Acta Polytechnica CTU Proceedings https://doi.org/10.14311/APP.2022.36.0099 Acta Polytechnica CTU Proceedings 36:99–108, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague ARTIFICIAL INTELLIGENCE - FINITE ELEMENT METHOD - HYBRIDS FOR EFFICIENT NONLINEAR ANALYSIS OF CONCRETE STRUCTURES Michael A. Krausa, b, ∗, Rafael Bischofa, Walter Kaufmanna, b, Karel Thomaa a ETH Zürich, Institute of Structural Engineering (IBK), Chair for Concrete Structures and Bridge Design, Stefano-Franscini-Platz 5, CH-8093 Zürich, Switzerland b ETH Zürich, Center for Augmented Computational Design in Architecture, Engineering and Construction, Design++ Initiative and Immersive Design Lab, Stefano-Franscini-Platz 5, CH-8093 Zürich, Switzerland ∗ corresponding author: kraus@ibk.baug.ethz.ch Abstract. Realistic structural analyses and optimisations using the non-linear finite element method are possible today yet suffer from being very time-consuming, particularly in case of reinforced concrete plates and shells. Hence such investigations are currently dismissed in the vast majority of cases in practice. The "Artificial Intelligence - Finite Element - Hybrids" project addresses the current unsatisfactory situation with an approach that combines non-linear finite element models for reinforced concrete shells with scientific machine learning algorithms to create hybrid AI-FEM models. The AI-based surrogate material model provides the material stiffness as well as the stress tensor for given concrete design parameters and the strain tensor. This paper reports on the current status of the project and findings of the calibration of the AI-based reinforced concrete material model. We successfully calibrated and evaluated k-nearest-neighbour, LGBM and ResNet algorithms and report their predictive capabilities. Finally, some light is shed on the future work of integrating the AI surrogate material models back into the finite element method in the course of the numerical analysis of reinforced concrete structures. Keywords: Concrete material model, machine and deep learning, nonlinear finite element method, surrogate modeling, uncertainty quantification. 1. Introduction Digital design and manufacturing methods, such as those to be developed at the new Immersive Design Lab (IDL) at ETH Zurich, offer great potential for sig- nificantly more efficient and sustainable construction. In order to ensure the structural safety, economic efficiency and sustainability of complex structures, reliable and powerful models for automatic analy- sis, optimisation and design are essential. However, such models are largely lacking to date. This is espe- cially true for concrete structures, as their behaviour is highly non-linear. Realistic structural analyses and optimisations using the non-linear finite element method (FEM) are possible today, yet being very time-consuming even for the case of extraordinary computational capacities. For reasons of temporal and monetary efficiency, established traditional and in many cases excessively conservative design methods without structural optimisation are still used in the vast majority of projects in practice today. The in- creased public awareness of and demand for a sustain- able built environment however urges civil engineers to make use of structural efficiency to the maximum level possible. To that end, a two-phased research program [1, 2] is proposed to address this unsatisfactory situation. In the first phase, non-linear FEM for reinforced con- crete plates and slabs developed at ETH Zurich [3–8] are combined with scientific machine learning (SciML) algorithms to create hybrid AI-FEM models, which are expected to be much more efficient compared to established analysis methods both in terms of the com- puting power required and the reliability of predicting the load-bearing behaviour. In the second phase of this project, the AI-FEM-Hybrids will be used within a novel Generative Design process for accelerated yet realistic conceptual design of bridges [1, 2]. Within a FE analysis, a material model has to pro- vide on the one hand the material stiffness matrix and on the other hand the current stress state. In the novel hybrid artificial intelligence finite element (AI- FEM) model implementation for material-nonlinear reinforced concrete slabs and plates elements, the mathematical description of the material model is per- formed using