Learning device, estimation device, learning method, estimation method, and program

Abstract

The objective is to improve the accuracy of total energy estimation by using neural network potential (NNP). One aspect of the present invention is a training device comprising a control unit for training an estimation model, which is a mathematical model for estimating the energy of atoms to be evaluated on the basis of feature quantities to be evaluated represented from first to u-th sums, wherein the u-th sum is the sum of: a three-dimensional wave function representing a u-th atomic orbit starting from an orbit having a maximum occupancy level as an atomic orbit of an atom to be evaluated located in a system comprising one or more atoms, a three-dimensional wave function of each u-th atomic orbit from an orbit having a maximum occupancy level with an atomic orbit of each atom that is within a predetermined distance range from the target to be evaluated among atoms that are different from the target to be evaluated in the system, and in the training, a three-dimensional wave function of each u-th atomic orbit from the orbit having the maximum occupancy level is obtained. The estimation model is updated such that the difference between the sum of the results obtained by executing the estimation model on each atom in the system and the energy of the system estimated by the density functional theory for the system is reduced.

Description

Technical Field

[0001] This disclosure relates to a learning device, an estimation device, a learning method, an estimation method, and a procedure.

[0002] This application is based on and claims priority to Japanese Patent Application No. 2024-056799, filed with the Japan Intellectual Property Office on March 29, 2024, the disclosure of which is incorporated herein by reference in its entirety. Background Technology

[0003] Density functional theory (DFT) is used to study the properties of materials in order to develop new drugs or improve the performance of semiconductor devices. However, DFT calculations are computationally expensive and limited to a finite cell size of about 200 atoms (the repeating unit lattice size of a crystal) that can be used to study composition, making it impossible to study the properties of materials in larger systems. For example, to reproduce the composition of a solid electrolyte with trace amounts of dopants, a cell size of 500 atoms or more is required, making it even more difficult to fully understand the material.

[0004] Therefore, numerous studies are underway to reduce computational costs by constructing neural network-based interatomic potential models (Patent Document 1 and Non-Patent Documents 1 to 3). Here, training data is generated from density functional theory calculations of relatively small systems, and a generalized mathematical model called a neural network potential (NNP) is created to accurately reproduce the results. Compared to density functional theory calculations, the use of NNPs significantly reduces the computational cost of interatomic forces and total energy. Therefore, it has been applied to molecular dynamics calculations of large systems.

[0005] <Related Technical Documents>

[0006] [Patent Literature]

[0007] Patent Document 1: US Patent Publication No. 2022 / 0207393

[0008] [Non-patent literature]

[0009] Non-Patent Literature 1: J. Behler and M. Parrinello, “Generalized Neural Network Representations of High-Dimensional Potential Energy Surfaces”, Physical Review Letters, Vol. 98, p. 146401 (2007)

[0010] Non-Patent Literature 2: AP Bartok et al., “Gaussian approximation potential: accuracy of quantum mechanics without electrons,” Physical Review Letters, Vol. 104, p. 136403 (2010).

[0011] Non-patent literature 3: AP Thompson et al., “Spectral nearest neighbor analysis for automatic generation of quantum accurate interatomic potentials”, Journal of Computational Physics, 285(2015): 316-330. Summary of the Invention

[0012] Technical issues

[0013] The neural network potentials (NNPS) proposed to date have low accuracy in predicting total energy, making it difficult to analyze the properties of materials involved in calculating total energy, such as the material's formation energy or elastic properties.

[0014] In view of the above, the present invention aims to provide a technique for improving the accuracy of total energy estimation by neural network potential (NNP).

[0015] Technical solution

[0016] One aspect of this disclosure provides a learning apparatus including a control unit for performing the learning of an estimation model, the estimation model being a mathematical model for estimating the energy of an analytical target atom based on evaluation target features indicating first and up to u, wherein u is the sum of the following two items: a three-dimensional (3D) wave function (u is an integer equal to or greater than 1 and equal to or less than U, and U is an integer 1 or greater) representing the u-th atomic orbital in the atomic orbitals of the evaluation target atom located in a system comprising one or more atoms, ordered by the occupied energy potential, and the 3D wave function of the u-th atomic orbital in the atomic orbitals of each of the other atoms in the system located within a predetermined distance from the evaluation target atom, ordered by the occupied energy potential.

