Multi-fidelity Models Research Articles

Physics-informed machine learning model for battery state of health prognostics using partial charging segments

The accurate and efficient estimation of battery state-of-health (SoH) is an ever-significant issue for applications of lithium-ion batteries (LIBs). Physics-of-failure (PoF) and machine-learning (ML) approaches have shown success in the field of prognostics and health management. However, accurate prediction of these models depends on clear understandings of complicated underlying physics, or full access to massive historical usage data of the system, which are not always obtainable in the engineering applications. Studies on physics-informed ML frameworks have gained attention because they embed physics knowledge into ML algorithms; recently, this method has demonstrated the potential to enhance performance over traditional model-based techniques, particularly when handling highly non-linear complex systems, such as LIBs. In this study, an advanced physics-informed ML (PIML) framework is proposed for the LIB SoH estimation, which includes three steps. The physics-based finite element (FE) model is firstly used to calculate the influences of the dominating aging mode, i.e., the solide electrolyte interface (SEI) growth on anode particle surfaces, and on capacity loss of LIB under various operating conditions. Then, the FE results are fused with experimental data from NASA Ames Prognostics Center of Excellence to construct a multi-fidelity model. Lastly, the Gaussian Process Regression (GPR) model is trained to create the mapping between voltage curves and the corresponding SEI thickness. A case study with the partial charging voltage segment as input is introduced to validate the PIML framework. Overall, the framework demonstrates a favorable performance for SoH estimation, as well as providing a basis for future online estimation frameworks.

Multifidelity Heating Prediction of Adaptable, Deployable Entry Placement Technology Vehicles

The objective of this work was to investigate a multifidelity modeling approach to accurately and efficiently predict the laminar convective, turbulent convective, and radiative heating on adaptable, deployable entry placement technology vehicles in Mars entry. A previously developed cokriging-based multifidelity modeling approach was used to model the heat fluxes at several surface locations along the vehicle, including the rib sections. The laminar convective heat flux multifidelity model was found to have a mean convective heat rate error of approximately 3% when compared with high-fidelity computational fluid dynamics (CFD) simulations. The turbulent convective heat flux multifidelity model was found to have a mean convective heat rate error of approximately 8% when compared with high-fidelity CFD simulations. The radiative heat flux multifidelity model was found to have a mean convective heat rate error of approximately 10% when compared with high-fidelity CFD simulations. Compared with a single-fidelity model, the multifidelity model required, at most, one-third the number of high-fidelity model evaluations to obtain the same accuracy level. The computational cost of evaluating the multifidelity model was approximately six orders of magnitude less than one high-fidelity model simulation.

An Integrated Surrogate Modeling Method for Fusing Noisy and Noise-Free Data

Abstract Many datasets in engineering applications are heterogeneous mixtures of noise-free data, noisy data with known noise variances, and noisy data with unknown noise variances. This article proposes a data fusion method called the multi-type data fusion (MTDF) model, which fully utilizes the information provided by each of these types of data. To capture the underlying trend implied in the multiple types of data, the method approximately interpolates the noise-free data, while regressing the noisy data. The prediction accuracy of the MTDF model is compared with those of various surrogate models (interpolation models, regression models, and multi-fidelity models) on both numerical and practical engineering problems. In the experiments, the proposed MTDF model demonstrates higher performance than the other benchmark models. The effects of noise level and sample size of the noise-free data on the model performance are investigated, along with the robustness of the MTDF model. The results demonstrate the satisfactory feasibility, practicality, and stability of the MTDF.