scientific Machine and Deep Learning algorithms, which act as functional approximators of the stress-strain relationship and the stiffness-strain relationship based on numerical simulations. The ba- sic data set for training, validation and testing of the AI algorithms is generated by extensive simula- tions of strain states for different reinforced concrete configurations with the material model for reinforced 99 https://doi.org/10.14311/APP.2022.36.0099 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en M. A. Kraus, R. Bischof, W. Kaufmann, K. Thoma Acta Polytechnica CTU Proceedings concrete as implemented in the mentioned USERMAT in ANSYS MECHANICAL APDL on the basis of the cracked membrane model (CMM) in combination with a Reissner-Mindlin layer model. This paper describes the process of data generation and calibration of the AI-based reinforced concrete material model for differ- ent Machine and Deep Learning algorithms for stress and stiffness tensor prediction. This paper is organised as follows: we first pro- vide background on the materials and methods from computational mechanics and scientific machine learn- ing used in this paper in sec. 2. Sec. 3 reports and presents selected numerical results of the SciML-based substitute concrete material models w.r.t. their ap- proximation accuracy and statistical qualities. Sec. 4 presents a discussion of current findings and sec. 5 sheds light on the future steps of incorporating the SciML-based substitute concrete material models into the FEM software ANSYS. 2. Materials and Methods This section reports on materials and methods for database generation, development of machine and deep learning (ML/DL) models, and the evaluation criteria for the performance of the developed models as a surrogate for a FEM. 2.1. Concept This research project explores the potential of us- ing ML/DL surrogate models of mechanically con- sistent nonlinear finite element material models for reinforced concrete plates and shells, developed at ETH Zurich, as a data-driven yet physics-informed AI method within a FEM workflow. The project is divided into two parts. The AI-FEM-Hybrids of phase one provide predic- tions of the material stiffness tensor as well as the stress tensor for a current strain state, where the SciML algorithms act as functional approximators of the stress-strain relationship and the stiffness-strain relationship, which was found to be applicable to a number of materials [9]. The basic data set for train- ing, validation and testing of the AI algorithms is generated by an extensive simulation of strain-stress- stiffness states for different reinforced concrete config- urations with the material model for reinforced con- crete as implemented in the mentioned USERMAT in ANSYS MECHANICAL APDL on the basis of the cracked membrane model (CMM) in combination with a layer model based on Reissner-Mindlin plate kinematics. The training, validation and testing of the SciML algorithms for stress and stiffness tensor pre- diction are presented and discussed in the following, where special consideration is given to model uncer- tainty quantification to allow for a future Eurocode- compliant use within the semi-probabilistic design philosophy. With the completion of the first phase, a validated software demonstrator of AI-FEM-Hybrids for the efficient non-linear analysis and design of rein- forced concrete structures is available. In the second project phase, the AI-FEM-Hybrids approach will be extended to structural optimisation of concrete bridges. The pilot project in this phase focuses on concrete bridges with a geometry defined by a few parameters. While the case study is specific, the de- veloped optimisation methodology is kept as general as possible to allow its application beyond parametric concrete bridge structures. At the end of the research project, a validated software demonstrator of the AI- FE-Hybrids will be available for the efficient non-linear analysis, design and optimisation of reinforced con- crete structures. In the future, this will then allow for (i) realistic structural analyses taking into account non-linearities and (ii) structural optimisation to be carried out much more efficiently. The outcomes of this research are potentially of high interest to indus- try and engineering practice, as structural concrete is the most widely used construction material world- wide and