[0017] During the learning process, the estimation model is updated to reduce the difference between the sum of the results of applying the estimation model to each atom in the system and the energy of the system estimated by density functional theory for the system.

[0018] One aspect of this disclosure provides an estimation apparatus comprising an estimation unit for performing estimation using a trained estimation model acquired by a learning device. The learning device includes a control unit for learning the estimation model, which is a mathematical model for estimating the energy of an analytical target atom based on evaluation target features indicating first and up to u, wherein u is the sum of two terms: a three-dimensional (3D) wavefunction (u is an integer equal to or greater than 1 and equal to or less than U, and U is an integer 1 or greater) of the u-th atomic orbital in the atomic orbitals of the evaluation target atom located in a system comprising one or more atoms, ordered by descending occupied energy potentials; and a 3D wavefunction of the u-th atomic orbital in the atomic orbitals of each of the other atoms in the system located within a predetermined distance from the evaluation target atom, ordered by descending occupied energy potentials.

[0019] In the learning process, the estimation model is updated to reduce the difference between the sum of the results of applying the estimation model to each atom in the system and the energy of the system estimated by density functional theory for the system.

[0020] One aspect of this disclosure provides a learning method performed by a learning device including a control unit for performing the learning of an estimation model, the estimation model being a mathematical model for estimating the energy of an analytical target atom based on evaluation target features indicating first and up to u, wherein u is the sum of the following two items: a three-dimensional (3D) wave function (u is an integer equal to or greater than 1 and equal to or less than U, and U is an integer 1 or greater) representing the u-th atomic orbital in the atomic orbitals of the evaluation target atom located in a system comprising one or more atoms, in descending order of occupied energy potential, and the 3D wave function of the u-th atomic orbital in the atomic orbitals of each of the other atoms in the system within a predetermined distance from the evaluation target atom, in descending order of occupied energy potential.

[0021] In the learning process, the estimation model is updated to reduce the difference between the sum of the results of applying the estimation model to each atom in the system and the energy of the system estimated by density functional theory for the system.

[0022] The learning method includes a control step in which the control unit performs the learning of the estimation model.

[0023] One aspect of this disclosure provides an estimation method performed by an estimation device comprising an estimation unit for performing estimation using a trained estimation model acquired by a learning device. The learning device includes a control unit for learning the estimation model, which is a mathematical model for estimating the energy of an analytical target atom based on evaluation target features indicating first and up to u, wherein u is the sum of two terms: a three-dimensional (3D) wavefunction (u is an integer equal to or greater than 1 and equal to or less than U, and U is an integer 1 or greater) of the u-th atomic orbital in the atomic orbitals of the evaluation target atom located in a system comprising one or more atoms, ordered by descending occupied energy potentials; and a 3D wavefunction of the u-th atomic orbital in the atomic orbitals of each of the other atoms in the system located within a predetermined distance from the evaluation target atom, ordered by descending occupied energy potentials.

[0024] In the learning process, the estimation model is updated to reduce the difference between the sum of the results of applying the estimation model to each atom in the system and the energy of the system estimated by density functional theory for the system.

[0025] The estimation method includes an estimation step in which the estimation unit performs the estimation using the trained estimation model.

[0026] One aspect of this disclosure provides a program for using a computer as a learning device.

[0027] One aspect of this disclosure provides a program for using a computer as an estimation device.

[0028] Beneficial effects

[0029] According to this disclosure, the accuracy of total energy estimation using neural network potentials (NNPs) can be improved. Attached Figure Description

[0030] Figure 1 This is a diagram illustrating the estimation system according to an implementation method.

[0031] Figure 2 This is a diagram illustrating an example of the hardware configuration of a learning device according to an embodiment.

[0032] Figure 3 This is a flowchart illustrating an example of a processing flow performed by a learning device according to an embodiment.

