Machine Learning for Contact Mechanics from Surface topography

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Machine Learning for Contact Mechanics from Surface topography

Shahin Salehi

Computer Science and Engineering, bachelor's level 2019

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

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Abstract

Being able to predict how different materials will perform in engineering applications is an essential insight to have before going into production. We achieve this with mathematical models that can simulate the expected behavior of these materials given arguments that describe them. The issue is that the complexity of the environments the mathematical model simulates causes high computation time. In this thesis, we avoid the time-consuming execution time of the mathematical model by implementing an artificial neural network (ANN) to predict the outcome based on labeled data. In addition to achieving a quick predicting ANN, the goal is to do so with acceptable accuracy.

We approach the issue of time and accuracy by implementing two different architectures of ANNs. The networks are feed labeled data which describe the environment, containing materials, and the given outcome in terms of the real area of contact.

The networks trains on this labeled data and as a result, make predictions. We observe the quality of the predictions in terms of time and accuracy to see if they are appropriate solutions.

Previous work made by rapetto et al. [8] is highly influential to this thesis. In their research paper, they apply an ANN to predict the contact mechanics of surfaces in a less complex environment. In this study, we reiterate the network they developed and compare it to a more modern network.

The results achieved in this thesis prove that modern ANNs are capable of quickly and accurately predicting the contact mechanics of 3D surfaces. Setting an example of the efficiency of ANNs in related problems.

During testing of the networks, we train both models with the same conditions and data; they are both subjected to the same optimization and regularization tech- niques. What makes them different is their size in neurons and hidden layers.

The results show that the shallow network representing an older model failed to learn from the data, given its average error of 848% on the test set with a running time of 2.3s. On the other hand, the deep ANN proved that it could make very accurate predictions, with an average error of 0.012% on the test set with a running time of 12.5s.

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Dedication

To my mother and my brother.

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Acknowledgements

Firstly, I want to thank my supervisor, Marcus Liwicki, for giving me this opportunity. When I asked to do my thesis at his department, I was swiftly put in to work with all the necessary tools to write this thesis.

I want to thank my Co-supervisor György Kovaks for often working with me to lay the foundation of my thesis.

I want to thank Andreas Almqvist for representing and informing me of the tribology aspect of this thesis. Our meetings were very helpful in making me understand the problems we faced and why they are important.

I want to thank Pedro Alonso for sharing his knowledge of deploying and training ANNs.

I want to thank my good friend for Marcus Lindner for teaching me how I should approach scientific writing. Proofreading my text and giving me tips while still letting me develop my own writing style.

Thank you, Malin Malm, for all the love and support.

I want to thank my good friend, Kammal Al-Kahwati, who has been like a big brother to me since my first days as a computer science student.

Finally, thanks to all my friends who have made this journey easier and thank you, Johannes, for the exciting discussions we have had over coffee.

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Chapter 1 Introduction

In the past decade, the number of articles about artificial intelligence published by the New York Times increased by approximately 315% [3]. While the majority of these articles share an optimistic view of AI, we can observe a growing trend in discussing concerns. Fear of losing control and the lack of ethical reasoning are two examples of such concerns. Although we do not discuss the concerns of the rise of AI in this thesis, the rise itself does affect it.

Approximately two decades ago, Influential papers such as [13] and [2] motivated the benefit of very large data sets when training their algorithms for better predictions.

With time passing more data has become available, something that artificial neural networks (ANNs) have taken advantage of and as a result, become more useful [4]. Over time these ANNs have grown in size with new computer hardware and software that can support it. With the increase in data sizes and computing power of the ANNs themselves, they have become powerful tools that can solve increasingly complex problems with increasing accuracy [4]. These factors have sparked the immense rise of AI research and related interest in AI assisted applications in recent years.

As mentioned earlier, this thesis is part of the scientific rise of attention to the topic.

Specifically, we study how ANNs perform when applied in a tribological setting.

Calculating how different surfaces behave in contact with each other is complex, and even more so when the surfaces are well defined. Given the complex description of the surfaces, using conventional methods to calculate their performance can be too time-consuming.