incorporation of these ideas allows for more economic, yet sustainable and reliable design. To achieve the objectives for phase one of the re- search project, the following sequence of steps was conducted: (1.) define relevant feature X and target variables Y for a reinforced concrete material model (cf. Tab. 1) (2.) create the database {X; Y} of FEM simulations (3.) calibrate different ML and DL algorithms given the database (4.) assess different metrics for accuracy and predic- tive capabilities of the ML resp. DL models (5.) implement a selection of ML resp. DL models for implementation in ANSYS (6.) verify efficiency and accuracy at industry scale via computation of example problems from reinforced concrete design 2.2. FEM Material Model Data Generation The dataset generation within the AI-FEM-Hybrid project was performed by nonlinear finite element analysis of a reinforced concrete structure utilising the CMM-USERMAT [3], a mechanically consistent material model for reinforced concrete implemented in ANSYS Mechanical APDL, cf. Figs. 1 and 2. Com- bined with a layer element (Shell181), the non-linear FEM analysis of reinforced concrete structures as girders or shells are possible. As presented in [3–5], an excellent agreement between experimental results and FEM analysis utilising the CMM-USERMAT is reported. In order to provide the necessary dataset for the training of the machine learning algorithms, a dummy girder bridge (cf. Figs. 1 and 2) was arbitrary loaded incrementally until failure, which enables the export of 100 vol. 36/2022 AI-FEM-Hybrids for Nonlinear Concrete Materials Aa A a 45.00 m A a 5.00 m A a 2.25 m A a 250 mm A a 250 mm Aa A a 400 mm A a 1'200 mm A a 495 mm A a 480 mm A a 437 mm q N q N A B C D b 0 L/2 L/2 q N y xz X Z Y - 8'600[kN] [kNm] V z M y [kN] [kNm] N x T x A z B z L/2 L/2 C z D z (a) (b) x z y w (x = L/2, S 1 ) w (x = L/4, S 1 ) w (x = L/2, S 2 ) w (x = L/4, S 2 ) q N = 1'911 kN/m z y x q 1 = 200 kN/m 11'250 -11'250 -4'500 4'500 -50'625 Aa A a = A a = A a = A a = A a = Aa A a = A a = A a = A a = A a = Z X Yq1·b0/2 q 1 Parameter: L b 0 h 0 t inf t sup Parameter: t w t end a X a Y a Z Figure 1. Dummy girder bridge used within ANSYS to generate the database. Features Xi Targets Yi Reinforcement Layer (ReLa) Normal Stress in X-Direction σx CMM Usermat Model (CMM) Normal Stress in Y-Direction σy Reinforcement Area as Shear Stress in XY-Direction τxy Reinforcement Diameter ds Stiffness Component K11 Effective Reinforcement Ratio for TCM Model ρs,ef f Stiffness Component K12 Reinforcement Yield Stress fy Stiffness Component K13 Reinforcement Ultimate Stress fu Stiffness Component K21 Reinforcement Ultimate Strain ϵsu Stiffness Component K22 Reinforcement Angle θs Stiffness Component K23 Concrete Compressive Strength fcc Stiffness Component K31 Concrete Ultimate Strain ϵcu Stiffness Component K32 Normal Strain in X-Direction ϵx Stiffness Component K33 Normal Strain in Y-Direction ϵy - Shear Strain in XY-Direction ϵxy - Table 1. List of Features and Targets for data generation and calibration of AI-FEM-Hybrids. Note that ϵxy is the tensorial component of the shear strain γxy , i.e., ϵxy =γxy /2. all necessary features (input parameters X: strain ten- sor, reinforcement properties, material constants, etc. acc. to Tab. 1) and results (stress and stiffness tensors acc. to Tab. 1) in every integration point (Gauss- points) for all converged load steps. This approach enables the collection of a considerable number of data points within one non-linear FEM computations, as in almost every integration point an individual data set exists. For simplicity, a dummy girder bridge was chosen to minimise the number of parameters, e.g. the diameter of the rebars or the material constants. As soon as the general framework of the AI-FEM-Hybrids is estab- lished (proof of concept), the results of any dummy experiment and the results of any FEM analysis us- ing the CMM-USERMAT can be used to train the machine learning algorithms. In total the data set consists of: • NS,wor = 10, 636, 800 samples of concrete elements without reinforcement (configuration "CF 0") • NS,wr = 443, 200 samples of concrete elements with reinforcement (configuration "CF 1") 2.3. Data Pre-Processing The data generated in the previous section is used to develop the ML resp. DL models. The dataset consists of over 11 million data points with 14 feature