[0033] Figure 4 This is a diagram illustrating an example of the hardware configuration of an estimation device according to an embodiment.

[0034] Figure 5 This is a flowchart illustrating an example of a processing flow performed by an estimation device according to an embodiment.

[0035] Figure 6 The first figure shows an example of experimental results according to the implementation method.

[0036] Figure 7 The second figure shows an example of experimental results according to the implementation method. Detailed Implementation

[0037] Figure 1 This is a diagram illustrating an estimation system 100 according to an embodiment. The estimation system 100 includes a learning device 1 and an estimation device 2. The estimation device 2 performs estimation using a trained mathematical model acquired by the learning device 1. Specifically, the trained mathematical model acquired by the learning device 1 is referred to as a trained estimation model as described below. The trained estimation model is a neural network potential (NNP).

[0038] The learning device 1 includes a control unit 11 having a processor 91 (e.g., a central processing unit (CPU), graphics processing unit (GPU), or neural network processing unit (NPU)) and a memory 92 connected via a bus. For example, the control unit 11 performs learning processing. Learning processing is the process of acquiring a trained estimation model by training an estimation model as a target mathematical model until a predetermined condition for the end of learning (hereinafter referred to as the "learning termination condition") is met.

[0039] The estimation model is a mathematical model that estimates the energy of the target atom based on the evaluation target characteristic quantities. The evaluation target characteristic quantities are quantities indicating the first sum to the u-th sum (u is an integer equal to or greater than 1 and equal to or less than U, where U is an integer of 1 or greater). The u-th sum is the sum of the three-dimensional (3D) wavefunction representing the orbital of the u-th central atom and the 3D wavefunction representing the orbital of the u-th peripheral atom.

[0040] The u-th central atomic orbital is the atomic orbital of the target atom located within the system to be analyzed (hereinafter referred to as the "analytical target system"), which includes one or more atoms, and is the u-th atomic orbital in descending order of occupied energy potential.

[0041] The u-th outermost atomic orbital is the atomic orbital of each of the surrounding atoms within a predetermined distance from the target atom, and is ordered in descending order of occupied energy potential. Surrounding atoms are atoms in the target system other than the target atom being evaluated.

[0042] For example, when the target system being analyzed is Li6PS5Cl and the target atom being evaluated is S, the surrounding atoms are, for example, Li, P, and Cl. In this case, when u=1, the u-th central atomic orbital is a 3p orbital, the u-th central atomic orbital of one of the surrounding atoms, Li, is a 2s orbital, and the u-th central atomic orbitals of the other surrounding atoms, P and Cl, are 3p orbitals.

[0043] <Example of evaluating target characteristic quantity>

[0044] The target feature quantity to be evaluated is a vector, for example, represented by equation (1) below. Meanwhile, in equation (1) where u=2, the number of elements in the vector is 2, but this does not depend on the value of u, and the number of elements is not limited to 2. The dimension of the vector indicating the target feature quantity to be evaluated is u. In the case of u=1, the target feature quantity to be evaluated is a scalar.

[0045] [Formula 1]

[0046]

[0047] [Equation 2]

[0048]

[0049] Here, (n, l) i Let n represent the principal quantum number n and the angular momentum quantum number l of the i-th atom. N represents the number of atoms within a sphere of a predetermined radius from the center of the i-th atom. The predetermined radius is, for example, 7. . ρ i Let ρ represent the wavefunction of the i-th atom. j Let r represent the wavefunction of the j-th atom. The j-th atom is different from the i-th atom. R represents the position in 3D space. i This represents the position of the i-th atom in 3D space.

[0050] For example, the target characteristic quantity for evaluating sulfur atoms where u=2 is represented by the following equation (3).

[0051] [Equation 3]

[0052]

[0053] In the learning process, for each atom in the target system being analyzed, the energy of the target atom is estimated based on the target characteristic quantities of each atom to be evaluated. The energy estimation process for the target atom based on the target characteristic quantities of the target atom is then performed by the estimation model.

[0054] In addition, during the learning process, the estimation model is updated to reduce the difference between the sum of the estimated energies (i.e., the results of performing the estimation model) and the energy of the analytical target system estimated by density functional theory.