1.1 Background

In a typical problem setting of tribology, there is a need to figure out what materials are going to be the best fit for an engineering application [6]. Being able to simulate how they perform in various environments where they interact with other materials is an essential insight to have before creating a product.

Simulating these environments and calculating the outcome can be done with complex mathematical models. Tribology researchers at Luleå University of Technology

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CHAPTER 1. INTRODUCTION

have developed one of these models [7, 10, 1], which this thesis uses. The model builds its environment and surfaces based on arguments that describe them, the more arguments its subjected to, the more detailed the description of the environment will be.

1.2 Motivation

The core problem that motivates this thesis has its origins in tribology. Calculating the contact mechanics of a specific material, with a given set of arguments that describe its surface topography, can be done with a mathematical model. The model will incorporate the arguments into its mathematical equations and calculate how the contact mechanics between the two surfaces will yield and return a result.

With this tool, we can know beforehand how our materials will perform in our given engineering application. This insight is beneficial from many standpoints, be it safety-critical, economic or otherwise.

The issue with this approach is when we increase the complexity of the environment we want to simulate, we also increase the number of arguments the model has to take into consideration. This increase in complexity will result in heavier computation and more time-consuming execution.

Solving this problem can serve as a step in the right direction of solving similar future problems where the environments we predict are increasingly complex.

1.3 Problem description

In this thesis, the focus is on predicting the real area of contact. It is one of three output variables produced by the mathematical model. Its a standard variable studied in tribology applications, and it is a value for how much two surfaces are touching. The fact is that surfaces are not perfectly smooth, and when looking close enough, its rough nature is apparent. Meaning that two contacting surfaces do not touch evenly; they have a real area of contact. Figure 1.1 illustrates how this might look.

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CHAPTER 1. INTRODUCTION

Figure 1.1: Real area of contact illustration

The increasing input arguments that describe the surfaces are causing high complexity, and the issue that we face is the time-consuming computation of the mathematical model. This thesis aims to solve this issue by implementing an ANN that will learn to map different surface topography arguments to the real area of contact With the goal of doing this much quicker than the mathematical model and with reasonable accuracy.

1.4 Delimitations

The mathematical model’s simulated environment pushes one generated surface against a flat surface with constant pressure and records the result. The model records the deformation of the surface over fourteen timesteps. In this study, we predict the outcome of the first timestep and not all fourteen.

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Chapter 2 Related Work

Being able to get fast and accurate predictions on how materials of different kinds perform in their intended environments are attracting researchers from many ar- eas.

In a previous study [11], authors have used neural networks with different configurations to predict how steel surfaces perform during milling — comparing different ANN structures and activation functions. During their experiments, they learn that the activation function sigmoid manages to get the best results for their application. They later conclude that the networks error is in an acceptable range and that neural networks can be used effectively in applications like theirs, to support the planning and modeling of technological processes.

In a similar study [5], an ANN is used to predict the wear loss quantities of some aluminum–copper–silicon carbide composite materials. The ANN effectively manages to predict the hardness, density, and porosity in the materials. When they compare the error of the network to the error that generally arises due to experimental vari- ation, the ANNs error is lower. They conclude in the paper that the ANN produces satisfactory results and that its results can be used instead of measured results, hence saving time and cost.

In another study made in 2002 [14], a simple older ANN is compared to a more complex one when trying to predict specific wear rate and frictional coefficient.

In this study, the results show that the complex ANN that learns on more data outperforms the older one. They conclude that they achieve increased performance when optimizing the ANN and feeding it more data. They recognize the more complex ANN as a helpful tool to have for material design, parameters studies, and property analysis of polymer composites.

Closely related to this problem and the inspiration of this thesis is the work done by Rapetto et al. [8]. A former researcher at Lulea University of technology that applied a simple neural network to data generated by a similar mathematical model to the one in this thesis. The main difference is that their model generates 2D profiles as opposed to the 3D surfaces used in the experiments of this thesis. Rapetto used ANNs to see their potential when applied to tribology problems, analyzes its results and concludes that there is value to applying ANNs to similar problems and that the paper should serve as a stepping stone for future research. With time passing

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CHAPTER 2. RELATED WORK

and the advancement of machine learning to today, it indeed serves as a stepping stone for this thesis.