Multi-fidelity Data Aggregation using Convolutional Neural Networks

Multi-fidelity data exist in almost every engineering and science discipline, which can be from simulation, experiments, and a hybrid form. High fidelity data are usually associated with higher accuracy and expense (e.g., high resolution experimental testing or finer scale simulation), while low-fidelity data are on the opposite side in terms of the accuracy and cost. Multi-fidelity data aggregation (MDA) in this study refers to the process of combining two or multiple sources of different fidelity data to have a high accuracy estimation and low computational cost. MDA has a wide range of application in engineering and science, such as multiscale simulation, multi-resolution imaging, and hybrid simulation-testing. This paper presents a novel framework named Multi-fidelity Data Aggregation using Convolutional Neural Networks (MDA-CNN) for multi-fidelity modeling. The MDA-CNN architecture has three components: multi-fidelity data compiling, multi-fidelity perceptive field and convolution, and deep neural network for mapping. This framework captures and utilizes implicit relationships between any high-fidelity datum and all available low-fidelity data using a defined local perceptive field and convolution. Most existing strategies rely on the collocation method and interpolation, which focuses on the single point relationship. The proposed method has several unique benefits. First, the proposed framework treats the multi-fidelity data as image data and processes them using CNN, which is very scalable to high dimensional data with more than two fidelities. Second, the flexibility of nonlinear mapping in neural network facilitates the multi-fidelity aggregation and does not need to assume specific relationships among multiple fidelities. Third, the proposed framework does not assume that multi-fidelity data are at the same order or from the same physical mechanisms (e.g., assumptions are needed for some error estimation-based multi-fidelity model). Thus, the proposed method can handle data aggregation from multiple sources across different scales, such as different order derivatives and other correlated phenomenon data in a single framework. The proposed MDA-CNN is validated using extensive numerical examples and experimental data with multi-source and multi-fidelity data. Discussions are given to illustrate the benefits and limitations of the proposed framework. Conclusions and future work are presented based on the observations in the current study.

Selection of Existing Sail Designs for Multi-Fidelity Surrogate Models

Velocity Prediction Programs (VPPs) are commonly used to help predict and compare the performance of different sail designs. A VPP requires an aerodynamic input force matrix which can be computationally expensive to calculate, limiting its application in industrial sail design projects. The use of multi-fidelity kriging surrogate models has previously been presented by the authors to reduce this cost, with high-fidelity data for a new sail being modelled and the low-fidelity data provided by data from existing, but different, sail designs. The difference in fidelity is not due to the simulation method used to obtain the data, but instead how similar the sail’s geometry is to the new sail design. An important consideration for the construction of these models is the choice of low-fidelity data points, which provide information about the trend of the model curve between the high-fidelity data. A method is required to select the best existing sail design to use for the low-fidelity data when constructing a multi-fidelity model. The suitability of an existing sail design as a low fidelity model could be evaluated based on the similarity of its geometric parameters with the new sail. It is shown here that for upwind jib sails, the similarity of the broadseam between the two sails best indicates the ability of a design to be used as low-fidelity data for a lift coefficient surrogate model. The lift coefficient surrogate model error predicted by the regression is shown to be close to 1% of the lift coefficient surrogate error for most points. Larger discrepancies are observed for a drag coefficient surrogate error regression.

Deep Reinforcement Learning for Robotic Control with Multi-Fidelity Models

Deep reinforcement learning (DRL) can be used for the development of robotic controllers. Complicated kinematic relationships can be learned by a DRL agent, which will result in a control policy that takes actions based on an observed state. However, a DRL agent typically goes through much trial and error before beginning to take appropriate actions. Therefore, it is often useful to leverage simulated robotic manipulators before performing any training or testing on actual hardware. There are several options for such simulation, ranging from simple kinematic models to more complex models seeking to accurately simulate the effects of gravity, inertia, and friction. The latter models can provide excellent representations of a robotic plant, but typically with a noticeably increased computational expense. Reducing the expense of simulating the robotic plant (while still maintaining a reasonable degree of accuracy) can accelerate an already expensive DRL training loop. In this work, we present a methodology for using a lower-fidelity model (based on Denavit-Hartenberg parameters) to initialize the training of a DRL agent for control of a Sawyer robotic arm. We show that the trained DRL policy can then be fine-tuned in a higher-fidelity simulation provided by the robot's manufacturer. We demonstrate the accuracy of the fully trained policy by transferring it to the actual hardware, demonstrating the power of DRL to learn complicated robotic tasks entirely in simulation. Finally, we benchmark the time required to train a policy using each level of fidelity.