[0055] Therefore, in training the estimation model, a pair of information indicating the evaluation target feature quantity of each atom in the target system (hereinafter referred to as "training input data") and information indicating the energy of the target system estimated by density functional theory (hereinafter referred to as "training labels") are used during training. The training labels are labels indicating the correct answer. Therefore, this pair is called labeled training data.

[0056] The density functional theory analysis of the target system can be performed by the control unit 11 during the learning process, or it can be performed by a device other than the learning device 1 before the learning process, and it can be performed synchronously by a device other than the learning device 1 and the learning process of the control unit 11.

[0057] Meanwhile, the 3D wave function can be, for example, a 3D radial wave function.

[0058] Meanwhile, the target system being analyzed can be, for example, a part of a semiconductor crystal. A semiconductor crystal is, for example, Li. 24 P4S 20 Cl4.

[0059] At the same time, the learning termination condition can include any condition related to the end of learning, such as the condition that the estimated model as the target mathematical model is updated a predetermined number of times, the condition that the change caused by the update of the estimated model is less than a predetermined change, or the condition that all the pre-prepared labeled or unlabeled training data has been used.

[0060] <Regarding the effectiveness of learning processing>

[0061] As mentioned above, the 3D wave function is used for learning. Meanwhile, the NNP technique described in Patent Document 1 uses the charge density (the sum of squares of the moduli of a one-dimensional (1D) radial wave function that is converted into 1D information by solid angle integration (in descending order of occupied potential energy from the maximum occupied energy level to the u-th (where u is an integer between 1 and U, and U is an integer of 1 or greater)) instead of the 3D wave function.

[0062] However, when the position of an atom changes in 3D space, the distribution of the wave function in 3D space also changes. Therefore, when learning device 1 learns using values ​​with information from the 3D wave function, a trained mathematical model that takes into account the position information of the atom is generated. Conversely, when charge density, converted to 1D information, is used instead of the 3D wave function, it is not possible to adequately represent the change in the distribution of the wave function in 3D space as the atom position changes, because it is not 3D.

[0063] Therefore, the learning device 1 can acquire a mathematical model with higher estimation accuracy than the technology described in Patent Document 1. Because the energy estimation accuracy of each atom is high, the energy estimation accuracy of the target system analyzed using the trained mathematical model acquired by the learning device 1 is also high.

[0064] <Example of hardware configuration for learning device 1>

[0065] Figure 2 This is a diagram illustrating an example of the hardware configuration of the learning device 1 according to an embodiment. The learning device 1 includes a control unit 11 having a processor 91 and a memory 92 connected via a bus, and executes a program. By executing the program, the learning device 1 functions as an apparatus including the control unit 11, an interface unit 12, and a storage unit 13.

[0066] More specifically, the processor 91 reads the program stored in the storage unit 13 and stores the read program in the memory 92. When the processor 91 executes the program stored in the memory 92, the learning device 1 functions as a device including the control unit 11, the interface unit 12, and the storage unit 13.

[0067] Control unit 11 controls the operation of each functional unit of learning device 1. For example, control unit 11 acquires a pair of training input data and training correct data through interface unit 12. For example, control unit 11 performs learning processing. Therefore, for example, control unit 11 executes an estimation model. For example, control unit 11 acquires information stored in storage unit 13. Specifically, the process for acquiring information stored in storage unit 13 is reading.

[0068] Interface unit 12 includes a communication interface for connecting learning device 1 to an external device. Interface unit 12 communicates with the external device via wired or wireless means. The external device is, for example, a transmitter source device providing pairs of training input data and training labels. In this case, interface unit 12 acquires pairs of training input data and training labels through communication with the transmitter source device providing the pairs of training input data and training labels.

[0069] Interface unit 12 may include an input device, such as a mouse, keyboard, or touch panel. Interface unit 12 can be configured to connect the input device to the interface of learning device 1. Interface unit 12 receives various types of information input about learning device 1 via the input device, either wired or wirelessly. Simultaneously, various types of data acquired through the communication interface of interface unit 12 can be input to the input device of interface unit 12, rather than through the communication interface of interface unit 12.