and 12 target variables, cf. Tab. 1. Some dimensions of the feature dataset are categorical (reinforcement layer, CMM-USERMAT model), while the remaining features are all of numerical nature. For the categori- cal features we use one-hot-encoding. All dimensions of the target dataset are of numerical nature. None of the attributes has any missing data since the data is generated via FEM analyses. The range of values 101 M. A. Kraus, R. Bischof, W. Kaufmann, K. Thoma Acta Polytechnica CTU Proceedings X Z Y Integration Poinst Shell Element SHELL181 L L b 0 h 0 a Y a Z a X a X Z X Y Y X Z (a) (b) (c) Figure 2. Detail of the CMM finite element within ANSYS to generate the database. of the features as well as the targets are extremely different, hence we employ normalization of the data. For each numerical variable vi we deduce its mean m and divide it by its standard deviation s (yielding its z-score) using: zi = vi − µi σi (1) For sake of brevity of this paper, we omit report- ing further statistical properties or histograms of the dataset, which would usually be delivered in the ex- ploratory data analysis step. 2.4. Machine Learning Model Development In this study, three ML resp. DL algorithms for re- gression are investigated for predictive capabilities for stress and stiffness target tensors given the strain ten- sor together with concrete and reinforcement features: • k-nearest-neighbours (kNN) • LightGBM • Artificial Neural Networks: Residual Neural Net- works (ResNet) While kNN is a strictly data-driven approach and hence needs to store some of the simulation data instances, LightGBM as well as ResNet are surrogate functions without the need to store the training data. Due to reasons of brevity of this paper, only selected results and implementation details can be reported. The ML models are developed using sklearn, Keras, and XGBoost (extreme gradient boosting) python libraries and run on the ETH Euler cluster1. 2.4.1. kNN The k-nearest-neighbours algorithm is a decision method originally developed for classification tasks [10]. At inference time, it compares the previously un- seen samples to all instances in a database and assigns classes by means of majority voting of the k nearest neighbours, where k and the distance metric are hy- perparameters. The algorithm was later extended to support regression tasks [11] by interpolating between the values of the k nearest neighbours (e.g. through inverse distance weighted average) to obtain values for new, unseen samples. 2.4.2. LightGBM LightGBM is a gradient boosting decision tree frame- work [12] that has found application in many different data mining, classification and regression tasks. The LightGBM algorithm variant used in this paper inte- grates a number of regression trees to approximate the dataset. It contains many of the advantages of other common gradient boosting decision tree algorithms, such as sparse optimization, parallel training, regu- larization, bagging, and early stopping. However, it grows trees leaf-wise by greedily choosing the leaf that will lead to the largest improvement. This tree-growth method, in combination with a histogram-based mem- ory and computation optimization, make LightGBM considerably more computationally efficient than other frameworks. 2.4.3. Artificial Neural Networks Fully-connected feed-forward neural networks (FFNN) consist of one or more layers, where each node is con- nected to every node in the following layer [13]. De- spite being very straight forward architectures, FFNNs are universal function approximators [14] and have the advantage of being easy to implement and efficient to train. On the other hand, the lack of implicit bias limits their expressiveness and capability to gener- alise. Furthermore, they are prone to pathologies like vanishing gradients [15]. We therefore extend the FFNN by adding skip con- nections [15] as well as batch normalisation [16] and dropout [17] , cf. Fig. 3. These additions allow us to build deeper architectures, which were shown to generalise better in practice [18]. We will hereafter be referring to this model by ResNet. 