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Chapter 3 Theory

This section gives an overview of the scientific methods used in this thesis and its inner functionality.

3.1 Artificial Neural Networks

Artificial neural networks (ANNs) are networks composed of different layers in which artificial neurons reside. Each neuron in the net has adaptive weights which configure the strength of the output sent to the neurons in the next layer. In the experiments done in this thesis, the neurons connect with directed links in a feedforward manner, forming a directed acyclic graph, as opposed to a recurrent architecture, where neurons can feed outputs back to its inputs [9].

Figure 3.1 illustrates the structure of an ANN, which contain an input layer, a configurable amount of hidden layers, and an output layer. The input layer size is determined by the number of input features, while the size and number of hidden layers are configurable. Last is the output layer, which outputs the weighted summation from previous layers.

The procedural steps of the feedforward neural network work as follows. Supply the network with labeled data to be learned, containing the input and output for each sample. When the network makes a prediction on the data, it will be assessed based

Figure 3.1: Representation of netural network Input

layer

Hidden layer

Output layer

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CHAPTER 3. THEORY

on the mean squared error (MSE) between the predicted output and the correct output. If the prediction does not match the desired result, the weights of the neurons will be adjusted to reduce the error.

3.1.1 Neurons, weights and activation functions

In each layer, there are a set of neurons. For every neuron in the layer, there is a connected weight, and for the entire layer, there is a bias. The first step of calculating the output is to take the dot product between the input from the previous layer, and respective weights of the current layer and then adding the bias. For the equation 3.1, z represents the output vector of a layer, W is the weight vector, X is the input from the previous layer, and b is the layers bias vector.

z = W^T · X + b (3.1)

Before feeding the output vector to the next layer, the system introduces nonlin- earity to the equation by applying an activation function on the vector z. Multiple activation functions are available, and the one used in this thesis is the Rectified Linear Unit (ReLU) shown in equation 3.2, because of its computational advantages [4].

f (x) = max(0, x) (3.2)

3.1.2 Loss Function

As mentioned earlier in 3.1, how accurate the model is, depends on the mean squared error. The M SE is what the network use as a metric to determine how to modify the weights to minimize loss. The goal is to alter the weight and bias parameters so that the loss is as small as possible for all training examples. Equation 3.3 demonstrates how the M SE is calculated. In the equation n is the number of examples, Y_i is the value we want to predict where i is the ith example, and ˆY_i is the prediction the network makes.

M SE = 1 n

n

X

i=0

(Y_i− ˆY_i)² (3.3)

3.1.3 Data scaling

Different scales may increase the difficulty when the network is learning and trying to adjust its weights to the data, and eventually lead to poor performance. That is why the data is scaled so that the neurons can avoid the problem of adapting to large variations. Python library sklearn’s function standard scaler scales the data.

It standardizes features by removing the mean and scaling to unit variance. The

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CHAPTER 3. THEORY

equation 3.4 represents the calculation. Where x is training examples, u is mean, and s is the standard deviation.

z = (x − u)/s (3.4)

3.1.4 Dropout

Dropout is a computationally cheap method that we implement into our network to regularize the data [4]. By randomly ignoring neurons in calculations [4], the subnetwork constructed as a result is forced to work without the left out neurons — making neurons that might have depended on the left out neuron before, to improve its performance. The weight matrices of the network become more evenly distributed as a result. Figure 3.2 illustrates the full network with no dropout enforced, and Figure 3.3 shows the sub-network constructed by temporarily dropping the second neuron in the second hidden layer.

Figure 3.2: Dropout Full Network Input

layer

Hidden layer

Output layer

Figure 3.3: Dropout Sub-network Input

layer

Hidden layer

Output layer

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CHAPTER 3. THEORY

3.2 Tribology

Tribology is a word that originates from the Greek word "tribos", which means rubbing or sliding [12]. The year 1967 is when the scientific field of tribology was defined; it is the study of wear, friction, and lubrication of interacting surfaces.

Making it a relatively young field, nevertheless an important one. It is a field where we analyze how different materials will perform in different applications. We do this by applying operational analysis to our problems in order to figure out how it will perform in terms of reliability, maintenance, and wear. This type of analysis is necessary for a wide range of applications, from household appliances to spacecrafts.