[0070] Interface unit 12 outputs various types of information. Interface unit 12 includes a display device, such as a cathode ray tube (CRT) display, a liquid crystal display (LCD), or an organic electroluminescent (EL) display. Interface unit 12 can be configured to connect the display device to the interface of learning device 1. For example, the display device output of interface unit 12 may be input to the communication interface or input device of interface unit 12.

[0071] Storage unit 13 includes a non-transitory computer-readable recording medium, such as a magnetic hard disk or semiconductor memory. Storage unit 13 stores various types of information about learning device 1. For example, storage unit 13 stores various types of information generated by the operation of control unit 11. For example, storage unit 13 stores information acquired by interface unit 12.

[0072] Figure 3 This is a diagram illustrating an example of the processing flow performed by the learning device 1 according to an embodiment. The control unit 11 acquires a pair of training input data and training labels (step S101). Subsequently, the control unit 11 performs a learning process (step S102). A trained estimation model is obtained by performing the learning process.

[0073] <Example of hardware configuration for estimation device 2>

[0074] Figure 4 This is a diagram illustrating an example of the hardware configuration of the estimation device 2 according to an embodiment. The estimation device 2 includes a control unit 21 (an example of an estimation unit) having a processor 93 and a memory 94 connected via a bus, and executes a program. By executing the program, the estimation device 2 functions as a device including the control unit 21, an interface unit 22, and a storage unit 23.

[0075] More specifically, the processor 93 reads the program stored in the storage unit 23 and stores the read program in the memory 94. When the processor 93 executes the program stored in the memory 94, the estimation device 2 acts as a device including the control unit 21, the interface unit 22, and the storage unit 23.

[0076] Control unit 21 controls the operation of each functional unit of estimation device 2. For example, control unit 21 executes a trained estimation model. Control unit 21 acquires, for example, information input to the trained estimation model (hereinafter referred to as "estimation input data").

[0077] For example, the control unit 21 acquires information stored in the storage unit 23. Specifically, the process for acquiring the information stored in the storage unit 23 is reading.

[0078] Interface unit 22 includes a communication interface for connecting estimation device 2 to an external device. Interface unit 22 communicates with the external device via wired or wireless means. The external device is, for example, a transmitter source device for estimating input data. In this case, interface unit 22 acquires the estimating input data through communication with the transmitter source device. The external device is, for example, a learning device 1. In this case, interface unit 22 acquires a trained estimation model through communication with the learning device 1.

[0079] Interface unit 22 may include an input device, such as a mouse, keyboard, or touch panel. Interface unit 22 may be configured to connect the input device to the interface of estimation device 2. Interface unit 22 receives various types of information input about estimation device 2 via the input device, either wired or wirelessly. Simultaneously, various types of data acquired by the communication interface of interface unit 22 may be input to the input device of interface unit 22, rather than to the communication interface of interface unit 22.

[0080] Interface unit 22 outputs various types of information. Interface unit 22 may include, for example, a display device such as a CRT monitor, a liquid crystal display, or an organic EL display. Interface unit 22 can be configured to connect the display device to the interface of estimation device 2. For example, the display device output of interface unit 22 may be information input to the communication interface or input device of interface unit 22.

[0081] Storage unit 23 includes a non-transitory computer-readable recording medium, such as a magnetic hard disk or semiconductor memory. Storage unit 23 stores various types of information about the estimation device 2. For example, storage unit 23 stores various types of information generated by the operation of control unit 21. For example, storage unit 23 stores information acquired by interface unit 22.

[0082] Figure 5 This is a flowchart illustrating an example of the processing flow performed by the estimation device 2 according to an embodiment. The control unit 21 acquires estimation input data (step S201). Subsequently, the control unit 21 executes a trained estimation model on the acquired estimation input data (step S202). Through the processing in step S202, the result of estimating the estimation input data is estimated. As described above, in step S202, the control unit 21 performs estimation using the trained estimation model.