Flexible cylinder flow-induced vibration

In this paper, we conducted a selective review on the recent progress in physics insight and modeling of flexible cylinder flow-induced vibrations (FIVs). FIVs of circular cylinders include vortex-induced vibrations (VIVs) and wake-induced vibrations (WIVs), and they have been the center of the fluid-structure interaction (FSI) research in the past several decades due to the rich physics and the engineering significance. First, we summarized the new understanding of the structural response, hydrodynamics, and the impact of key structural properties for both the isolated and multiple circular cylinders. The complex FSI phenomena observed in experiments and numerical simulations are explained carefully via the analysis of the vortical wake topology. Following up with several critical future questions to address, we discussed the advancement of the artificial intelligent and machine learning (AI/ML) techniques in improving both the understanding and modeling of flexible cylinder FIVs. Though in the early stages, several AL/ML techniques have shown success, including auto-identification of key VIV features, physics-informed neural network in solving inverse problems, Gaussian process regression for automatic and adaptive VIV experiments, and multi-fidelity modeling in improving the prediction accuracy and quantifying the prediction uncertainties. These preliminary yet promising results have demonstrated both the opportunities and challenges for understanding and modeling of flexible cylinder FIVs in today's big data era.

Multi-Fidelity Modeling and Analysis of a Pressurized Vessel-Pipe-Safety Valve System Based on Moc and Surrogate Modeling Methods

Multi-Fidelity Modeling and Analysis of a Pressurized Vessel-Pipe-Safety Valve System Based on Moc and Surrogate Modeling Methods

Multi-Fidelity Model-Based Size Optimization of Electric Machines

The multi-modal problem and high computational cost represent challenges in the optimization of electric machines owing to their highly nonlinear electromagnetic response. To overcome these challenges, this paper proposes a multi-fidelity model-based sequential optimization method in which both low- and high-fidelity models are employed in two phases. In phase 1, the reluctance network (RN) is adopted as the low-fidelity model and mainly contributes to alleviating the abovementioned challenges. To overcome the low accuracy of the RN, the optimal design is obtained using a finite element model (FEM) in phase 2. The multi-start strategy and gradient-based algorithm are utilized instead of a heuristic algorithm in all phases to avoid excessive calculations. This multi-fidelity model concept presents a novelty compared to previous research that focused only on algorithm development. The effectiveness of the proposed method is validated with two examples, consisting of the TEAM workshop problem 25 and the torque ripple minimization of an interior permanent magnet synchronous motor. The optimal designs and computational time resulting from the proposed method are compared with those of the conventional method, where only a FEM is used during optimization. The results show that the proposed method is remarkable in finding superior optimal designs, while ignoring unimportant local optima. Additionally, it can save up to 90% of the computational time required by the conventional method.

Adaptive multi-fidelity (AMF) modelling of progressive damage in notched composite laminates

In recent years, high-fidelity simulations with discrete crack modelling techniques have been shown to accurately model the mechanics of progressive failure in composite laminates. However, these require high computational effort and are currently not practical beyond small coupon-sized specimens with a few tens of plies. In this paper, an adaptive multi-fidelity (AMF) modelling approach is proposed, wherein actively damaging areas are modelled with high-fidelity three-dimensional (3D) brick elements and discrete cracks, while dormant and inactive sites are modelled with lower-fidelity shell elements and smeared cracks. The transition criteria between the two levels of modelling are studied in order to preserve as much fidelity to the physics as necessary while improving computational efficiency. The results are benchmarked with that of high-fidelity simulations and experimental data of open-hole tension (OHT) of cross-ply and quasi-isotropic laminates in the literature.

Multi-fidelity regression using artificial neural networks: Efficient approximation of parameter-dependent output quantities

Highly accurate numerical or physical experiments are often very time-consuming or expensive to obtain. When time or budget restrictions prohibit the generation of additional data, the amount of available samples may be too limited to provide satisfactory model results. Multi-fidelity methods deal with such problems by incorporating information from other sources, which are ideally well-correlated with the high-fidelity data, but can be obtained at a lower cost. By leveraging correlations between different data sets, multi-fidelity methods often yield superior generalization when compared to models based solely on a small amount of high-fidelity data. In the current work, we present the use of artificial neural networks applied to multi-fidelity regression problems. By elaborating a few existing approaches, we propose new neural network architectures for multi-fidelity regression. The introduced models are compared against a traditional multi-fidelity regression scheme — co-kriging. A collection of artificial benchmarks are presented to measure the performance of the analyzed models. The results show that cross-validation in combination with Bayesian optimization leads to neural network models that outperform the co-kriging scheme. Additionally, we show an application of multi-fidelity regression to an engineering problem. The propagation of a pressure wave into an acoustic horn with parametrized shape and frequency is considered, and the index of reflection intensity is approximated using the proposed multi-fidelity models. A finite element, full-order model and a reduced-order model built through the reduced basis method are adopted as the high- and low-fidelity, respectively. It is shown that the multi-fidelity neural networks return outputs that achieve a comparable accuracy to those from the expensive, full-order model, using only very few full-order evaluations combined with a larger amount of inaccurate but cheap evaluations of the reduced order model.

