ABSTRACT Magnetic skyrmions are potential candidates for high-density storage and logic devices because of their inherent topological stability and nanoscale size. Two-dimensional (2D) Janus

transition metal chalcogenides (TMDs) are widely used to induce skyrmions due to the breaking of inversion symmetry. However, the experimental synthesis of Janus TMDs is rare, which

indicates that the Janus configuration might not be the most stable MXY structure. Here, through machine-learning-assisted high-throughput first-principles calculations, we demonstrate that

not all MXY compounds can be stabilized in Janus layered structure and a large proportion prefer to form other configurations with lower energy than the Janus configuration. Interestingly,

INTERPRETABLE MACHINE LEARNING FOR ATOMIC SCALE MAGNETIC ANISOTROPY IN QUANTUM MATERIALS Article Open access 18 May 2025 DFT-AIDED MACHINE LEARNING-BASED DISCOVERY OF MAGNETISM IN FE-BASED

BIMETALLIC CHALCOGENIDES Article Open access 25 February 2023 HIGH-THROUGHPUT DESIGN OF PERPENDICULAR MAGNETIC ANISOTROPY AT QUATERNARY HEUSLER AND MGO INTERFACES Article Open access 13 July

2023 INTRODUCTION As one of the most typical chiral magnetic structures, magnetic skyrmions have been regarded as promising candidates for future spintronic devices1,2,3. Microscopically,

the Dzyaloshinskii–Moriya interaction (DMI), which favors canted spin configurations, plays an essential role in the formation of skyrmions. The existence of DMI is generally believed to

require both magnetism and strong spin-orbit coupling (SOC), as well as a system with broken inversion symmetry4,5. Several materials with DMI have been found, such as MnSi, FeGe,

BaTiO3/SrRuO3 superlattices and 2D Janus transition metal chalcogenides (TMDs)6,7,8. Among them, 2D Janus TMDs are an archetypal class of graphene analogs in which one atomic layer is

replaced by other atoms. This unique structure breaks the inversion symmetry and is considered as an excellent carrier for generating skyrmions9,10. Several Janus skyrmion systems have been

widely studied theoretically, such as CrXTe (X = S,Se)11, MnXY (X,Y = S, Se, Te, X ≠ Y)12,13 and VXSe (X = S,Te)14. These works highlight the possibility of tuning basic magnetic parameters

and even inducing distinct spin textures in Janus magnets. While the number of theoretical studies on Janus TMDs continues to increase, experimental characterization efforts are lacking. To

date, only two syntheses have focused on Janus MoSSe and WSSe15,16, while no other successful syntheses of Janus chalcogenide systems have been reported. One of the possible reasons for the

lack of experimental data is that the Janus configuration might not be more stable than the other MXY configurations. Previous works have focused only on the ordered layered arrangement of

atoms9, while other arrangements of X and Y atoms have not yet been considered. Thus, previous theoretical works are insufficient to guide experimental synthesis. Searching for 2D Janus

materials that can be synthesized requires us to clarify whether the Janus structure is the most stable structure among all the possible AXY structures at the same atomic ratio. Generally,

finding the most stable structure requires high-throughput calculations. However, the computational cost is high. This is especially true when the number of structure candidates is large.

For example, searching for the optimal structure for one type of MXY with a 4 × 4 × 1 supercell requires tens of thousands of computational tasks, and the calculation time for each task

using the Vienna ab initio simulation package (VASP)17,18 is 3 hours with a 96-core node. Thus, it is necessary to determine an efficient way to reduce the computational cost. An alternative

way to overcome this time-consuming problem is to introduce machine learning techniques. Finding the optimal structure from different atom arrangements is similar to that from different

defect positions or doping positions, and several works have successfully used machine learning methods to determine the distribution of defects or doped atoms18,19,20,21,22,23. These works

suggested that machine learning methods can be used to construct surrogate models and assist in time-consuming Density Functional Theory (DFT) calculations. In this work, we constructed

massive possible structures for 2D MXY TMDs and compared their energies calculated from DFT. A machine learning method, i.e., the support vector regression method, is adopted to accelerate

this process. The Janus structure has the lowest energy in only a few chemical compositions, while most other chemical compositions favor two anti-Janus structures. Although some TMDs do not

possess a Janus structure, a considerable DMI still exists among them. Furthermore, based on the magnetic exchange parameters calculated from DFT, we obtained richer spin textures than