2.4.4. Hyperparameter Tuning We employ Bayesian Optimization in order to avoid running extensive grid-search over the entire hyper- parameter space [19][20]. At first, 25 random points 1https://scicomp.ethz.ch/wiki/Euler 102 vol. 36/2022 AI-FEM-Hybrids for Nonlinear Concrete Materials FU L LY - C O N N E C T E D B A T C H N O R M A L IS A T IO N D R O PO U T FU L LY - C O N N E C T E D + Figure 3. ResNet: Neural Network with skip connec- tions on the hyperparameter space are evaluated. Gaussian Processes then serve as prior distribution in order to approximate the unknown function and a posterior distribution is maintained as 75 more observations are made, where Expected Improvement (EI) is used as exploration strategy [21]. The hyperparameters together with optimization intervals are reported in Tabs. 2, 3, and 4. Hyperparameter Range Number of neighbours k [1, 10] Power of Minkowski metric p [1, 3] Table 2. Hyperparameters for training settings to- gether with ranges as used for Bayesian Optimisation of kNN Hyperparameter Range Log-scaling Learning Rate [10−5, 10−1] yes Max Depth [3, 50] no Min Child Weight [0, 10] no Number Estimators [30, 300] no Number Leaves [10, 100] no Min Child Samples [10, 30] no Table 3. Hyperparameters for training settings to- gether with ranges as used for Bayesian Optimisation of LightGBM Hyperparameter Range Log-scaling Learning Rate [10−5, 10−1] yes Depth [3, 50] no Width [0, 10] no Activation {ReLU, no LeakyReLU} Table 4. Hyperparameters for training settings together with ranges for Bayesian Optimisation of ResNets For the optimization procedure, we split the dataset into a training, validation and test set at ratios of (50; 25; 25)% of the whole dataset. The regressor is trained on the training set using the hyperparameters chosen by the Bayesian Optimization algorithm. Once the model loss converged, it is evaluated on the validation data set and the result serves as feedback for the Bayesian Optimization step to refine its posterior and select new hyperparameters. Finally, the test set is used to estimate the model’s capability of generalizing. 2.4.5. Model Quantitative Performance Metrics and AI Surrogate Model Selection Model selection refers to the process of seeking for a model in a set of candidate models, which delivers the best balance between model fit and complexity [22]. For regression modeling, the quantitative performance metrics used with this research are the mean absolute error (MAE), root mean squared error (RMSE), R- squared (R2) value as well as the data variance V ar (which is the square of the standard deviation (SD)), cf. Tab. 5. The MAE shows the average difference between the actual values of the output variable in the original data vs. the predicted output values via the ML models. The lower the MAE, the more precise the performance of the model is in predicting future occurrences of the output. The RMSE is defined as the standard deviation of the response variable. Val- ues of R2 range from 0 to 1, where 1 is a perfect fit, and 0 means there is no gain by using the model over using fixed background response rates. It esti- mates the proportion of the variation in the response around the mean that can be attributed to terms in the model rather than to random error. For compar- ing different models, we provide Taylor diagrams [23] and additionally report R2, RMSE and MAE values. Taylor diagrams are used to quantify the degree of correspondence between the modeled and observed data using the Pearson correlation coefficient, RMSE, and standard deviation. A model with the highest R2 and the lowest RMSE and MAE is preferred. The details of how MAE, RMSE, and R2 are calculated are shown in Tab. 5. MAE 1 N ∑N i=1 |ŷi − yi| RMSE √ 1 N ∑N i=1(ŷi − yi)2 R2 1 − ∑ N i=1 (ŷi−yi)2∑ N i=1 (ŷi−ȳ)2 , ȳ = 1 N ∑N i=1 |yi| Table 5. Regression performance evaluation metrics. where N is the sample size, ŷi is the predicted and yi is the true target value, SSE is the sum of the square of error and SST is the total sum of squares. 