Multi-fidelity meta modeling using composite neural network with online adaptive basis technique

A composite neural network (NN) is a way to improve the reliability of a prediction field by implementing a multi-fidelity model when high-fidelity data are extremely limited. The reliability of the composite NN depends highly on the quantity and quality of low-fidelity data. Satisfying both issues simultaneously is difficult in engineering practices even if low-fidelity data are considered. With this motivation, we suggest a strategy for ameliorating low-fidelity data to efficiently improve the prediction fields of a composite NN. In the proposed strategy, a reduced-order modeling (ROM) is used to supply a large number of low-fidelity data with relatively low computational costs. To decrease ROM errors that may decisively hinder the training of a cross-correlation between low-fidelity and high-fidelity data, the low-fidelity data are updated using an efficient adaptive basis technique. The adaptive basis in this work is calculated by optimizing the local error data to avoid huge computational costs. As a result, the low-fidelity data can better reflect the trend while satisfying the condition in which the low-fidelity data should be sufficiently provided through an efficient procedure. Furthermore, the strategy applied in this study is conducted without any modifications of the given high-fidelity data. Hence, even if high-fidelity data may no longer be obtained, it is possible to efficiently obtain the high-quality prediction field of the composite NN. A detailed strategy is proposed herein, and its performance is evaluated through various numerical examples.

Multifidelity modeling for Physics-Informed Neural Networks (PINNs)

Multifidelity simulation methodologies are often used in an attempt to judiciously combine low-fidelity and high-fidelity simulation results in an accuracy-increasing, cost-saving way. Candidates for this approach are simulation methodologies for which there are fidelity differences connected with significant computational cost differences. Physics-informed Neural Networks (PINNs) are candidates for these types of approaches due to the significant difference in training times required when different fidelities (expressed in terms of architecture width and depth as well as optimization criteria) are employed. In this paper, we propose a particular multifidelity approach applied to PINNs that exploits low-rank structure. We demonstrate that width, depth, and optimization criteria can be used as parameters related to model fidelity and show numerical justification of cost differences in training due to fidelity parameter choices. We test our multifidelity scheme on various canonical forward PDE models that have been presented in the emerging PINNs literature.

A multi-fidelity surrogate modeling approach for incorporating multiple non-hierarchical low-fidelity data

Multi-fidelity (MF) surrogate models have been widely used in simulation-based design problems to reduce the computational cost by integrating the data with different fidelity levels. Most of the existing MF modeling methods are only applicable to the problems with hierarchical low-fidelity (LF) models, namely the fidelity levels of multiple LF models can be identified. However, the fidelity levels of the LF models that are obtained from different simplification methods often vary over the design space. To address this challenge, a non-hierarchical Co-Kriging modeling (NHLF-Co-Kriging) method that can flexibly handle multiple non-hierarchical LF models is developed in this work. In the proposed method, multiple LF models are scaled by different scale factors, and a discrepancy model is utilized to depict the differences between the HF model and the ensembled LF models. To make the discrepancy Gaussian process (GP) model easy to be fitted, an optimization problem whose objective is to minimize the second derivative of the prediction values of the discrepancy GP model is defined to obtain optimal scale factors of the LF models. The performance of the NHLF-Co-Kriging method is compared with the extended Co-Kriging model and linear regression MF surrogate model through several analytical examples and an engineering case. Results show that the proposed method selects more reasonable scale factors for the multiple LF models and provides more accurate MF surrogate models under a limited computational budget.

