Adaptive exploration under localization uncertainty using multi-fidelity Gaussian processes

A cost-effective solution to address the challenges posed by sensitive instrumentation in next-gen turbomachinery components is to reduce the number of measurement samples required to assess complex flows. This study investigates Gaussian Process (GP) modeling approaches within the framework of a data-driven hybrid measurement technique for turbomachinery applications. Three different modeling approaches—Baseline GP, CFD to Experiments GP, and Multi-Fidelity GP—are evaluated, and their performance in predicting mean flow characteristics and associated uncertainties on a low aspect ratio axial compressor stage, representative of the last stage of a high-pressure compressor, are focused on. The Baseline GP demonstrates robust accuracy, while the integration of CFD data in CFD into Experiments GP introduces complexities and more errors. The Multi-Fidelity GP, leveraging both CFD and experimental data, emerges as a promising solution, exhibiting enhanced accuracy in critical flow features. A sensitivity analysis underscores its stability and accuracy, even with reduced measurements. The Multi-Fidelity GP, therefore, stands as a reliable data fusion method for the proposed hybrid measurement technique, offering a potential reduction in instrumentation effort and testing times.

This paper evaluates the potential of two machine learning approaches i.e. Support vector machine (SVR) and Gaussian processes (GP) regression to model the oblique load capacity of batter pile groups. Linear regression was used to compare the performance of both SVR and GP based regression approaches to model the oblique load. Data set used consists of 147 samples obtained from the laboratory experiments. Out of the total sample size, 105 randomly selected samples were used for training whereas remaining 42 were used for testing the models. Input data set consist of angle of oblique load, pile length, sand relative density, number of vertical piles, number of batter piles where as oblique load was considered as output. Two kernel functions i.e. Polynomial and radial based kernel function were used with both SVR and GP regression. A comparison of results suggest that radial basis function based SVR approach works well in comparison to GP and linear regression based approaches and it could successfully be employed in modelling the oblique load capacity of batter pile groups. Parametric analysis and sensitivity analysis suggest that loading angle, pile length and number of batter pile were important in prediction of oblique load.

Autonomous marine vehicles are deployed in oceans and lakes to collect spatio-temporal data. GPS is often used for localization, but is inaccessible underwater. Poor localization underwater makes it difficult to pinpoint where data are collected, to accurately map, or to autonomously explore the ocean and other aquatic environments. This paper proposes the use of multifidelity Gaussian process regression to incorporate data associated with uncertain locations. With the proposed approach, an adaptive sampling algorithm is developed for exploration and mapping of unknown scalar fields. The reconstruction performance based on the multifidelity model is compared to that based on a single-fidelity Gaussian process model that only uses data with known locations, and to that based on a single-fidelity Gaussian process model that ignores the localization error. Numerical results show that the proposed multifidelity approach outperforms both single-fidelity approaches in terms of the reconstruction accuracy.

Bayesian optimization is a powerful technique for finding extrema of an objective function, a closed-form expression of which is not given but expensive evaluations at query points are available. Gaussian Process (GP) regression is often used to estimate the objective function and uncertainty estimates that guide GP-Upper Confidence Bound (GP-UCB) to determine where next to sample from the objective function, balancing exploration and exploitation. In general, it requires an auxiliary optimization to tune the hyperparameter in GP-UCB, which is sometimes not easy to carry out in practice. In this paper we present a simple practical method which improves GP-UCB, especially in cases where the objective function is not smooth with sharp peaks and valleys. We first present a geometric interpretation of GP-UCB on which we base our development of the clustering-guided method to select the next observation. Clustering is applied to two-dimensional vectors whose entries correspond to the posterior mean and standard deviation computed by GP regression, which is followed by utility maximization with GP-UCB, in order to determine where next to sample from the objective function. Experiments on various functions demonstrate our method alleviates the chance of being trapped in local extrema, making small efforts for auxiliary optimization.

In this paper, we formulate Gaussian process regression with observations under the localization uncertainty. In our formulation, effects of observations, measurement noise, localization uncertainty and prior distributions are all correctly incorporated in the posterior predictive statistics. The analytically intractable posterior predictive statistics are proposed to be approximated by Laplace approximations in different degrees of complexity. Such approximations have been carefully tailored to our problems and their approximation errors and complexity are analyzed. Simulation results demonstrate that the proposed approaches perform much better than approaches without considering the localization uncertainty correctly.

We consider a scenario in which an autonomous vehicle equipped with a downward facing camera operates in a 3D environment and is tasked with searching for an unknown number of stationary targets on the 2D floor of the environment. The key challenge is to minimize the search time while ensuring a high detection accuracy. We model the sensing field using a multi-fidelity Gaussian process that systematically describes the sensing information available at different altitudes from the floor. Based on the sensing model, we design a novel algorithm called Expedited Multi-Target Search (EMTS) that (i) addresses the coverage-accuracy trade-off: sampling at locations farther from the floor provides wider field of view but less accurate measurements, (ii) computes an occupancy map of the floor within a prescribed accuracy and quickly eliminates unoccupied regions from the search space, and (iii) travels efficiently to collect the required samples for target detection. We rigorously analyze the algorithm and establish formal guarantees on the target detection accuracy and the detection time. We illustrate the algorithm using a simulated multi-target search scenario.