3. Results This section describes the results obtained with kNN, LightGBM, and ResNet and evaluates their perfor- mance as a surrogate for the FEM data generated and used within this study, cf. Sec. 2.3. All three ML models were trained, validated and tested using the data from the total of 12 million 103 M. A. Kraus, R. Bischof, W. Kaufmann, K. Thoma Acta Polytechnica CTU Proceedings data points. Comparing the R2 values, all models have high values of around 0.98. This indicates that the ML algorithms are able to describe over 98% of variations in the data, and are highly predictive of the output based on the feature variables used in the study. Regarding the MAE and RMSE, the kNN model performs one order of magnitude better than the other two ML models for the CF 0 configuration (without reinforcement) at a level of 0.3% resp. 0.05%, while for the CF 1 configuration (with reinforcement) the ResNets has the lowest RMSE value of 2.4% and the LGBM has the lowest MAE value of 1.8%. The results suggest that all ML models developed for predicting the stress (all in unit: MPa) and stiffness (all in unit: MNm2) tensor components are promising, with ResNet being the most predictive model amongst the three. The model performances, described by the mean absolute error MAE, root mean squared error RMSE and R-squared have converged to a stable minimum during training. 3.1. kNN Model Results Fig. 4 provides selected results for the kNN algorithm in the two configurations "CF 0" (without reinforce- ment) and "CF 1" (with reinforcement). It is important to note, that the size of the trained kNN model is of around 1.13 GB, as it needs to store all training samples in order to make new predictions. 3.2. LGBM Model Results Fig. 5 provides selected results for the LGBM al- gorithm in the two configurations "CF 0" (without reinforcement) and "CF 1" (with reinforcement). 3.3. ResNet Model Results Fig. 6 provides selected results for the ResNet al- gorithm in the two configurations "CF 0" (without reinforcement) and "CF 1" (with reinforcement). 3.4. Overall Model Comparison Results The performances of different AI models according to the mentioned criteria are reported in Tab. 6 for configuration "CF 0" (without reinforcement) and in Tab. 7 for configuration "CF 1" (with reinforcement). Model RMSE MAE R2 V ar KNN 0.00045 0.00310 0.99955 0.00044 LGBM 0.00530 0.01837 0.99440 0.00560 ResNet 0.00212 0.01576 0.99760 0.00238 Table 6. Performance of different models on test set without reinforcement For model selection purposes, we provide selected results for the Taylor diagrams in the two configu- rations "CF 0" (without reinforcement) and "CF 1" (with reinforcement) in Fig. 7. Model RMSE MAE R2 V ar KNN 0.06678 0.01579 0.93789 0.06678 LGBM 0.02783 0.02329 0.97390 0.02783 ResNet 0.02417 0.03301 0.97715 0.02416 Table 7. Performance of different models on test set with reinforcement 4. Discussion The choice for the three investigated AI algorithms was done under three aspects: modeling bias, com- puting efficiency and prediction precision. The kNN and LGBM belong to the family of non-parametric models, whereas ResNet is a parametric model. Non- parametric ML algorithms promise greater perfor- mance at the cost of higher data requirements and training times together with a risk of overfitting. Para- metric ML/DL models on the other hand come with high modeling bias towards the functional relation, yet neural networks provide a great enough expressiveness for modeling the database. Concerning computational efficiency in the prediction stage (which is called a lot of times during FEM analysis in each iteration step) kNN requires to store the dataset and call it at predic- tion time, leading to impractical lengthy procedures. LGBM and ResNet however are much more time effi- cient and hence to be preferred for an implementation into the FEM analysis. The AI models perform better in CF 0 (without reinforcement) than in CF 1 accross the different qual- ity measures. Looking at the R2 values, all models have high values of approx. 0.99 (CF 0) resp. 0.97 (CF 1). This means, that the ML algorithms are able to describe over 97% of variations in the data, and hence are highly predictive for the outputs. For CF 0 RMSE is around 0.35% and MAE is around 1.5%, while for CF 1 RMSE as well as MAE lie about 3% and are thus one magnitude bigger than for CF 0. It is noteworthy that on average, every target is pre- dicted well, while the prediction’s standard deviation is dependent on the respective target quantity. Given all model results, especially the stiffness tensor terms K22 and K23 show severe model predictive deviations. For model selection, inspection of the Taylor dia- grams in Fig. 7 suggests that all described AI models possess great approximation quality with slight dif- ferences, where kNN outperforms ResNet and LGBM (in the order of decreasing approximation quality). The results in summary suggest that the AI models developed for predicting the stress and stiffness re- sponses of the CMM USER-MAT are promising