[0083] <Experimental Results>

[0084] An example of experimental results validating the accuracy of the estimation using a trained estimation model is shown. The number of wave functions used in each element is 2 (u=2). The experiments validated the estimation accuracy of each of the NNP (hereinafter referred to as the "first technique"), NNP (hereinafter referred to as the "second technique"), and the trained estimation model. The NNP estimates the energy of the target system based on the Bayreu-Palinello type descriptor, and the NNP estimates the energy of the target system based on electron density. In the case of using charge density, it is a single function, i.e., u=1. Meanwhile, the second technique estimates the energy based on electron density, but more specifically, it estimates the energy based on 3D electron density by processing the electron density in three dimensions.

[0085] Figure 6 The first figure shows an example of experimental results according to an implementation method, and Figure 7 This is a second figure illustrating an example of experimental results according to an implementation method. More specifically, Figure 6 This is a graph showing the root mean square error of the estimation results for each of the first technique, the second technique, and the trained estimation model. Figure 7 The determination coefficients R0 of each of the first technique, the second technique, and the trained estimation model are shown. 2 .

[0086] exist Figure 6 and 7 In this context, "Belle-Parinello" represents the first technique, "density" represents the second technique, and "this method" represents the trained estimation model. Figure 6 and 7 In the text, the "Components" column indicates the target system being analyzed. Therefore, for example, the row "Li..." 24 P4S 20"Cl4" indicates that the target system for analysis is a semiconductor crystal (LI). 24 P4S 20 CL4.

[0087] Figure 6 The results show that the estimation accuracy of the trained estimation model is higher than that of the first and second techniques, regardless of the target system being analyzed. This is because higher R0... 2 To provide higher accuracy, therefore Figure 7 The results show that the estimation accuracy of the trained estimation model is higher than that of the first and second techniques.

[0088] Learning device 1 acquires a trained estimation model through a learning process. The learning process provides the effects described in <Regarding the Effects of the Learning Process>. Therefore, learning device 1 can improve the accuracy of total energy estimation via NNP.

[0089] Furthermore, estimation device 2 uses the trained estimation model acquired by learning device 1 to perform estimation. Therefore, the accuracy of the estimation of total energy via NNP can be improved.

[0090] Furthermore, the estimation system 100 includes a learning device 1. Therefore, the accuracy of the total energy estimation performed via NNP can be improved.

[0091] (Variation)

[0092] Meanwhile, the target feature quantity being evaluated indicates the sum of the 3D wave functions as described above. That is, the target feature quantity being evaluated indicates the sum of the wave functions at each location in 3D space. Coordinates are needed to represent each location in space. The coordinates used to represent the target feature quantity being evaluated, which indicates the sum of the 3D wave functions, can be called local coordinates.

[0093] Local coordinates are the defined coordinates of each atom within the target system being analyzed, with each atom serving as the origin. The axes of the local coordinates may include, for example, an axis connecting the nearest atom at the origin to the origin (hereinafter referred to as the "first axis"), an axis connecting the second nearest atom at the origin to the origin (hereinafter referred to as the "second axis"), and an axis orthogonal to the first and second axes (hereinafter referred to as the "third axis"). The local coordinate axes may, for example, consist of three axes in total, including a pair of orthogonal axes different from the first and second axes within the plane spanned by the first and second axes, and the third axis.

[0094] Meanwhile, each of the learning device 1 and the estimation device 2 can be installed using multiple information processing devices that are connected to enable communication via a network. In this case, each process executed by the control unit 11 and each process executed by the control unit 21 can be executed in a distributed manner by multiple information processing devices.

[0095] Meanwhile, all or some of the functions of the learning device 1 and the estimation device 2 can be implemented using hardware such as application-specific integrated circuits (ASICs), programmable logic devices (PLDs), or field-programmable gate arrays (FPGAs). The program can be recorded on a computer-readable recording medium. Such media include, for example, media designed to be portable and easily moved, such as flexible disks, magneto-optical disks, ROMs, or CD-ROMs, or storage devices such as hard disks built into a computer system. The program can be transmitted via electrical communication lines.