In this paper, we propose distributed Gaussian process regression (GPR) for resource-constrained distributed sensor networks under localization uncertainty. The proposed distributed algorithm, which combines Jacobi over-relaxation (JOR) and discrete-time average consensus (DAC), can effectively handle localization uncertainty as well as limited communication and computation capabilities of distributed sensor networks. We also extend the proposed method hierarchically using sparse GPR to improve its scalability. The performance of the proposed method is verified in numerical simulations against the centralized maximum a posteriori (MAP) solution and a quick-and-dirty solution. We show that the proposed method outperforms the quick-and-dirty solution and achieve an accuracy comparable to the centralized solution.

Propagation of uncertainty in the mechanical and biological response of growing tissues using multi-fidelity Gaussian process regression

Abstract. The rivers of High-mountain Asia provide freshwater to around 1.9 billion people. However, precipitation, the main driver of river flow, is still poorly understood due to limited in situ measurements in this area. Existing tools to interpolate these measurements or downscale and bias-correct precipitation models have several limitations. To overcome these challenges, this paper uses a probabilistic machine learning approach called multi-fidelity Gaussian processes (MFGPs) to downscale the fifth ECMWF climate reanalysis (ERA5). The method is first validated by downscaling ERA5 precipitation data over data-rich Europe and then data-sparse upper Beas and Sutlej river basins in the Himalayas. We find that MFGPs are simpler to implement and more applicable to smaller datasets than other state-of-the-art machine learning methods. MFGPs are also able to quantify and narrow the uncertainty associated with the precipitation estimates, which is especially needed over ungauged areas and can be used to estimate the likelihood of extreme events that lead to floods or droughts. Over the upper Beas and Sutlej river basins, the precipitation estimates from the MFGP model are similar to or more accurate than available gridded precipitation products (APHRODITE, TRMM, CRU TS, and bias-corrected WRF). The MFGP model and APHRODITE annual mean precipitation estimates generally agree with each other for this region, with the MFGP model predicting slightly higher average precipitation and variance. However, more significant spatial deviations between the MFGP model and APHRODITE over this region appear during the summer monsoon. The MFGP model also presents a more effective resolution, generating more structure at finer spatial scales than ERA5 and APHRODITE. MFGP precipitation estimates for the upper Beas and Sutlej basins between 1980 and 2012 at a 0.0625° resolution (approx. 7 km) are jointly published with this paper.

Wind turbines are growing bigger to becomemore cost-efficient. This does increase the severity of the vibrations that are present in the turbine blades, both due to predictable effects like wind shear and tower shadow, and due to less predictable effects like turbulence and flutter. If wind turbines are to become bigger and more cost-efficient, these vibrations need to be reduced. This can be done by installing trailing-edge flaps to the blades. Because of the variety of circumstances which the turbine should operate in, this results in large uncertainties. As such, we need methods that can take stochastic effects into account. Preferably we develop an algorithmthat can learn from online data how the flaps affect the wind turbine and how to optimally control them. A simple prior analysis can be done using a linearized version of the system. In this case it is important to know not only the expected cost (damage) that will be incurred by the wind turbine in various situations, but also the spread of this cost. This can for instance be done by looking at the variance of the cost function. Various expressions are available to analytically calculate this variance. Alternatively, we can prescribe a degree of stability for the system. Due to the limitations of linear approximations of systems, it is more effective to apply nonlinear regression methods. A promising one is Gaussian Process (GP) regression. Given a training set (X, y) it can predict function values f (x¤) for test points x¤. It has its basis in Bayesian probability theory, which allows it to not only make this prediction, but also give information (the variance) about its accuracy. The usual way in which GP regression is applied has a few important limitations. Most importantly, it is computationally intensive, especially when applied to constantly growing data sets. In addition, it has difficulties dealing with noise present in the training input points x. There are methods to solve either of these issues, but these tricks generally do not work well together, or their combination requires many computational resources. However, by making the right approximations, like Taylor expansions and at times even linearizations, Gaussian process regression can be applied efficiently, in an online way, to data sets with noisy input points. This enables GP regression to be used for system identification problems like online non-linear black-box modeling. Another limitation is that it can be difficult to find the optimum of a Gaussian process. The reason is that the optimum of a Gaussian process is not a fixed point but a random variable. The distribution of this optimum cannot be calculated analytically, but we can use particle methods to approximate it. We can subsequently use this principle to efficiently explore an unknown nonlinear function, trying to locate its optimum. To do so, we sample a point x from the optimum distribution, measure what the function value f (x) at this point is, update the Gaussian process approximation of the function, update the optimum distribution and repeat this process until the distribution has converged. Finding the optimum of a function like this has shown to have competitive performance at keeping the cumulative regret low, compared to similar algorithms. In addition, it allows wind turbines to tune the gains of a fixed-structure controller so as to optimize a nonlinear cost function like the damage equivalent load. All these improvements are a step forward in the application of Gaussian process regression to wind turbine applications. But as is always the case with research, there are still many things left to improve further.