and may be used within a FEM. Given the outlines in the beginning of this section towards computational effi- ciency, the ResNet and LGBM are chosen for further implementation into the FEM analysis process. Another interesting thought is to use the calibra- tion of AI algorithms on such numerical datasets for non-intrusive verification purposes of the implementa- 104 vol. 36/2022 AI-FEM-Hybrids for Nonlinear Concrete Materials (a) (b) (c) (d) Figure 4. kNN results for: (a) stress component σx in CF 0, (b) stiffness component K11 in CF 0, (c) stress component σx in CF 1, and (d) stiffness component K11 in CF 1. (a) (b) (c) (d) Figure 5. LGBM results for: (a) stress component σx in CF 0, (b) stiffness component K11 in CF 0, (c) stress component σx in CF 1, and (d) stiffness component K11 in CF 1. 105 M. A. Kraus, R. Bischof, W. Kaufmann, K. Thoma Acta Polytechnica CTU Proceedings (a) (b) (c) (d) Figure 6. ResNet results for: (a) stress component σx in CF 0, (b) stiffness component K11 in CF 0, (c) stress component σx in CF 1, and (d) stiffness component K11 in CF 1. (a) (b) (c) (d) Figure 7. Taylor diagrams for: (a) stress component σx in CF 0, (b) stiffness component K11 in CF 0, (c) stress component σx in CF 1, and (d) stiffness component K11 in CF 1. 106 vol. 36/2022 AI-FEM-Hybrids for Nonlinear Concrete Materials tion of the ground truth data generating mechanism. Especially the data points in far distance to the diag- onal line in Figs. 4, 5,6 give rise to outlier detection methods. It is currently under investigation whether the identified outliers are truly faulty data (i.e. the generating FEM material model is not correct) or the AI algorithm with its current state is not able to approximate the respective quantities well. 5. Conclusions and Outlook This paper presented intermediate results of the first phase of a two-stage research project conducted cur- rently at ETH Zürich. It reports on calibrating AI- based surrogates for a highly nonlinear reinforced concrete plate and shell model upon the CMM. All three AI algorithms have proven to furnish as valid candidates for surrogate models to predict the stress and stiffness tensors used within a FEM. The data- driven kNN algorithm performed slightly better that the LGBM and ResNet model for prediction standard deviation and RMSE, whereas on average all models show almost full correlation. Future research is concerned with further improving predictive capabilities by training an ensemble model using the kNN, ResNet and LGBM models. Finally, the non-intrusive verification analysis of the ANSYS USER-MAT by AI surrogate modelling will shed light on generalisation of this idea of physics informed ML [24]. Acknowledgements The authors would like to acknowledge the facilities at ETH Zürich as well as the Design++ resp. Immersive Design Lab for providing computational resources for the Machine and Deep Learning parts of this research project. This research was funded through the ETH Foundation grant No. 2020-HS-388 (provided by Kollbrunner/Rodio). References [1] M. Kraus, W. Kaufmann, K. Thoma. (Research Project): KI-basierte Analyse und Optimierung von Betonstrukturen, 2020. [2] M. A. Kraus, M. Drass, B. Hörsch, et al. Künstliche Intelligenz - multiskalen und cross-domänen Synergien von Raumfahrt und Bauwesen. In K. Bergmeister, F. Fingerloos, J.-D. Wörner (eds.), Beton-Kalender. John Wiley & Sons, schwerpunk edn., 2021. [3] K. Thoma, P. Roos, M. Weber. Finite-Elemente- Analyse von Stahlbetonbauteilen im ebenen Spannungszustand: Scheiben- und Plattenberechnungen auf der Grundlage des gerissenen Scheibenmodells. Beton- und Stahlbetonbau 109(4):275–283, 2014. https://doi.org/10.1002/best.201300087. [4] K. Thoma, P. Roos, G. Borkowski. Finite Elemente Analyse von Stahlbetonplatten: Versuchsnachrechnungen von Platten mithilfe des gerissenen Scheibenmodells. Beton- und Stahlbetonbau 109(12):895–904, 2014. https://doi.org/10.1002/best.201400047. [5] K. Thoma. Finite element analysis of experimentally tested RC and PC beams using the cracked membrane model. Engineering Structures 167:592–607, 2018. https://doi.org/10.1016/j.engstruct.2018.04.010. [6] J. Kollegger. Algorithmus zur Bemessung von Flächentragwerkelementen unter Normalkraft- und Momentenbeanspruchung. Beton- und Stahlbetonbau 86(5):114–119, 1991. https://doi.org/10.1002/best.199100230. [7] P. Marti, M. Alvarez, W. Kaufmann, V. Sigrist. Tension