A fast multi-fidelity method with uncertainty quantification for complex data correlations: Application to vortex-induced vibrations of marine risers

We develop a fast multi-fidelity modeling method for very complex correlations between high- and low-fidelity data by working in modal space to extract the proper correlation function. We apply this method to infer the amplitude of motion of a flexible marine riser in cross-flow, subject to vortex-induced vibrations (VIV). VIV are driven by an absolute instability in the flow, which imposes a frequency (Strouhal) law that requires a matching with the impedance of the structure; this matching is easily achieved because of the rapid parametric variation of the added mass force. As a result, the wavenumber of the riser spatial response is within narrow bands of uncertainty. Hence, an error in wavenumber prediction can cause significant phase-related errors in the shape of the amplitude of response along the riser, rendering correlation between low- and high-fidelity data very complex. Working in modal space as outlined herein, dense data from low-fidelity data, provided by the semi-empirical computer code VIVA, can correlate in modal space with few high-fidelity data, obtained from experiments or fully-resolved CFD simulations, to correct both phase and amplitude and provide predictions that agree very well overall with the correct shape of the amplitude response. We also quantify the uncertainty in the prediction using Bayesian modeling and exploit this uncertainty to formulate an active learning strategy for the best possible location of the sensors providing the high fidelity measurements.

A screening-based gradient-enhanced Gaussian process regression model for multi-fidelity data fusion

The prediction accuracy of multi-fidelity models can be enhanced by incorporating gradient formation. However, the computational complexity would increase dramatically as the number of design variables increase. In this work, a gradient-enhanced multi-fidelity Gaussian process model using a portion of gradients (PGEMFGP) is proposed. To be specific, a Bayesian Gaussian process regression model for multi-fidelity (MF) data fusion is developed, which incorporates high-fidelity (HF) and low-fidelity (LF) responses, as well as the corresponding gradients. A screening technique based on distance correlation is applied to select a portion of gradients of the low-fidelity model so that the modeling complexity can be greatly reduced. The merit of the proposed method is tested with six numerical examples ranging from 10-D to 30-D, as well as an aerodynamic airfoil case with 18 design variables. The proposed method is compared to two other existing gradient-enhanced Gaussian process-based models. It is shown that the modeling efficiency of the proposed model is dramatically improved compared to the original gradient-enhanced multi-fidelity Gaussian process model, while the loss of the prediction accuracy can be almost negligible. In consequence, it can be a promising approach for gradient-enhanced models dealing with multi-fidelity data.

Improving the Film Cooling Performance of a Turbine Endwall With Multi-Fidelity Modeling Considering Conjugate Heat Transfer

Abstract The gas turbine endwall is bearing extreme thermal loads with the rapid increase of turbine inlet temperature. Therefore, the effective cooling of turbine endwalls is of vital importance for the safe operation of turbines. In the design of endwall cooling layouts, numerical simulations based on conjugate heat transfer (CHT) are drawing more attention as the component temperature can be predicted directly. However, the computation cost of high-fidelity (HF) CHT analysis can be high and even prohibitive especially when there are many cases to evaluate such as in the design optimization of cooling layout. In this study, we established a multi-fidelity (MF) framework in which the data of low-fidelity (LF) CHT analysis were incorporated to help the building of a model that predicts the result of HF simulation. Based upon this framework, MF design optimization of a validated numerical turbine endwall model was carried out. The high- and LF data were obtained from the computation of fine mesh and coarse mesh, respectively. In the optimization, the positions of the film cooling holes were parameterized and controlled by a shape function. With the help of MF modeling and sequentially evaluated designs, the cooling performance of the model endwall was improved efficiently.

Nonlinear Unsteady Aerodynamics Reduced Order Model of Airfoils Based on Algorithm Fusion and Multifidelity Framework