Janus structures by spin dynamic simulation and successfully predicted skyrmions in those structures. RESULTS LATTICE STRUCTURE OF JANUS TMDS Janus TMDs have two main polytypes depending on

the arrangement of chalcogen atoms24. As the transition metal atom is bonded with the six nearest chalcogen atoms, a triangular prism or octahedron can be formed; the left one is termed

1T-phase, and the right one is termed 2H-phase, with space groups _P-_3_m_1 (space group 164) and _P-_6_m_2 (space group 187), as shown in Fig. 1a and b, respectively. The stabilities of the

1T-phase and 2H-phase in 2D TMDs have been discussed in the literature9,11,12,14,25,26,27,28,29,30. Therefore, we fixed the phase of 2D TMDs while searching for the configurations with the

lowest energy. To verify our hypothesis that the Janus structure has the lowest energy among the structures with the same components (i.e., a 1:1 ratio of chalcogen atoms), we selected 11

transition metals, Ti, V, Cr, Mn, Ni, Zr, Nb, Hf, W, Re, and Pd. We started our simulation with a unit cell, which does not have a disordered arrangement of atoms. Our calculated ground

states are consistent with those of previous studies9,11,12,14,25,26,27,28,29,30. Next, we constructed 2 × 2 × 1, 3 × 3 × 1 and 4 × 4 × 1 supercells and generated all possible irreducible

structures via _disorder_ software31. With increasing size, the number of irreducible structures increases greatly. A 2 × 2 × 1 supercell produces ten possible structures, while a 3 × 3 × 1

supercell contains more than 500 structures. When the size of the supercell reaches 4 × 4 × 1, the number of structures exceeds tens of thousands. Such a large number makes it unrealistic to

determine the most stable structure by calculating all of their energies. Thus, it is necessary to use machine learning to assist this computational process. SVR ASSISTED DFT CALCULATIONS

FOR SCREENING NEW STRUCTURES Here, we employed the support vector regression (SVR) method to predict the energy of each structure instead of performing DFT calculations, as shown in Fig. S16

in the Supplemental Material. For each type of MXY, we performed the following procedure to obtain the structure with minimum energy. First, we trained the SVR model with 100 random

structures and predicted 20 structures with the lowest energy. Then, we used DFT calculations to verify these predictions. If the energy sequence of these 20 structures from the DFT results

does not agree with the SVR model, we add 100 more structures, which are predicted by the SVR model to have the lowest energy, to the training data. Once the DFT results provide an energy

sequence similar to that of the SVR model, we stop the loop and obtain the structure with the lowest energy. The locations of the X and Y atoms are regarded as descriptors, and the

structures with the lowest energy are obtained through the above procedure. We first verified the effectiveness of this procedure on CrSeTe, whose energies of all the possible structures

were obtained from DFT calculations. Fig. 2a–c show the comparison between the predictions from the SVR model and DFT calculations, while the training data increase from 200 to 400. With

increasing training data, the SVR model becomes increasingly accurate. The energy sequences of the different structures from the SVR predictions are almost the same as those from the DFT

calculations, especially in the low-energy region. Furthermore, with the increasing of training data from 300 to 400, the most stable structure predicted with SVR model remain unchanged.

Subsequently, we examined the energy sequence in the low energy region and calculated R2 for the 20 points predicted to have lowest structures. Fig. 2d shows the energy sequence in the

low-energy region predicted with the SVR model and its comparison with DFT calculations. The R2 for these 20 points is larger than 0.9. Thus, we can regard our predicting procedure is

converged. The SVR model successfully predicted the most stable structure of CrSeTe and the energy comparison between the prediction from SVR model and DFT calculations for other magnetic

materials can be found in Fig. S17-S22 and Table S4 in the Supplemental Material. These results demonstrate that the SVR model is sufficient for identifying the most stable structure of MXY.