Heat transfer analysis of cheese cooling incorporating uncertainty in temperature measurement locations: Model development and validation

High-dimensional surrogate modeling with limited high-fidelity data poses a major challenge in uncertainty quantification. Classical supervised dimension reduction methods often fail in this setting due to insufficient accurate observations, while low-fidelity data are abundant but biased. In this work, we propose a Rotated Multi-Fidelity Gaussian Process (RMFGP) framework that enables reliable dimension reduction and surrogate construction under severe data scarcity. The proposed method integrates nonlinear multi-fidelity Gaussian process regression with sliced average variance estimation (SAVE) to iteratively identify informative input directions. Low-fidelity data are first used to extract coarse structural information, which is exploited to rotate the input space prior to multi-fidelity model training. Predictions generated by the trained RMFGP surrogate are then used to refine the dimension reduction, allowing accurate estimation of the central sufficient dimension reduction subspace even when high-fidelity data are scarce. A Bayesian active learning strategy based on predictive uncertainty is further incorporated to adaptively select new high-fidelity samples. Numerical examples, including stochastic partial differential equations, demonstrate that RMFGP significantly improves prediction accuracy, convergence, and uncertainty propagation compared to existing Gaussian process-based dimension reduction approaches, while requiring substantially fewer high-fidelity evaluations.

Gaussian Process (GP) models are widely used for function approximation, with their performance heavily dependent on the selection of informative training points. Traditional uncertainty-based sampling methods prioritize regions with high predictive variance, capturing local uncertainty but failing to optimize information gain on a global scale. This paper introduces a novel entropy-based adaptive sampling strategy, which selects data points that maximize entropy reduction, thereby enhancing global model informativeness. To further improve efficiency, a hybrid entropy-uncertainty sampling method is proposed that balances global information gain and local uncertainty reduction. The effectiveness of entropy as a selection criterion is validated through its correlation with mean squared error (MSE). Experimental results demonstrate that combining entropy-based and uncertainty-based sampling outperforms both individual methods, achieving faster MSE reduction. Two case studies confirm that the optimal balance between entropy and uncertainty is problem-dependent, highlighting the need for adaptive weighting strategies. These findings establish entropy-based and hybrid sampling as valuable tools to improve the accuracy and efficiency of GP regression.

Unveiling the rheological properties of fiber suspensions is of paramount interest to many industrial applications. There are multiple factors, such as fiber aspect ratio and volume fraction, that play a significant role in altering the rheological behavior of suspensions. Three-dimensional (3D) numerical simulations of coupled differential equations of the suspension of fibers are computationally expensive and time-consuming. Machine learning algorithms can be trained on the available data and make predictions for the cases where no numerical data are available. However, some widely used machine learning surrogates, such as neural networks, require a relatively large training dataset to produce accurate predictions. Multi-fidelity models, which combine high-fidelity data from numerical simulations and less expensive lower fidelity data from resources such as simplified constitutive equations, can pave the way for more accurate predictions. Here, we focus on neural networks and the Gaussian processes with two levels of fidelity, i.e., high and low fidelity networks, to predict the steady-state rheological properties, and compare them to the single-fidelity network. High-fidelity data are obtained from direct numerical simulations based on an immersed boundary method to couple the fluid and solid motion. The low-fidelity data are produced by using constitutive equations. Multiple neural networks and the Gaussian process structures are used for the hyperparameter tuning purpose. Results indicate that with the best choice of hyperparameters, both the multi-fidelity Gaussian processes and neural networks are capable of making predictions with a high level of accuracy with neural networks demonstrating marginally better performance.

Semantic perception can provide autonomous robots operating under uncertainty with more efficient representation of their environment and better ability for correct loop closures than only geometric features. However, accurate inference of semantics requires measurement models that correctly capture properties of semantic detections such as viewpoint dependence, spatial correlations, and intra- and inter-class variations. Such models should also gracefully handle open-set conditions which may be encountered, keeping track of the resultant model uncertainty. We propose a method for robust visual classification of an object of interest observed from multiple views in the presence of significant localization uncertainty and classifier noise, and possible dataset shift. We use a viewpoint dependent measurement model to capture viewpoint dependence and spatial correlations in classifier scores, showing how to use it in the presence of localization uncertainty. Assuming a Bayesian classifier providing a measure of uncertainty, we show how its outputs can be fused in the context of the above model, allowing robust classification under model uncertainty when novel scenes are encountered. We present statistical evaluation of our method both in synthetic simulation, and in a 3D environment where rendered images are fed into a Deep Neural Network classifier. We compare to baseline methods in scenarios of varying difficulty showing improved robustness of our method to localization uncertainty and dataset shift. Finally, we validate our contribution w.r.t. localization uncertainty on a dataset of real-world images.