# Deep Learning Sequence Methods in Multiphysics Modeling of Steel Solidification

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Thermo-Mechanical Model of Steel Solidification

**b**is the body force density vector. The rate representation of total strain in this thermo-elastic-viscoplastic model is given by Equation (3):

_{sol}= 1482.4 °C, and T

_{liq}= 1520.5 °C, whose solidification path is shown in the pseudo-binary iron-carbon phase diagram shown in Figure 1.

## 3. Deep Learning Models

#### 3.1. Dense Feedforward Neural Network

**W**and

**b**. The output ${\widehat{\mathit{Y}}}^{[l]}$ for a layer l is calculated as:

**W**

^{[l]}(n

_{l}× n

_{l}

_{−1)}and

**b**

^{[l]}(n

_{l}

_{−1}× 1) are matrix of weights and vector of biases, respectively which are updated after every training pass. f

^{[l]}is the activation function that transforms

**Z**into output for every neuron in the layer l. Activation functions in neural networks are nonlinear functions such as Rectified Linear Unit (ReLu), Sigmoid, and Hyperbolic Tangent. They enable the neural network to learn almost any complex functional relationship between inputs and outputs. At the end of each feed-forward pass, the loss function L compares the predictions to the targets by calculating a loss value that measures how well the network’s predictions match what was expected. Then, in a backpropagation process, the optimizer minimizes loss value iteratively with gradient descent in Equation (7) and other similar optimization techniques. The gradients of loss function L are calculated with respect to the weights in the last layer, and the weights are updated for each node before the same process is done for the previous layer and backward until all of the layers have had their weights adjusted. Then, a new k iteration with forward propagation starts again. After a reasonable amount of iterations, the series

**W**should converge toward the minimum loss value location. The parameter γ is called the learning rate.

^{k}#### 3.2. Recurrent Neural Networks

**W**(hidden-to-hidden) and

**U**(input-to-hidden) weight connections, respectively,

**f**is the activation function, and

**b**is a bias. Output prediction at time step t is calculated from ${\mathit{s}}_{t}$ using

**V**(hidden-to-output) weights and

**c**bias. Similarly, to the dense feedforward network, all weights and biases are updated from their loss gradients by backpropagations. However, recurrent neural networks often consist of very deep computational graphs repeatedly (recurrently) applying the identical operation at each time step of a long-time sequence. This may cause the vanishing or exploding gradient problems during backpropagation and makes it challenging to train long sequence data with RNNs. To address the vanishing gradient problems of traditional RNN, the long short-term memory (LSTM) [25] and gated recurrent unit (GRU) [26] are devised. Hidden state (memory) cells in the LSTM and GRU advanced recurrent neural networks are designed to dynamically “forget” some old and unnecessary information via select gated units that control the flow of information inside a memory cell, thus avoiding multiplication of a long sequence of numbers during temporal backpropagation. GRU has a simpler gated unit architecture than LSTM and generally learns faster than LSTM. A recent work [19] has shown that GRU’s formulation is less prone to overfitting in materially non-linear sequence learning and allows faster training due to the smaller number of trainable parameters. A comprehensive mathematical overview of LSTM and GRU networks can be found here [27].

#### 3.3. Temporal Convolutional Neural Network

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Pseudo-binary iron-carbon phase diagram with solidification path for 0.09 wt%C low carbon steel grade.

**Figure 2.**Temperature dependent material properties for 0.09 wt%C steel grade: (

**a**) elastic modulus and enthalpy and (

**b**) thermal conductivity and thermal expansion.

**Figure 3.**(

**a**) Continuous caster with solidifying slice finite element domain and (

**b**) thermal and mechanical boundary conditions.

**Figure 10.**Output results, ground-truth, and deep learning predictions for the 4 control points, corresponding to the testing input sample in Figure 7.

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Koric, S.; Abueidda, D.W. Deep Learning Sequence Methods in Multiphysics Modeling of Steel Solidification. *Metals* **2021**, *11*, 494.
https://doi.org/10.3390/met11030494

**AMA Style**

Koric S, Abueidda DW. Deep Learning Sequence Methods in Multiphysics Modeling of Steel Solidification. *Metals*. 2021; 11(3):494.
https://doi.org/10.3390/met11030494

**Chicago/Turabian Style**

Koric, Seid, and Diab W. Abueidda. 2021. "Deep Learning Sequence Methods in Multiphysics Modeling of Steel Solidification" *Metals* 11, no. 3: 494.
https://doi.org/10.3390/met11030494