The computational cost was reduced from thousands to 400, which significantly accelerated our investigation of MXY materials. Based on machine-learning-assisted high-throughput

calculations, we obtained the most stable structure of MXY (M=Ti, V, Cr, Mn, Ni, Zr, Nb, Hf, W, Re, Pd; X, Y = S, Se, Te). The most stable structure does not require atoms to be arranged in

layers, and in addition to the Janus structure, there are two alternative structures that can have the lowest energy. We named the newly discovered structures “anti-Janus” due to the

opposite atomic stacking method. The Janus structures exhibit order in the out-of-plane z-direction, while the anti-Janus structures exhibit in-plane order. In this work, we used Structure I

(S1) to represent the Janus structure, while the anti-Janus structures are represented by Structure II (S2) and Structure III (S3). S2 and S3 in the 1-T phase are illustrated in Fig. 3a, b.

The S2 and S3 atoms in the 2-H phase are illustrated in Fig. 3c, d. The X atoms are clearly arranged alternately with the Y atoms, which is different from the Janus structure. The energies

of MXY in the S1, S2 and S3 structures are shown in Fig. 4, and they are divided into three categories according to the ground state. The color represents the energy relative to the ground

state. The Janus structure (S1) has the lowest energy only in TiXY, ZrXY, and CrXY (X, Y = S, Se, Te, X ≠ Y), while either the S2 or S3 structure is the ground state in the other systems.

The energy differences between different structures range from 0 to 0.5 eV/unit cell. Although we have predicted that the ground state of WSSe is an S3 structure, the energy difference

between the Janus structure and ground state is less than 0.1 eV/unit cell. Thus, it is still possible to synthesize these materials in Janus structures. The energy differences between the

Janus structure and ground state in HfSSe and HfSeTe are also less than 0.1 eV/unit cell, which means that they can also be synthesized in experiments. SPIN DYNAMICS OF JANUS AND ANTI-JANUS

STRUCTURE After we obtained the ground state of different MXY compounds, we focused on their magnetic properties and calculated their electronic band structures. In Figs. S1-S11, all the

electronic band structures are illustrated along the symmetry lines of the Brillouin zone (BZ). Ferromagnetism only exists in eight systems (e.g., CrSTe, CrSeTe, VSSe, VSTe, VSeTe, MnSSe,

MnSTe, and MnSeTe). The electronic band structures of the magnetic system are illustrated in Figs. S1 and S2. The spin-up and spin-down bands are distinguished in different colors. It should

be noted that the conduction band in these systems crosses the Fermi level, which indicates a clear metallic state. Moreover, the spin-up and spin-down bands of CrSeTe behave as conductors

and inductors, respectively, which indicates that CrSeTe is a half-metal. The narrow gap between the valence band and conduction band in MnSSe indicates that MnSSe is a semimetal, as shown

in Fig. S2d. The systems without magnetic field are illustrated in Figs. S3-S9. We also calculated the total and partial spin-polarized density of states (DOS) of the magnetic MXY, as shown

in Fig. 5, which confirms that the magnetism of these structures mainly originates from the Mn-d, Cr-d and V-d orbitals. The ground states of all these magnetic structures are ferromagnetic

order states, and the details are shown in Fig. S11. Solid magnetic behavior is beneficial for skyrmions; thus, we only focused on magnetic systems. To explore the magnetic properties of

these three structures, we calculated the parameters of the following Hamiltonian model, which describes the spins of magnetic atoms in the hexagonal structure: $$H=-\sum _{ < i,j >

}{D}_{{ij}}\,{{\cdot }}\,\left({S}_{i}\times {S}_{j}\right)-J\sum _{ < i,j > }\left({S}_{i}\,{{\cdot }}\,{S}_{j}\right)-K\sum _{i}{\left({S}_{i}^{z}\right)}^{2}-\mu B\sum

_{i}{S}_{i}^{z}$$ (1) where _S_i (_S_j) is the unit vector representing the orientation of the spin of the _i-_th (_j-_th) site, and _<i, j_> refers to nearest-neighbor magnetic atom

pairs. \({D}_{{ij}}\), \(J\) and \(K\) are the parameters for the DMI, Heisenberg exchange coupling and single-ion anisotropy, respectively. The last term is the Zeeman interaction, where

\(\mu\) and \(B\) represent the magnetic moment of the magnetic atoms and the external magnetic field, respectively. Note that we have only considered the first nearest-neighboring

interaction of the magnetic atomic pairs. According to Moriya’s symmetry rules4,5, the Dzyaloshinskii –Moriya vector can be written as

