Neural network as a tool for design of amorphous metal alloys with desired elastoplastic properties

The development and implementation of the methods for designing amorphous metal alloys with desired mechanical properties is one of the most promising areas of modern materials science. Here, the machine learning methods appear to be a suitable complement to empirical methods related to the synthesis and testing of amorphous alloys of various compositions. In the present work, it is proposed a method to determine amorphous metal alloys with mechanical properties closest to those required. More than $50\,000$ amorphous alloys of different compositions have been considered, and the Young's modulus $E$ and the yield strength $\sigma_{y}$ have been evaluated for them by the machine learning model trained on the fundamental physical properties of the chemical elements. Statistical treatment of the obtained results reveals that the fundamental physical properties of the chemical element with the largest mass fraction are the most significant factors, whose values correlate with the values of the mechanical properties of the alloys, in which this element is involved. It is shown that the values of the Young's modulus $E$ and the yield strength $\sigma_{y}$ are higher for amorphous alloys based on Cr, Fe, Co, Ni, Nb, Mo and W formed by the addition of semimetals (e.g. Be, B, Al, Sn), nonmetals (e.g. Si and P) and lanthanides (e.g. La and Gd) than for alloys of other compositions. Increasing the number of components in alloy from $2$ to $7$ and changing the mass fraction of chemical elements has no significantly impact on the strength characteristics $E$ and $\sigma_{y}$. Amorphous metal alloys with the most improved mechanical properties have been identified. In particular, such extremely high-strength alloys include Cr$_{80}$B$_{20}$ (among binary), Mo$_{60}$B$_{20}$W$_{20}$ (among ternary) and Cr$_{40}$B$_{20}$Nb$_{10}$Pd$_{10}$Ta$_{10}$Si$_{10}$ (among multicomponent).


Introduction
Amorphous metal alloys are the promising materials for the automotive, aerospace, energy, electronics and medical technology industries [1,2,3,4]. High corrosion resistance, high magnetic permeability, superior mechanical strength, high fracture toughness, high elastic strain limit and high formability are just some of the unique set of properties that make amorphous metal alloys widely applicable [5,6,7]. Such the combination of properties is directly due to the absence of structural order accompanied by defects that is typical for crystalline analogues [8,9,10,11].
However, despite all the advantages of amorphous metal alloys, their production is complicated by the fact that the formation of a stable disordered structure depends strongly on alloy composition (i.e. number of components, type of added chemical elements) and its preparation protocol (i.e. cooling and compression procedures, initial and final melt temperatures) [12,13,14,15,16].
Amorphous metal alloys are actively studied for more than 80 years, beginning, in particular, with the works of Kramer [17,18]. One of the first methods of practical formation of alloys with amorphous structure was based on the so-called electrodeposition process. Later, in the 60's of the 20th century, the first works related with formation of amorphous metal films by rapid cooling of the corresponding melts were appeared [19,20]. As it turned out later, amorphization of metallic melts of almost any composition is possible if extremely fast cooling is used. The next stage in the development of the amorphous alloy formation methodology concerned the consideration of alloys in eutectics, where it was found that bulk amorphous samples of more than 1 mm thickness can be formed [21]. Further attention in this area focuses on some aspects. Namely, the mechanical properties of bulk amorphous metal alloys are strongly dependent on alloy composition and chemical purity of raw material. The strength properties of amorphous alloys can be significantly reduced due to the presence of impurities. Moreover, bulk amorphous metal alloys are inherently fragile.
Therefore, in the early 2000's, studies were aimed to improve the alloy hardening methods as well as to determine the relationship between the key mechanical properties of amorphous metal alloys, which include the Young's modulus E, the yield strength σ y and the strength σ f [22,23]. It has been shown that the relationship between the hardness H (by Vickers method), the strength σ f , the Young's modulus E and the yield strength σ y of amorphous metal alloys is close to linear and can be reproduced, for example, by Tabor's relation H = Kσ y , by Johnson's model H = σ y (a+b ln[cE/σ y ]) and by relation σ f = dE 1/2 (here, K, a, b, c and d are constants) [24,25,26]. These studies found that amorphous metal alloys with large values of E and σ y are characterized by high hardness H and strength σ f .
The synthesis of amorphous metal alloy with desired mechanical properties may require listing various combinations of compositions followed by mechanical testing. This makes the process of synthesizing new alloys extremely difficult and significantly increases the costs. Then, methods of computer design seem to be a suitable support for empirical methods at the stage of determining amorphous metal alloys with desired mechanical properties [27,28]. In recent decades, rapid development of information technologies as well as automation of data collection and storage processes contribute to accumulation and systematization of information about the physical and mechanical properties of bulk amorphous metal alloys glasses [29,30,31,32]. The methods of machine learning operate with large arrays of the data and allow us to determine the relationship between composition and properties of alloys both already known and not previously known [33,34,35,36]. For example, Xiong and co-authors have been developed a machine learning model that can predict the glass-forming ability and elastic moduli of bulk metallic glasses based on the fundamental atomic properties, chemical and physical properties obtained from experiments or density functional theory simulations [37]. These results find the importance of individual chemical element properties and macroscopic properties in determining the strength characteristics of amorphous alloys. The results obtained by Khakurel et al. established that the average concentration of valence electrons, the atomic radius and the melting temperature are the key properties, which are correlated with the Young's modulus of compositionally complex alloys [38]. The results of this work can also be extended to amorphous metal alloys, as it is confirmed in Refs. [39,40]. In addition, as it was found in Ref. [41] using a machine learning model, the Young's modulus of metal alloys under normal conditions correlates with the yield strength and with the glass transition temperature. In this case, the specificity of "chemical formula" of alloy, which is determined by the molar mass and the number of components, is not as important as is usually expected. Johnson and Samwer have found that the mechanical properties (elastic constants, compressive yield strength, elastic strain limit) of 30 bulk metallic glasses as functions of the scaled temperature T R /T g obey the universal law ∝ a − b(T R /T g ) 2/3 , where a and b are the constants, T R is the room temperature, T g is the glass transition temperature [42]. The results of this work systematize existing knowledge about the mechanical properties of amorphous alloys. An artificial neural network has created by Jeon and co-authors for designing Fe-based amorphous metal alloys with the desired crystallization temperature and glass transition temperature [43]. Thus, all these studies show that the machine learning methods are suitable tool to find new amorphous alloys with required physical and mechanical properties. Despite the significant number of such studies, little attention has been paid to the development of methods for determining previously unknown amorphous alloys with the desired mechanical properties.
The present work proposes a new method for determining amorphous metal alloys of arbitrary composition based on a large set of empirical data. The originality of this method is that it is based on a machine learning model capable of predicting the Young's modulus and the yield strength of amorphous alloys taking into account the fundamental properties of each chemical element that forms the alloys. It is quite significant that the obtained results lead to new knowledge, which will contribute to the determination of amorphous metal alloys that maximally satisfy the required mechanical properties.