Chord Model for Structural Concrete. Structural Engineering International: Journal of the International Association for Bridge and Structural Engineering (IABSE) 8(4):287–298, 1998. https://doi.org/10.2749/101686698780488875. [8] W. Kaufmann, P. Marti. Structural Concrete: Cracked Membrane Model. Journal of Structural Engineering 124(12):1467–1475, 1998. https://doi.org/10.1061/(asce)0733-9445 (1998)124:12(1467). [9] M. A. Kraus. Machine Learning Techniques for the Material Parameter Identification of Laminated Glass in the Intact and Post-Fracture State. Ph.D. thesis, Universität der Bundeswehr München, 2019. [10] E. Fix, J. L. Hodges. Discriminatory analysis - nonparametric discrimination: Consistency properties. International Statistical Review 57:238, 1989. [11] N. S. Altman. An introduction to kernel and nearest-neighbor nonparametric regression. The American Statistician 46(3):175–185, 1992. https://www.tandfonline.com/doi/pdf/10.1080/ 00031305.1992.10475879 https://doi.org/10.1080/00031305.1992.10475879. [12] G. Ke, Q. Meng, T. Finley, et al. Lightgbm: A highly efficient gradient boosting decision tree. In I. Guyon, U. V. Luxburg, S. Bengio, et al. (eds.), Advances in Neural Information Processing Systems, vol. 30. Curran Associates, Inc., 2017. [13] Y. LeCun, Y. Bengio, G. Hinton. Deep learning. nature 521(7553):436–444, 2015. [14] K. Hornik, M. Stinchcombe, H. White. Universal approximation of an unknown mapping and its derivatives using multilayer feedforward networks. Neural Networks 3(5):551–560, 1990. https://doi.org/10.1016/0893-6080(90)90005-6. [15] A. Emin Orhan, X. Pitkow. Skip Connections Eliminate Singularities. arXiv e-prints arXiv:1701.09175, 2017. 1701.09175. [16] S. Ioffe, C. Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. arXiv e-prints arXiv:1502.03167, 2015. 1502.03167. [17] N. Srivastava, G. Hinton, A. Krizhevsky, et al. Dropout: A simple way to prevent neural networks from overfitting. Journal of Machine Learning Research 15(56):1929–1958, 2014. [18] M. Telgarsky. Benefits of depth in neural networks. arXiv e-prints arXiv:1602.04485, 2016. 1602.04485. 107 https://doi.org/10.1002/best.201300087 https://doi.org/10.1002/best.201400047 https://doi.org/10.1016/j.engstruct.2018.04.010 https://doi.org/10.1002/best.199100230 https://doi.org/10.2749/101686698780488875 https://doi.org/10.1061/(asce)0733-9445 (1998)124:12(1467) https://doi.org/10.1061/(asce)0733-9445 (1998)124:12(1467) https://www.tandfonline.com/doi/pdf/10.1080/00031305.1992.10475879 https://www.tandfonline.com/doi/pdf/10.1080/00031305.1992.10475879 https://doi.org/10.1080/00031305.1992.10475879 https://doi.org/10.1016/0893-6080(90)90005-6 1701.09175 1502.03167 1602.04485 M. A. Kraus, R. Bischof, W. Kaufmann, K. Thoma Acta Polytechnica CTU Proceedings [19] J. Snoek, H. Larochelle, R. P. Adams. Practical bayesian optimization of machine learning algorithms. In Proceedings of the 25th International Conference on Neural Information Processing Systems - Volume 2, NIPS’12, pp. 2951–2959. Curran Associates Inc., Red Hook, NY, USA, 2012. [20] F. Nogueira. Bayesian Optimization: Open source constrained global optimization tool for Python, 2014–. [21] J. Mockus, V. Tiesis, A. Zilinskas. The application of Bayesian methods for seeking the extremum, vol. 2, pp. 117–129. 2014. [22] K. P. Burnham, D. R. Anderson. Multimodel inference: Understanding aic and bic in model selection. Sociological Methods & Research 33(2):261–304, 2004. https://doi.org/10.1177/0049124104268644. [23] K. E. Taylor. Summarizing multiple aspects of model performance in a single diagram. Journal of Geophysical Research: Atmospheres 106(D7):7183–7192, 2001. [24] R. Bischof, M. Kraus. Multi-objective loss balancing for physics-informed deep learning. arXiv preprint arXiv:211009813 2021. 108 https://doi.org/10.1177/0049124104268644 Acta Polytechnica CTU Proceedings 36:99–108, 2022 1 Introduction 2 Materials and Methods 2.1 Concept 2.2 FEM Material Model Data Generation 2.3 Data Pre-Processing 2.4 Machine Learning Model Development 2.4.1 kNN 2.4.2 LightGBM 2.4.3 Artificial Neural Networks 2.4.4 Hyperparameter Tuning 2.4.5 Model Quantitative Performance Metrics and AI Surrogate Model Selection 3 Results 3.1 kNN Model Results 3.2 LGBM Model Results 3.3 ResNet Model Results 3.4 Overall Model Comparison Results 4 Discussion 5 Conclusions and Outlook Acknowledgements References