\({{\boldsymbol{d}}}_{{ij}}={d}_{{||}}({\boldsymbol{z}}\times {{\boldsymbol{u}}}_{{ij}})+{d}_{\perp }{\boldsymbol{z}}\), where \({\boldsymbol{z}}\) is the unit vector normal to the plane,

and \({{\boldsymbol{u}}}_{{ij}}\) is the unit vector points from site _i_ to site _j_. \({d}_{{||}}\) is the in-plane microscopic DMI component, and \({d}_{\perp }\) is the out-of-plane

component. The contribution of \({d}_{\perp }\) to the Néel skyrmions is negligible; hence, we only evaluate the values of \({d}_{{||}}\). The DMI for one magnetic atom from the nearest six

magnetic atoms is equal to \(\pm \frac{3}{2}{d}_{{||}}\) (the sign ± corresponds to the left-hand or right-hand spin-spiral configuration, as shown in Fig. 1a). For a 4 × 1 × 1 supercell

with four magnetic atoms, we have \({E}_{L}-{E}_{R}=4\times \left[\frac{3}{2}{d}_{{||}}-\left(-\frac{3}{2}{d}_{{||}}\right)\right]\). The in-plane component can be derived as

\({d}_{{||}}=\,\frac{{E}_{L}-{E}_{R}}{12}\)32. This spin spiral approach is effective for calculating the DMI in ferromagnetic/heavy-metal (FM/HM) heterostructures, such as Co/Pt32 and

Fe/Ir(111)33 thin films, and B20-type chiral magnets, such as FeGe34. Its disadvantage lies in the fact that spin spiral calculations rely on the spiral state with collective spin rotations,

and one must consider several spirals with different wavelengths to obtain the DMI beyond the nearest neighbors, which is often very cumbersome35. The specific method used to calculate _J_

and _K_ is described in the Supplemental Material. In the present work, the results of the previous magnetic Janus systems are reproduced, as shown in Table 1, and the magnetic parameters of

the corresponding lowest energy S2 and S3 are calculated, as shown in Table 2. Notably, MnSTe-S2 and MnSeTe-S2 still have strong DMI, and the magnitudes of the DMIs reach 4.86 and 4.21 meV,

respectively, which are even greater than those of the Janus configuration. After confirming the stability of MnXY-S2 by their phonon spectrum (Fig. 6), we analyzed the contributions of

different atoms to the spin–orbit coupling. The atomic-layer-resolved localization of the DMI associated SOC energy for Janus MnXY and MnXY-S2 are shown in Fig. 7. MnXY-S2 enhanced the

contribution from Te atoms to spin–orbit coupling and led to a stronger DMI. The DMI values obtained in this work are comparable to those of many state-of-the-art FM/HM heterostructures with

chiral spin textures, e.g., Co/Pt (∼3.0 meV)32 and Fe/Ir(111) (∼1.7 meV)33 thin films. Therefore, it is possible to realize chiral spin textures in MnXY-S2 monolayer structures. To further

verify the aforementioned scenario, spin dynamics simulations based on the magnetic parameters extracted from DFT were performed for the MnXY-S2 case. For MnSTe-S2, our simulations reveal

that the coexistence of helical domains and a small number of skyrmions occurs under zero magnetic field, as shown in Fig. 8a, demonstrating a rich spin texture. More importantly, we found

that the application of an external magnetic field can induce a large number of skyrmions, as in many magnetic systems with skyrmions36,37,38. Applying a field shrinks the helical domains

and causes the size and quantity of skyrmions to first peak and then decrease until the external magnetic field is large enough to annihilate the magnetic state into a single domain.

Compared with that of the anti-Janus structure, the spin texture of the Janus MnSTe is relatively monotonous at zero field, and the critical magnetic field for the annihilation of skyrmions

is relatively small, as shown in Fig. 8b. The skyrmions in Janus MnSTe almost disappear as the external magnetic field reaches 3 T, while in MnSTe-S2, skyrmions still exist even when the

external magnetic field is as large as 10.0 T. A similar scenario can also be seen in MnSeTe, as shown in Fig. S10. The critical magnetic field of MnSeTe-S2 is above 5.0 T, and at the same

time, a skyrmions bag phenomenon occurs at 0 T. Meanwhile, to verify that skyrmions are stable rather than metastable, we obtained the magnetic textures of MnSeTe through Monte Carlo