General strategy of the method
The developed method for determining amorphous metal alloys is based on a machine learning model, which is an artificial neural network of direct propagation. The main advantage of this method is the possibility to calculate the Young's modulus E and the yield strength σ y both for known amorphous metal alloys and for alloys that are yet to be synthesized. The developed method makes it possible to determine E and σ y of alloys, whose number of components varies in the range from 2 to 7. Note that such the number of components is ordinary for the majority of known metal alloys. In addition, the proposed method can be adapted to identify alloys with large number of components at the presence of appropriate data for neural network training. The composition and mass fraction of chemical elements in the generated alloys are the control parameters, which allow us to construct a diverse set of compounds.
The general strategy for determining amorphous metal alloys implemented in this work consists of four main stages [see Figure 1]: • Stage I. This stage includes the process of data collection and systematization of information about the properties of multicomponent amorphous metal alloys based on Al, Au, Ca, Co, Cu, Fe, La, Hf, Mg, Ni, Pd, Pt, Sc, Ti, W, Zr, etc., as well as information about the properties of the other additional chemical elements involved in the formation of these alloys. Among these properties are the atomic mass m a , the covalent radius r c , the ionization energy E i and the electronegativity χ, which characterize the nature of the chemical element [see Table 1]. This choice is due to the following reasons. First, these parameters most clearly define the possible physical and chemical bonds between the elements, which can either promote or inhibit the formation of an amorphous structure. For example, according to the empirical rule proposed by Inone et al. in the early 1990's [44], the difference in atomic sizes must be greater than 12 % for good amorphization of a liquid. Secondly, most of the intrinsic properties of chemical elements (especially of the same type) are correlated. In addition, the thermal conductivity λ, the specific heat capacity C s , the density ρ, the melting temperature where "Value min " and "Value max " are the smallest and largest known values of the "Property".
Moreover, all these listed properties correlate with the mechanical properties of materials. considered in a machine learning model [37]. In addition, the results obtained by Wang based on the analysis of a large set of empirical data for amorphous alloys allow one to establish the existence of correlation between elastic moduli (i.e. Young's modulus, shear modulus, bulk modulus), microstructural features, rheological properties, the glass transition temperature, the melting temperature and the boson peak [47,48].  Table   S1 in Supplementary data. The mass fraction of the chemical elements in a generated alloy is also set randomly so that the total mass fraction of all chemical elements is equal to 100 %.
A set of physical properties is created for each chemical element [see Table 1].
• Stage III. Information about the alloy composition and the physical properties of all the chemical elements is processed by the pre-trained neural network. This neural network evaluates the Young's modulus E and the yield strength σ y for all generated alloys. The training pro-cedure of the neural network is discussed in more details in the subsection "Machine learning model: structure and training".
• Stage IV. Statistical interpretation of machine learning results is performed.
Thus, the proposed method makes it possible to perform a complete cycle of alloy design and determine its mechanical properties: from obtaining the correct alloy composition to calculating the correct values of E and σ y .  Calculation of the values of all neurons is carried out by expression [51]: Here, n  The neural network is trained using the backpropagation algorithm [53,54]. The values of the weight coefficients are adjusted as follows: where ξ is the squared error between the output neuron and the desired value of the mechanical property; γ is the training rate. In the present work, the training rate is γ = 0.3, which is optimal for the created neural network. At the training rate γ = 0.   Thus, the results of the machine learning model are reliable and predictable.

