSLPDB sleep database. We then trained and tested a convolutional neural network for sleep stage classification, and found that our surrogate-based augmentation i…
Virginia Tech（弗吉尼亚理工学院）；Argonne National Laboratory（阿贡国家实验室）；Université Côte d’Azur（蔚蓝海岸大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；LJAD（朱西厄分析、几何与应用实验室）
In recent years, active subspace methods (ASMs) have become a popular means of performing subspace sensitivity analysis on black-box functions. Naively applied, however, ASMs require gradient evaluations of the target function. In the event of noisy, expensive, or stochastic simulators, evaluating gradients via finite differencing may be infeasible. In such cases, often a surrogate model is employed, on which finite differencing is performed. When the surrogate model is a Gaussian process, we show that the ASM estimator is available in closed form, rendering the finite-difference approximation…
We present a probabilistic deep learning methodology that enables the construction of predictive data-driven surrogates for stochastic systems. Leveraging recent advances in variational inference with implicit distributions, we put forth a statistical inference framework that enables the end-to-end training of surrogate models on paired input-output observations that may be stochastic in nature, originate from different information sources of variable fidelity, or be corrupted by complex noise processes. The resulting surrogates can accommodate high-dimensional inputs and outputs and are able …
Sorbonne University（索邦大学）；University of Reading（雷丁大学）；University of Utrecht（乌得勒支大学）；École des Ponts ParisTech（巴黎高科路桥学院）；EDF R&D（法国电力集团研发部）；Université Paris-Est（巴黎东大学）
A novel method, based on the combination of data assimilation and machine learning is introduced. The new hybrid approach is designed for a two-fold scope: (i) emulating hidden, possibly chaotic, dynamics and (ii) predicting their future states. The method consists in applying iteratively a data assimilation step, here an ensemble Kalman filter, and a neural network. Data assimilation is used to optimally combine a surrogate model with sparse noisy data. The output analysis is spatially complete and is used as a training set by the neural network to update the surrogate model. The two steps ar…
Uncertainty quantification in mechanics: A unified Bayesian perspective
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Institute of Applied Mechanics, Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学应用力学研究所）；Institute of Biomechanics, Graz University of Technology（格拉茨工业大学生物力学研究所）；Department of Structural Engineering, Norwegian University of Science and Technology (NTNU)（挪威科技大学结构工程系）；Department of Mechanical Engineering, Stanford University（斯坦福大学机械工程系）
Optimizing expensive black-box systems with limited data is an extremely challenging problem. As a resolution, we present a new surrogate optimization approach by addressing two gaps in prior research -- unimportant input variables and inefficient treatment of uncertainty associated with the black-box output. We first design a new flexible non-interpolating parsimonious surrogate model using a partitioning-based multivariate adaptive regression splines approach, Tree Knot MARS (TK-MARS). The proposed model is specifically designed for optimization by capturing the structure of the function, be…