A reduced order modeling method based on algorithm fusion and multifidelity framework for nonlinear unsteady aerodynamics is proposed to obtain a low-cost and high-precision unsteady aerodynamic model. This method integrates the traditional algorithm, intelligent algorithm, and multifidelity data fusion algorithm. In this method, the traditional algorithm is based on separated flow theory, the intelligent algorithm refers to the nonlinear autoregressive (NARX) method, and the multifidelity data fusion algorithm uses different fidelity data for aerodynamic modeling, which can shorten the time cost of data acquisition. In the process of modeling, firstly, a multifidelity model with NARX description provides a general intelligent algorithm framework for unsteady aerodynamics. Then, based on the separated flow theory, the correction equation from low-fidelity model to high-fidelity result is constructed, and the cuckoo algorithm based on chaos optimization is used to identify the parameters. In order to verify the effectiveness of the method, an unsteady aerodynamic model of NACA0012 airfoil is established. Three kinds of data with low, medium, and high fidelity are used for modeling. The low-fidelity and medium-fidelity data is obtained from the CFD-Euler solver and CFD-RANS solver, respectively, while the high-fidelity data comes from the experimental results. Then, the model is established, and its prediction of unsteady aerodynamic coefficients is in good agreement with the CFD results and the experimental data. After that, the model is applied to a two-dimensional aeroelastic system, and the bifurcation and limit cycle response analysis are compared with the experimental results, which further shows that the model can accurately capture the main flow characteristics in the flow range of low speed and high angle of attack. In addition, the convergence of the model is studied; the accuracy and generalization ability as well as applicability scope of the model are compared with other aerodynamic models and finally discussed.

Bayesian learning of orthogonal embeddings for multi-fidelity Gaussian Processes

Uncertainty propagation in complex engineering systems often poses significant computational challenges related to modeling and quantifying probability distributions of model outputs, as those emerge as the result of various sources of uncertainty that are inherent in the system under investigation. Gaussian Processes regression (GPs) is a robust meta-modeling technique that allows for fast model prediction and exploration of response surfaces. Multi-fidelity variations of GPs further leverage information from cheap and low fidelity model simulations in order to improve their predictive performance on the high fidelity model. In order to cope with the high volume of data required to train GPs in high dimensional design spaces, a common practice is to introduce latent design variables that are typically projections of the original input space to a lower dimensional subspace, and therefore substitute the problem of learning the initial high dimensional mapping, with that of training a GP on a low dimensional space. In this paper, we present a Bayesian approach to identify optimal transformations that map the input points to low dimensional latent variables. The “projection” mapping consists of an orthonormal matrix that is considered a priori unknown and needs to be inferred jointly with the GP parameters, conditioned on the available training data. The proposed Bayesian inference scheme relies on a two-step iterative algorithm that samples from the marginal posteriors of the GP parameters and the projection matrix respectively, both using Markov Chain Monte Carlo (MCMC) sampling. In order to take into account the orthogonality constraints imposed on the orthonormal projection matrix, a Geodesic Monte Carlo sampling algorithm is employed, that is suitable for exploiting probability measures on manifolds. We extend the proposed framework to multi-fidelity models using GPs including the scenarios of training multiple outputs together. We validate our framework on three synthetic problems with a known lower-dimensional subspace. The benefits of our proposed framework, are illustrated on the computationally challenging aerodynamic optimization of a last-stage blade for an industrial gas turbine, where we study the effect of an 85-dimensional shape parameterization of a three-dimensional airfoil on two output quantities of interest, specifically on the aerodynamic efficiency and the degree of reaction.

Adaptive reliability analysis for multi-fidelity models using a collective learning strategy

In many fields of science and engineering, models with different fidelities are available. Physical experiments or detailed simulations that accurately capture the behavior of the system are regarded as high-fidelity models with low model uncertainty; however, they are expensive to run. On the other hand, simplified physical experiments or numerical models are seen as low-fidelity models that are cheaper to evaluate. Although low-fidelity models are often not suitable for direct use in reliability analysis due to their low accuracy, they can offer information about the trend of the high-fidelity model thus providing the opportunity to explore the design space at a low cost. This study presents a new approach called adaptive multi-fidelity Gaussian process for reliability analysis (AMGPRA). Contrary to selecting training points and information sources in two separate stages as done in state-of-the-art mfEGRA method, the proposed approach finds the optimal training point and information source simultaneously using the novel collective learning function (CLF). CLF is able to assess the global impact of a candidate training point from an information source and it accommodates any learning function that satisfies a certain profile. In this context, CLF provides a new direction for quantifying the impact of new training points and can be easily extended with new learning functions to adapt to different reliability problems. The performance of the proposed method is demonstrated by three mathematical examples and one engineering problem concerning the wind reliability of transmission towers. It is shown that the proposed method achieves similar or higher accuracy with reduced computational costs compared to state-of-the-art single and multi-fidelity methods. A key application of AMGPRA is high-fidelity fragility modeling using complex and costly physics-based computational models.