Wear plays a significant role in the sustainability of materials, with friction being the primary cause of the wear. The friction causes the material to lose mechanical performance and eventually become dysfunctional and in need of replacement. Un- derstanding how the wear affects the materials and reducing the wear we expose it to is essential for performance and can also result in considerable savings [12].

In this thesis, we focus on the friction aspect, which in turn leads us to the real area of contact. We analyze at what points the materials are in contact and cause friction. Our tool for analysis is an ANN; we can use it in order to predict the contact mechanics of materials. We do this to achieve a more lightweight solution to accurately predict contact mechanics and increase the robustness and sustainability of future technical equipment.

3.3 Evaluation methods

In this thesis, we are using artificial neural networks for a regression problem. Un- like a classification problem, we try to predict continuous values and not discrete classifications. In order to understand how accurate the predictions are, we need to apply an evaluation method.

There are two primary components used to analyze the performance of the ANNs.

First, we calculate the mean squared error (MSE) between the predicted real area of contact (Ar) values the ANNs make and the actual values for the given surfaces.

As a result of the MSE, we have a scalar that demonstrates the magnitude of the mean error. Second, to understand how big the MSE is relative to the dataset, we divide the MSE with the average size of the dataset, that way we end up with a percentage on the MSE relative to the dataset. If this percentage is low, then the error is small, which means that we have an ANN that is performing well. On the other hand, if the percentage is large, that gives us a high error and a poorly performing ANN.

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Chapter 4 Data

This chapter will explain how the data is generated and treated before its feed into the artificial neural network.

4.1 Simulator

The mathematical model generates environments where a surface is pushing against another flat surface with a certain pressure. How the surface performs in this simulation is recorded and stored as a result. In the results, the arguments describing the surface topography and how it performed in terms of the real area of contact is stored. This data will later be extracted and fed into the ANN.

After a complete run, the model produces a total of 720 training examples of different surfaces and their environments with their respective arguments. On the CPU Intel CoreR ^TM i7-7500U, one run takes approximately 8 hours to complete. Figure 4.1 illustrates a typical surface that is generated by the model.

Figure 4.1: Example of simulated surface

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CHAPTER 4. DATA

4.2 Feature Selection

It is important to analyze the features of the data by checking its linear correlations between the input and output to see if there are any features that don’t contribute to the solution. The correlation between the surface topography arguments and the real area of contact (Ar) is given by applying the Pearson correlation coefficient on the data, and Figure 4.2 illustrates the results. The heatmap shows that the input variable Sku has zero correlation to the output Ar. Meaning that when the network treats this input, it will be considering it as noise, since its change cannot be mapped appropriately to the output. Therefore, moving forward with the implementation, Sku was eliminated as an input.

Figure 4.2: Example of simulated surface

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CHAPTER 4. DATA

4.3 Data Organization

The data generated from the mathematical model is extracted and labeled such that the surface topography arguments represent the input and real area of contact represents the output. The 13 features that remain after we have eliminated Sku all represent different aspects of the surface, and as a consequence, have different scales. We use equation 3.4 to achieve a uniform scale over all input data and to avoid weights becoming biased to larger features.

To prepare the data for the ANN, we split it into training, validation, and test sets.

The training set is what defines the model, sets the standard for all input-output mappings, and becomes the basis for all future predictions. The validation set is used to see if the network can generalize well to unseen data. It is an opportunity to adjust the hyperparameters of the ANN to achieve best possible performance.

It is important to acknowledge that the changes made to the ANN during the validation phase can make the network biased to the validation data. Meaning that although it is unseen data, there is still bias. That is why we make the final evaluation of the network on the test set. Table 4.1 demonstrates the percentages of the split.

Table 4.1: Percentage of examples in different sets train validation test

(81%) (11%) (8%)

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Chapter 5 Implementation

This chapter describes the technical details of the project and the different architectures of ANNs that were implemented and tested in the experiments.

5.1 Neural Network Configurations

The programming language used in this project is Python. Keras is the library that supported the construction of the ANN, and TensorFlow is running in the background. The CPU Intel CoreR ^TM i7-7500U hosted the training of all different configurations of the ANNs.