simulations. As shown in Fig. S13, the results of both are almost identical, proving the stability of skyrmions. These results indicate that the ground states of MnSTe and MnSeTe are more

suitable for spin-orbitronic devices under intense magnetic fields that rely on magnetic skyrmions than Janus MnSTe and MnSeTe. DISCUSSION In summary, we present machine-learning-assisted

high-throughput calculations to explore the ground-state configuration of MXY compounds and search for TMDs with strong DMIs. Not all configurations can be stabilized in the Janus layered

configuration, and two anti-Janus structures (S2 and S3) can have a lower energy state. This result can be used to explain why 2D Janus TMDs materials are difficult to synthesize

experimentally and to guide future works on synthetic TMDs. Next, we predicted 8 TMDs, and the comparison of the energies of different magnetic orders indicated that all of them had

ferromagnetic metallic states. The DMI coefficients are obtained via first-principles calculations, and we find that the DMI in S2 is even larger than that in the Janus configuration. Spin

dynamics simulations indicate that the anti-Janus structure has a richer spin texture in the absence of an external magnetic field, and skyrmions still exist under strong external magnetic

fields. The present work provides an effective way to discover 2D magnets with strong DMI and suggests two anti-Janus ferromagnetic materials that are desirable for spin-orbitronic and

memory devices. METHODS FIRST-PRINCIPLES CALCULATIONS All first-principles calculations were performed within the framework of spin-polarized density functional theory (DFT) as implemented

in the Vienna ab initio simulation package (VASP)17,18. The electron-core interaction is described by the projected augmented wave (PAW) method39,40,41. The exchange correlation effects are

treated with the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof (PBE)42. To address the strong on-site Coulomb interaction of 3d metals, we employed the GGA + U

method in all calculations43. The U values were tested for all the compounds by comparing the resulting magnetic moment with those obtained by the Heyd–Scuseria–Ernzerhof (HSE) hybrid

functional43,44. For most of the cases, we find that U = 2.2 eV for Mn, U = 2.4 eV for Cr and U = 1.3 eV for V yield magnetic moments in good comparison with the HSE, details in Table S1-S3.

Additionally, we also tested the band structure comparison with HSE, as shown in Fig. S14. The energy cutoff for plane wave expansion is set to 500 eV, and a Γ-centered 20 × 20 × 1 k-point

mesh is adopted for Brillouin zone integration. All the structures are simulated by a slab model with a 20 Å vacuum layer. An energy convergence of 10−6 eV is used, and the structures are

fully relaxed until the Hellmann–Feynman forces on all atoms are less than 10−3 eV/Å. In the subsequent calculations, 9 × 9 × 1 and 7 × 7 × 1 k-points are used for the 2 × 2 × 1 and 3 × 3 × 

1 supercells, respectively. SUPPORT VECTOR REGRESSION As a type of support vector machine (SVM), SVR was introduced by Corinna Cortes in 199545. The aim of SVR is to find a hyperplane that

can minimize the total deviation between sample points and the hyperplane, and only the deviation of sample points out of the decision boundary is considered. The hyperplane in this work can

be written as: $$f\left({\boldsymbol{x}}\right)={{\boldsymbol{\omega }}}^{T}\left({\boldsymbol{x}}\right)+b$$ (2) where \({\boldsymbol{x}}\) is the feature of our problem. In this work, we

select the coordinates of the X or Y atoms in MXY as features, and \(f\left({\boldsymbol{x}}\right)\) is the energy of the structure. Our regression problem can be transformed into finding

the minimum of: $$\frac{1}{2}{\left||{\boldsymbol{\omega }}|\right|}^{2}+C\mathop{\sum }\limits_{i}^{k}l\left(f\left({{\boldsymbol{x}}}_{{\boldsymbol{i}}}\right)-{y}_{i}\right)$$ (3) where

\(\left|\left|{\boldsymbol{\omega }}\right|\right|\) is the L2 norm of \({\boldsymbol{\omega }}\), \(C\) is the penalty coefficient, and \(l\) is the \(\epsilon\)-insensitive loss function,

which can be written as: $$l\,\left(z\right)=\left\{\begin{array}{cc} {0} & {{if}\left|z\right| < \epsilon} \\ {\left|z\right|-\epsilon} & {{otherwise}} \end{array} \right.$$ (4)

