**1. Introduction**

There is currently a surge in the application of machine learning algorithms in various fields of materials mechanics. In general, scientific and industrial research groups focus on the identification and utilization of one or more relationships along the process– structure–property–performance (p-s-p-p) chain [1]. In this domain, the application of machine learning techniques can be a key enabler for accelerated identification, characterization, understanding and optimization of processes, materials and parameters [2]. For instance, unique material descriptors can be qualified and quantified for material

**Citation:** Bock, F.E.; Keller, S.; Huber, N.; Klusemann, B. Hybrid Modelling by Machine Learning Corrections of Analytical Model Predictions towards High-Fidelity Simulation Solutions. *Materials* **2021**, *14*, 1883. https:// doi.org/10.3390/ma14081883

Academic Editor: Moncef L. Nehdi

Received: 11 March 2021 Accepted: 4 April 2021 Published: 10 April 2021

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characterization [3–5]. Optimization and rapid design of novel manufacturing methods and involved materials [6,7] can be achieved, and inaccurate measurement techniques can be corrected [8]. The generation of knowledge and understanding to enable improved predictions of mechanical properties and performances, among others, can be acquired on the basis of experimental and/or numerical data in combination with machine learning models [9,10]. Furthermore, the integration of well-established physical laws into data-driven machine-learning models can be very beneficial to perform highly accurate predictions and inferences of involved phenomena [11,12]. However, besides these physics-informed machine learning methodologies, Chinesta et al. [13] introduced a hybrid modelling approach, where an efficient physics-based model shows some prediction errors that are corrected by a subsequent data-driven model to ultimately reach the anticipated solution.

Deployment of only either data-driven predictive models or calibrated physics-based models is accompanied with respective disadvantages based on each approach. Calibration of physics-based models can be difficult, expensive and time-costly even for domain experts, as it can be challenging or even impossible for physical quantities of interest to be accessible through experimental measurements. It is almost unattainable to represent the reality via such models only through data assimilation [14]. For purely data-driven approaches, the relevant relationships between input and output variables are required to be satisfactorily represented in the data set, as there is an absence of internal physics-related variables [15]. This creates the demand for a comprehensive database for the learning algorithm to represent those relationships. For problems that are still largely unknown, this can be a suitable approach; however, when some relations are already known, it is inefficient to create the need of a big-data-set for ensuring it represents all relevant aspects of the underlying physical laws that are required to be learned "from scratch" by the machine-learning algorithm [16]. In a study by Liu et al. (2020) [17], a data-driven surrogate model to predict the plane-strain stress intensity factor at the crack tip during fracture toughness tests is built with an adaptability and efficiency that is comparable to an analytical or empirical solution within their physical problem domains. In [17], high-fidelity numerical simulations are used to create the data-base for correlation of dimensionless inputs and outputs. However, due to the purely data driven approach, a vast number of computationally expensive simulation solutions are required for sufficient training of the surrogate model, which could create challenges for accuracy and generalization when switching to an experimental data source for training. Purely data-driven approaches can be beneficial for those problems where few relationships are identified, as they can help to detect hidden relationships in data; however, when established physical-laws apply and available data is scarse or biased, the utilization of physically-related data-driven approaches can be countervailing and utile [18,19].

Consequently, studies are focused on the aim to represent physical problems and their associated behaviour through physics-based models as well as on the pursuit to account for the deviation between those models and the reality via data-driven corrections. González et al. (2019) [20] performed corrections for hyperelastic models based on data-driven machine learning, whereas Ibáñez et al. (2018) [21] implemented a hybrid approach consisting of constitutive modelling and data-driven machine learning correction of plasticity models. In a manufacturing application example for metal forming production, Havinga et al. (2020) [22] performed real-time predictions via a hybrid modelling approach that contains physics-based simulations those predictive deviations to the real process are eliminated via an additional corrective model. Overall, the specific employment of machine learning models alongside governing physics-based relationships allows for highly valid predictions within materials mechanics and its related fields.

Generally, physics-based models might show prediction errors but as these deviations are systematic and not owed to noise, they can be accounted for separately. In combination, physics-based models and deviation models can be used to correctly predict a real system's behaviour. The advantages of using a calibrated model based on well-established physics, even when it shows deviations to reality, are that the compensating corrective model

applied for achieving high prediction accuracy requires fewer samples and less complexity to approximate the deviation, since it is usually considerably less non-linear than the problem itself. This opens up the possibility to easily correct a physics-based model with a relatively simple correction model towards true/desired data points to assure an adequate representation of the behaviour by the system of interest [13]. Chupakhin et al. [8] introduced a corrective artificial neural network (ANN) for the hole drilling method, where residual stresses are determined based on measurements of elastic material behaviour, which are corrected towards the solution of a plasticity-including finite element (FE) model by an ANN. Thus, as opposed to correcting numerical models by empirical observations, in this case, biased experimental measurements can be successfully corrected through an ANN driven by physics-based numerical data.

The objective of this study is to build a hybrid model, consisting of a physics-based model and a data-driven corrective model, with low prediction errors even when training data is scarce. A semi-analytical model, originally proposed by Hu et al. [23], is employed as low-fidelity physics-based model, including a number of simplifications and a subsequent ANN is used to correct this solution towards a true reference solution provided by an FE model considered as high-fidelity. As example use-case, laser shock peening (LSP)-induced residual stress distributions over the specimen depth in aluminium alloy AA2024 are considered. In particular, since the representation of the relationships between residual stress distributions in dependence of LSP-generated pressure pulses over time is severely simplified in its semi-analytical model solution, we aim for the complementary corrective approach. Ultimately, high-fidelity approximation of the desired system behaviour is achieved by combining semi-analytical and ANN-correction models, which are both computationally efficient. In addition, when the data used for training, validation and testing is reduced, the predictions obtained via this hybrid modelling approach exhibit less errors than a purely data driven model. We propose a hybrid process model consisting of data-driven correction-learning of an LSP process model, which also shows good generalization ability, even when the parameter space of the training region is expanded and the available data becomes scarce.