Two main ANN architectures are implemented and experimented with in this study.

The first being a simple shallow network, inspired by older related research, and a more modern deep and wide network, representing current advancement in the area.

The deep network has 14 hidden layers that contain 128 neurons each to achieve good adaptation to complex input variables. The shallow network has two hidden layers, the first one containing 14 neurons and the second one containing 4 neurons, representing a simpler model.

For each layer in both ANNs, dropout is added and the dropout rate is set to 0.5. The activation function used is ReLU since it has computational benefits, as mentioned in 3.1.1.

The optimization function used is Adam. Out of the available functions that Keras provides, Adam was the most successful in generalizing to the training data and minimizing error. The learning rate α is set to 1.1 ∗ 10⁻7.

The network trains on batches of size 512 for 170 epochs, after 170 epochs the MSE converges and running for more epochs only increases the running time without improving performance.

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CHAPTER 5. IMPLEMENTATION

5.2 Summary of architectures

To get the best fit of hyperparameters for our ANNs, we use empirical methods to test and observe. We then implement the parameters proven most beneficial for the performance of the ANNs. Tables 5.1 and 5.2, both represent a summary of the networks structures.

Table 5.1: Summary of deep network hyperparameters Hyperparameters

Width 14

Depth 128

Activation ReLU

Dropout rate 0.5

Optimizer Adam

learning rate 0.0000011

Batch size 512

Epochs 170

Table 5.2: Summary of shallow network hyperparameters Hyperparameters

Width 2

Depth 14-4

Activation ReLU

Dropout rate 0.5

Optimizer Adam

learning rate 0.0000011

Batch size 512

Epochs 170

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Chapter 6 Evaluation

This chapter will describe how the results are calculated and illustrate the performance of the networks.

6.1 Test Setup

For every run, the ANN stores its predictions for Ar in the prediction vector ˆY . The size of the data set determines the vector ˆY ’s length n. As an example, If there are 500 training examples, there will be 500 predictions, meaning n = 499. To calculate the M SE of the predictions, we use equation 3.3 on the prediction vector ˆY and the vector with the correct values of Ar, Y . 6.1 illustrates how the vectors with the prediction values and actual values look.

Y =ˆ





 ˆ y₀ ˆ y₁ ... ˆ y_n







Y =





 y₀ y₁ ... y_n







(6.1)

The M SE itself is not enough to know how accurate the predictions are. To understand the ANNs accuracy, we first calculate the mean of the vector Y with 6.2.

Then to understand how large the MSE is with respect to the average value of the vector Y , we use the equation given in 6.3. This equation returns how large the MSE is as a percentage of the average value of Y .

Y_mean= 1 n

n

X

i=0

(Y_i) (6.2)

M SE_norm = M SE

Y_mean ∗ 100 (6.3)

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CHAPTER 6. EVALUATION

To visualize the accuracy of the ANNs, we demonstrate their predictions with plots.

The x-axis of the plots represent the observed value of Ar, and the y-axis represents the predicted value of Ar. Every red dot on the plot represents a prediction, and to further visualize the accuracy of the red dots, we add an identity line to the figure. The closer the red dots are to the identity line, the more accurate their predictions are. So when a red dot sits perfectly on the identity line, that means that the prediction for that example perfectly matches the actual value of Ar, i.e., Yˆ_i = Y_i.

To get a better understanding of how the two networks perform in general, we run the networks ten times and take the average on the time, accuracy, and MSE.

Equations 6.4, 6.5, and 6.6 calculate the averages.

time_mean= 1 n

n

X

i=0

(time) (6.4)

M SE_norm_mean = 1 n

n

X

i=0

(accuracy) (6.5)

M SEmean= 1 n

n

X

i=0

(M SE) (6.6)

6.2 Results

6.2.1 Shallow Network Results

Figure 6.1 and 6.2 illustrate the shallow networks predictions after a run on the training and test set respectively.

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Figure 6.1: Training set predictions

Figure 6.2: Test set predictions

Tables 6.1 and 6.2 both show the average performance of the network on the training and test set respectively.