After introducing slack variables \({\xi }_{i}\) and \({\hat{\xi }}_{i}\) to Eq. (3), it can be rewritten as: $$min \frac{1}{2}{\left|\left|{\boldsymbol{\omega

}}\right|\right|}^{2}+C\mathop{\sum }\limits_{i}^{k}({\xi }_{i}+{\hat{\xi }}_{i})$$ (5) s.t. \(f\left({{\boldsymbol{x}}}_{{\boldsymbol{i}}}\right)-{y}_{i}\le \epsilon +{\xi }_{i}\),

\({y}_{i}-f\left({{\boldsymbol{x}}}_{{\boldsymbol{i}}}\right)\le \epsilon +{\hat{\xi }}_{i}\), \({\xi }_{i}\ge 0\), \({\hat{\xi }}_{i}\ge 0\). To solve the objective function, we introduced

Lagrange multiplier method. The Lagrange multiplier coefficients are \({\alpha }_{i}\), \({\hat{\alpha }}_{i}\), \({\mu }_{i}\), \({\hat{\mu }}_{i}\). The formula of Lagrange function can be

written as: $$L\left({\boldsymbol{\omega }}{\boldsymbol{,}}\,b{\boldsymbol{,}}{{\xi }_{i},{\hat{\xi }}_{i},\alpha }_{i}{,\hat{\alpha }}_{i},\,{\mu }_{i},\,{\hat{\mu

}}_{i}\right)=\frac{1}{2}{\left|\left|{\boldsymbol{\omega }}\right|\right|}^{2}+C\mathop{\sum }\limits_{i}^{k}({\xi }_{i}+{\hat{\xi }}_{i})-\,\mathop{\sum }\limits_{i}^{k}{\mu }_{i}{\xi

}_{i}-\mathop{\sum }\limits_{i}^{k}{\hat{\mu }}_{i}{\hat{\xi }}_{i}+\mathop{\sum }\limits_{i}^{k}{\alpha }_{i}\left(f\left({{\boldsymbol{x}}}_{{\boldsymbol{i}}}\right)-{y}_{i}-{\xi

}_{i}-\epsilon \right)+\mathop{\sum }\limits_{i}^{k}{\hat{\alpha }}_{i}({y}_{i}-f\left({{\boldsymbol{x}}}_{{\boldsymbol{i}}}\right)-{\hat{\xi }}_{i}-\epsilon )$$ (6) The partial derivatives

of Lagrange function to \({\boldsymbol{\omega }}{\boldsymbol{,}}\,b\), \({\xi }_{i}\) and \({\hat{\xi }}_{i}\) are zero, and we can have: $$\left\{\begin{array}{c}\frac{\partial L}{\partial

{\boldsymbol{\omega }}}=0\,\to {\boldsymbol{\omega }}=\mathop{\sum }\limits_{i}^{k}\left({\hat{\alpha }}_{i}-{\alpha }_{i}\right){{\boldsymbol{x}}}_{{\boldsymbol{i}}}\,\\ \frac{\partial

L}{\partial b}=0\,\to \mathop{\sum }\limits_{i}^{k}\left({\hat{\alpha }}_{i}-{\alpha }_{i}\right)=0\\ \frac{\partial L}{\partial {\xi }_{i}}=0\to \,C={\alpha }_{i}+{\mu }_{i}\\

\frac{\partial L}{\partial {\hat{\xi }}_{i}}=0\to C={\hat{\alpha }}_{i}+{\hat{\mu }}_{i}\end{array}\right.$$ (7) With the above equations, we can transfer the origin issue into a dual

problem, and we can have: $$\max \mathop{\sum }\limits_{i=1}^{k}{y}_{i}\left({\hat{\alpha }}_{i}-{\alpha }_{i}\right)-\epsilon \left({\hat{\alpha }}_{i}+{\alpha }_{i}\right)-\mathop{\sum

}\limits_{i=1}^{k}\mathop{\sum }\limits_{j=1}^{k}\left({\hat{\alpha }}_{i}-{\alpha }_{i}\right)\left({\hat{\alpha }}_{j}-{\alpha

}_{j}\right){{\boldsymbol{x}}}_{i}^{T}{{\boldsymbol{x}}}_{j}$$ (8) $$s.t.\,\left\{\begin{array}{c}\mathop{\sum }\limits_{i=1}^{k}\left({\hat{\alpha }}_{i}-{\alpha }_{i}\right)=0\\ 0\le