[0096] Embodiments of this disclosure have been described in detail above with reference to the accompanying drawings. However, specific configurations are not limited to the disclosed embodiments, and this disclosure includes any other designs without departing from its scope.

Claims

1. A learning device, the learning device comprising: a control unit configured to perform learning of an estimation model that is a mathematical model for energy estimation of an analysis target atom based on evaluation target feature amounts indicating a first sum to a u-th sum, wherein the u-th sum is a sum of a three-dimensional 3D wave function representing a u-th atomic orbital in descending order of an occupation energy potential of an evaluation target atom located in a system including one or more atoms (u is an integer equal to or greater than 1 and equal to or less than U, and U is an integer of 1 or more), and a 3D wave function representing a u-th atomic orbital in descending order of an occupation energy potential of each of other atoms in the system within a predetermined distance range from the evaluation target, wherein, in the learning, the estimation model is updated to reduce a difference between a sum of results of executing the estimation model for each atom in the system and an estimated energy of the system obtained by a density functional theory.

2. The learning device according to claim 1, wherein the 3D wave function is a 3D radial wave function.

3. The learning device according to claim 1, wherein the system is a portion of a semiconductor crystal.

4. An estimation device comprising: an estimation unit configured to perform estimation by a trained estimation model acquired by a learning device, the learning device including a control unit configured to perform learning of the estimation model that is a mathematical model for energy estimation of an analysis target atom based on evaluation target feature amounts indicating a first sum to a u-th sum, wherein the u-th sum is a sum of a three-dimensional 3D wave function representing a u-th atomic orbital in descending order of an occupation energy potential of an evaluation target atom located in a system including one or more atoms (u is an integer equal to or greater than 1 and equal to or less than U, and U is an integer of 1 or more), and a 3D wave function representing a u-th atomic orbital in descending order of an occupation energy potential of each of other atoms in the system within a predetermined distance range from the evaluation target, wherein, in the learning, the estimation model is updated to reduce a difference between a sum of results of executing the estimation model for each atom in the system and an estimated energy of the system obtained by a density functional theory.

5. A learning method executed by a learning device, the learning device including a control unit for executing learning of an estimation model that is a mathematical model for energy estimation of an analysis target atom based on evaluation target characteristic amounts indicating a first and an u-th evaluation target characteristic amount, wherein the u-th sum is a sum of a three-dimensional 3D wave function representing a u-th atomic orbital in descending order of an occupation energy potential of an evaluation target atom located in a system including one or more atoms (u is an integer equal to or greater than 1 and equal to or less than U, and U is an integer of 1 or more), and a 3D wave function representing a u-th atomic orbital in descending order of an occupation energy potential of each of other atoms in the system within a predetermined distance range from the evaluation target, wherein, in the learning, the estimation model is updated to reduce a difference between a sum of results of executing the estimation model for each atom in the system and an estimated energy of the system obtained by a density functional theory, and wherein the learning method includes a control step of performing, by the control unit, learning of the estimation model.

6. An estimation method performed by an estimation device including an estimation unit operable to perform estimation by a trained estimation model acquired by a learning device including a control unit operable to perform learning of an estimation model that is a mathematical model for energy estimation of an analysis target atom based on evaluation target characteristic amounts indicative of a first and an u-th evaluation target characteristic amount, wherein the u-th and is a sum of a three-dimensional 3D wave function of a u-th atomic orbital in descending order of an occupation energy potential of an evaluation target atom located in a system including one or more atoms (u is an integer equal to or greater than 1 and equal to or less than U, and U is an integer of 1 or more), and a 3D wave function of a u-th atomic orbital in descending order of an occupation energy potential of each of other atoms in the system within a predetermined distance range from the evaluation target, wherein, in learning, the estimation model is updated to reduce a difference between a sum of results of executing the estimation model for each atom in the system and an estimated energy of the system obtained by density functional theory, and wherein the estimation method includes an estimation step of performing estimation by the estimation unit through the trained estimation model.

7. A program for causing a computer to function as the learning device according to any one of claims 1 to 3.

8. A program for causing a computer to function as the estimation device according to claim 4.

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