Multi-fidelity meta modeling using composite neural network with online adaptive basis technique
Introduction
Deep learning approaches have recently become an essential tool for predicting physical phenomena in many types of engineering practices [1], [2], [3]. In general, large numbers of high-fidelity data are needed to obtain a reliable prediction field when using a deep learning approach. However, in real cases, the high-fidelity data may not be sufficient, and thus the reliability of the prediction field may not be adequately secured [4], [5].
The multi-fidelity approach can readily handle the issue caused by a lack of the high-fidelity data [6], [7]. In this approach, low-fidelity data are additionally invoked to generate a cross-correlation between the low- and high-fidelity data. As a result, the quality of low-fidelity data is significantly enhanced by using the acquired cross-correlation, and thus the effect of obtaining a large number of high-fidelity data can be similarly implemented. Hence, the multi-fidelity approach can dramatically ameliorate the prediction field in comparison to a single-fidelity approach [6], [7].
In the multi-fidelity approach, two issues should be addressed if high-fidelity data are already given: first, how to find the cross-correlation between the low- and high-fidelity data, and second, how to obtain a large number of low-fidelity data in an efficient way. The former issue can be accurately handled by a composite neural network (NN) [4], [8], [9]. In a composite NN, the cross-correlation is approximated using a machine learning approach. The excellent performance of a composite NN has been proven in many multi-fidelity applications [4], [8], [9]. For this reason, we implement a multi-fidelity approach through a composite NN. It should be noted that in the traditional modeling point of a view, the low-fidelity modeling is about building an approximate model, and then solving the problem accurately. On the other hand, the high-fidelity modeling is about constructing the model precisely, and then handling the problem with well-assumed conditions [10]. However, in this work, the low- and high-fidelity data are classified according to the data quality because we employ the concept of the fidelity for the composite NN used in the previous studies [4], [11].
The quantity issue of low-fidelity data can be handled using reduced-order modeling (ROM). ROM techniques have been widely used to solve large and complex models with extremely low numerical costs in mono- and/or multi-physics problems [12], [13], [14], [15], [16], [17]. Using a ROM method, more response fields with various parameters can be computed than in the case of the entire model over a given period of time. [12], [13], [14], [15], [16], [17]. Therefore, the condition of the low-fidelity data, which should be plentifully supplied to the composite NN, can be easily achieved with relatively low computational costs through the ROM method [8], [9]. There are several methods used to take the ROM basis in the construction of a reduction model. Among these methods, a proper orthogonal decomposition (POD) has been actively employed because the basis returned by this method can mathematically compress the data of interest [18], [19].
Despite the outstanding results achieved by combining the ROM methods with a composite NN, several challenges remain. This is because the multi-fidelity approach assumes that the low-fidelity data should reflect trends regarding high-fidelity data [4], [11]. However, if ROM errors are significant, the ROM results may be insufficient to properly explain the trends of the high-fidelity data [20], [21], [22]. In particular, these errors often occur when the ROM basis is inadequate to describe various parameter fields [21], [22]. In this case, the quality of the prediction field may be untrustworthy because the ROM errors may interfere with finding the cross-correlation between a given small number of high-fidelity data and overall ROM results. Basis correction techniques are additionally employed to treat such problems. However, it may not be easy to pursue both reliable ROM results and an efficient computational procedure if the ROM basis is fixed during the response calculation. This is because it is not easy to confirm that the fixed basis obtained during the offline stage can suitably describe the unknown response fields [20], [21], [22]. Although adaptive online-based methods may be alternatives, these methods are based on the global computational process with large computational costs [20], [21], [22].
To better reflect the information of high-fidelity data to low-fidelity data while preventing huge computational costs, we suggest a strategy for a composite NN using an efficient online adaptive procedure. With the proposed strategy, the role of the ROM results corresponds to the low-fidelity data in the composite NN. However, unlike existing approaches for a composite NN, an additional basis is adaptively computed to update the low-fidelity data during the online stage. Here, the purpose of this basis is to directly reduce the errors that may significantly interfere with connecting the multi-fidelity data in the composite NN. Moreover, to avoid large calculation costs during the process of updating low-fidelity data, the additional basis is algebraically determined by optimizing some of the errors for the low-fidelity data, but not all. As a result, low-fidelity data can more accurately contain information for high-fidelity data than the existing approaches [8], [9], and thus the composite NN can more accurately identify the relationship between the low- and high-fidelity data within the whole domain. In addition, because the inverse operation required for the additional basis is calculated within the local region, the computational costs can be much lower than the entire model case. In other words, the purpose of employing the ROM method in a composite NN can be still maintained. Consequently, through the proposed strategy, the reliability of the composite NN can be dramatically improved through an efficient numerical procedure.
The remainder of the present manuscript is organized as follows: the multi-fidelity approach and its composite NN are reviewed in Section 2. Section 3 describes the ROM method used to generate low-fidelity data. The proposed strategy for the composite NN is introduced in Section 4. Section 5 shows the performance of the suggested method through various numerical examples. Finally, conclusions are given in Section 6.
Section snippets
Multi-fidelity modeling and its composite neural network (NN)
In multi-fidelity modeling, the high-fidelity data can be expressed using low-fidelity data and a correlation function as follows [6], [7], [23]: where denotes the input or domain, and denotes the correlation function of the multi-fidelity method. Moreover, indicates the data or responses to the problem of interest. Subscripts and represent the low- and high-fidelity part, respectively.
In Eq. (1), the input and data for the low- and high-fidelity part can be expressed as
Reduced-order modeling (ROM) for preparing low-fidelity data
In this section, we briefly describe the projection-based ROM. In addition, we review the POD method to illustrate how to obtain the low-fidelity data in the conventional composite NN approaches [8], [9].
Composite neural networks with online adaptive basis technique
With the online adaptive basis technique, the fixed basis is redefined when the ROM response does not converge during the response calculation. In particular, adding another basis to the fixed basis can extend the ROM subspace, and thus the ROM results can be closer to those of the full model [20], [21], [22]. Hence, low-fidelity data using the extended ROM results can more easily reflect the trends of the high-fidelity data than before. The expanded ROM basis can be written
Numerical examples
In this section, we use various numerical examples to evaluate the prediction field with the proposed strategy (Algorithm 3). Here, three examples are covered: a Poisson equation, a coupled viscous Burgers’ equation, and a commercial gearbox-housing model in the structural dynamics [30], [31], [32]. Note that if the low-fidelity data in the composite NN are obtained from Algorithm 1, it is then called a conventional method for convenience [8], [9]. In the numerical examples, is selected
Conclusion
This study introduced a data strategy for effectively increasing the accuracy of the multi-fidelity model implemented by the composite NN. The proposed strategy includes an online adaptive technique used to ameliorate the low-fidelity data for the composite NN when the ROM method is invoked to acquire a large number of low-fidelity data in an efficient manner. With the proposed strategy, the low-fidelity data are updated based on the terms of the additional basis obtained through the efficient
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This research was supported by the Basic Science Research Programs of the National Research Foundation of Korea funded by the Ministry of Science, South Korea (NRF-2021R1A2C4087079).
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