{\hat{\alpha }}_{i},{\alpha }_{i}\le C\end{array}\,\right.$$ (9) In nonlinear problems, SVR applies a kernel function to map the nonlinear regression to a higher latitude space. The SVR

regression equation can be written as: $$f\left(x\right)=\mathop{\sum }\limits_{i=1}^{k}\left({\hat{a}}_{i}-{a}_{i}\right)K\left({x}_{i},x\right)+b$$ (10) where K is the kernel function, and

Here, Gaussian kernel function \(K\left({x}_{i},{x}_{j}\right)=\exp \left(-\frac{{{||}{x}_{i}-{x}_{j}{||}}^{2}}{2{\sigma }^{2}}\right)\) is used. SPIN DYNAMIC SIMULATIONS Using the magnetic

parameters determined by first-principles calculations, we performed spin dynamics simulations using the VAMPIRE package to simulate the steady state and dynamics of the magnetization

textures46. During the simulation, we use a 100 × 100 supercell with periodic boundary conditions, and the size of the supercell is large enough to eliminate finite-size effects. An external

magnetic field is applied in the z-direction from 0 T to 10 T. DATA AVAILABILITY The data that support the findings of the work is in the manuscript’s main text and Supplementary

Information. Additional data are available from the corresponding author upon request. CODE AVAILABILITY The central codes used in this paper are VASP and VAMPIRE. Source code of machine

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nanomaterials. _J. Phys. Condens. Matter._ 26, 103202 (2014). Article  CAS  PubMed  Google Scholar  Download references ACKNOWLEDGEMENTS This work was supported by the National Program on

Key Basic Research Project (Grant No. 2022YFB3807601) and the National Natural Science Foundation of China (Grant Nos. 12302207, 12272338 and 12192214). Y.Z. acknowledges the financial

support from the National Natural Science Foundation of China (Grant 12102157) and computational support from the Center for Computational Science and Engineering of Lanzhou University.

AUTHOR INFORMATION Author notes * These authors contributed equally: Peng Han, Jingtong Zhang. AUTHORS AND AFFILIATIONS * Department of Engineering Mechanics and Key Laboratory of Soft

Machines and Smart Devices of Zhejiang Province, Zhejiang University, 38 Zheda Road, Hangzhou, 310027, China Peng Han, Shengbin Shi & Jie Wang * Research Center for New Materials

Computing, Zhejiang Laboratory, Hangzhou, 311100, Zhejiang, China Jingtong Zhang, Yunhong Zhao & Jie Wang * Key Laboratory of Mechanics on Disaster and Environment in Western China,

Ministry of Education of China, Lanzhou University, Lanzhou, Gansu, 730000, China Yajun Zhang * Department of Mechanics and Engineering Sciences, College of Civil Engineering and Mechanics,

Lanzhou University, Lanzhou, Gansu, 730000, China Yajun Zhang Authors * Peng Han View author publications You can also search for this author inPubMed Google Scholar * Jingtong Zhang View

author publications You can also search for this author inPubMed Google Scholar * Shengbin Shi View author publications You can also search for this author inPubMed Google Scholar * Yunhong

Zhao View author publications You can also search for this author inPubMed Google Scholar * Yajun Zhang View author publications You can also search for this author inPubMed Google Scholar *

Jie Wang View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS J.W. and Y.Z. conceived and supervised this project. P.H., J.Z., S.S. and Y.Z.

carried out the calculation and analysis of the result. P.H. and J.Z. wrote the paper. P.H. and J.Z. are listed as co-first author. All authors discussed the results and reviewed the

manuscript. CORRESPONDING AUTHORS Correspondence to Yajun Zhang or Jie Wang. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. ADDITIONAL INFORMATION

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ARTICLE Han, P., Zhang, J., Shi, S. _et al._ Machine learning assisted screening of two dimensional chalcogenide ferromagnetic materials with Dzyaloshinskii Moriya interaction. _npj Comput

Mater_ 10, 232 (2024). https://doi.org/10.1038/s41524-024-01419-y Download citation * Received: 20 March 2024 * Accepted: 06 September 2024 * Published: 03 October 2024 * DOI:

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