Properties importance scores
The analysis of the importance scores shows that all the considered physical properties (λ, T b ,   the MRE decreases from ∼ 58 % to ∼ 13 % for E and from ∼ 33 % to ∼ 11 % for σ y . A rapid decrease of the error is observed when the temperatures T m and T b as well as the quantities E i and χ have been added, which may be due to their multicollinearity. Figure 3(d) shows that the Pearson correlation coefficients for the considered properties take both positive and negative values in the range from −1 to 1 [60]. For example, the positive correlation between the temperatures T m and T b is due to the fact that an increase in the melting temperature leads to an increase in the boiling temperature [61]. An increase in the atomic mass m a of the alloy components usually leads to an increase in its density ρ, which leads to a positive correlation between m a and ρ [62,63].
The presence of the pronounced negative correlation between the pairs r c , E i and r c , χ is due to the fact that a decrease in the covalent radius r c leads to an increase E i and χ by increasing the electron density in the atom [64,65].

Statistical interpretation of the results
In the present study, 50 000 different amorphous metal alloys were obtained by the proposed In the statistical interpretation, the results reveal that E and σ y depend mainly on the properties of the chemical element with the largest mass fraction. As seen in Figure 4, changing the number of components in alloy has no significant effect on values of E and σ y . In the array of 50 000 different alloys obtained by the machine learning model, some alloys with the highest Young's modulus E and the yield strength σ y were selected [see Table 2 and Co 60 B 35 Ta 5 it was experimentally established that E > 250 GPa and σ y > 5.0 GPa [50,41].
Obtained results reveal that the alloys based on Cr, Mo and W from the group VI-B of the Periodic  [66,67,68]. Then, their significant presence in an alloy improves its strength. Note that this fact is also known in metallurgy, where these metals are widely used to increase the hardness of steel alloys, to increase wear resistance and to form wear-resistant coatings (e.g. alloys Cr-Co, Cr-Fe, Mo-Fe, Mo-Cr-Fe, W-Fe, W-Ni-Co) [69,70]  In  Table 3]. The doping with refractory metals, nonmetals and lanthanides (e.g. B, Si, Gd, La) makes it possible to increase the strength of these alloys, which is actively used in modern metallurgy to produce heat-resistant alloys [71,72]. This simple quantitative analysis confirms that the properly selected composition and physical properties of the main chemical elements are most important in determining the alloys that best match the required mechanical properties.

Conclusions
In the present study, the machine learning model was applied to predict the Young's modulus E and the yield strength σ y of amorphous metal alloys with different compositions. More than 50 000 different alloys were determined as well as E and σ y were evaluated for each of them. It was found that the artificial neural network trained on the basis of information about the atomic number of a chemical element, its atomic mass, covalent radius, ionization energy, electronegativity, thermal conductivity, specific heat capacity, density, melting temperature and boiling temperature allows us to correctly determine of E and σ y of amorphous metal alloys consisting 2 to 7 components and containing chemical elements with atomic numbers from Z = 3 to Z = 79. Here, the mean relative error is ∼ (12 ± 1)% that is the good accuracy for the direct propagation multilayer neural network.
The results of the statistical treatment made it possible to determine the chemical elements with the largest mass fraction, whose presence in the alloy leads to a significant increase in the strength of alloys. These chemical elements are B, Cr, Fe, Co, Ni, Nb, Mo, Pd and W. At the same time, the quantities E and σ y show a weak dependence on the number of components in alloy. Thus, the most significant factors in the synthesis of alloys with the desired mechanical properties are the properly selected composition and the physical properties of the basic chemical element of alloy.