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Table 6.1: Average performance training data time_mean M SE_mean Y_mean accuracy_mean

2.3s 1.6 ∗ 10⁻3 19 ∗ 10⁻5 873%

Table 6.2: Average performance test data time_mean M SE_mean Y_mean accuracy_mean

2.3s 1.5 ∗ 10⁻3 19 ∗ 10⁻5 848%

6.2.2 Deep Network Results

The plots 6.3 and 6.4 represent a prediction run on the training and test sets respectively on the deep network.

Figure 6.3: Training set predictions

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Figure 6.4: Test set predictions

Tables 6.3 and 6.4 both show the average performance of the network on the training and test set respectively.

Table 6.3: Average performance training data time_mean M SE_mean Y_mean accuracy_mean

12.5s 2.7 ∗ 10⁻8 19 ∗ 10⁻5 0.014 %

Table 6.4: Average performance test data time_mean M SE_mean Y_mean accuracy_mean

12.5s 2.3 ∗ 10⁻8 19 ∗ 10⁻5 0.012 %

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Chapter 7 Discussion

7.1 Shallow Network Performance

Looking at the results in tables 6.1 and 6.2 we can see that the network has a M SE_avg that is more than eight times as big as Y_mean for both training and test data. From these results, it is obvious that the simple network failed to learn from the data. Further inspection of figures 6.1 and 6.2 show interesting behaviour of the network. Instead of learning the patterns in the data the network learns a specific value to predict for every input, so that it can achieve as good M SE as possible.

Further confirming that the network has failed to understand the data.

7.2 Deep Network Performance

The results from table 6.3 and 6.4 show that the deep network has a very small M SE_avg. When comparing it to the shallow network, the superiority of the deep network is clear. When we look at the figures 6.3 and 6.4, we can observe that the predictions have become significantly better and that the network has effectively learned from the data and can make accurate mappings from input to output.

The performance of the deep network is very good. Managing to make highly accurate predictions with an average running time of 12.5 seconds makes it the network that solves the problem defined in 1.3. The accuracy is in range of acceptable accuracy and the running time of 12.5 seconds is small compared to the 8 hours of the mathematical model.

7.3 Final Remarks

In this application, the deep model is superior to the shallow model. The shallow model’s lack of complexity keeps it from learning the relations between input and output and renders it useless. This does not mean that a deep model is always the right choice. While their complexity makes them adapt to complex problems,

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CHAPTER 7. DISCUSSION

they could be unnecessarily complex and time-consuming for simple problems. Even so complex that it learns the data too well and fails to generalize to unseen data, making it a bad implementation to use. Simple models can work well for simple problems. As seen in related work [8], they use a simple model and manage to get good results. The simple network learns to map the input to output appropriately and generalizes well to unseen data. Even though the deep models are powerful, one should not consider it a solution to all problems.

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Chapter 8 Conclusion and Outlook

Mathematical models can calculate how materials perform in different environments.

While they make accurate calculations, the downside is that the more complex the environment is, the more time-consuming the execution becomes. The purpose of this thesis is to avoid that problem by implementing two variations of an ANNs to see if they can make accurate predictions based on labeled data.

We construct two networks, one deep and one shallow. They are both feed labeled data that describe the characteristics of the given materials and how they performed in their environment. The predictions the ANNs make are measured and evaluated based on their accuracy. After running both networks, the deep network architecture is superior to the shallow network architecture. Although the shallow network runs in 2.3 seconds and the deep network runs in 12.5s on average, the error rate of the deep network is significantly lower. The deep network on average having an error of 0.012% compared to the shallow network with an average error of 848%. With these figures, we can conclude that the deep network is a suitable architecture for this problem.

Given the results from the deep ANN, we achieve the goals set out in this thesis;

solving the problem of computationally heavy and time-consuming execution, with a highly accurate and quick predictor, the deep network. The work achieved with the deep ANN serves as a stepping stone for future research, where we apply AI to solve even more complex problems. Going beyond three-dimensional surfaces and predicting how the object will perform in that given environment with sophisticated neural networks.

To further showcase the capabilities of the network, and make the solution applicable in a broader, more general case. More examples from different simulations should be made available, to test different configurations of the network on more complex problems.

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Machine Learning for Contact Mechanics from Surface topography