Information processing device, information processing method, and information processing program

Abstract

An information processing device according to an embodiment is provided with at least one memory and at least one processor. The at least one processor: calculates, at each of a plurality of positions in a system to be calculated, a descriptor based on values obtained by convolving results obtained by performing a linear operation on respective eigenfunctions of a plurality of electrons in an equation used in a density functional theory; inputs at least the descriptor to a trained model at each of the plurality of positions and outputs the exchange correlation energy density of each of the plurality of electrons; applies the exchange correlation energy density to the equation at each of the plurality of positions and calculates the eigenfunction of each of the plurality of electrons and the eigenenergy of each of the plurality of electrons; and calculates the energy of the system by using the eigenfunctions and the eigenenergies.

Description

Information processing device, information processing method, and information processing program

[0001] The embodiments of the present disclosure relate to an information processing device, an information processing method, and an information processing program.

[0002] Density Functional Theory (hereinafter referred to as DFT) is a theory for efficiently calculating the state of electrons in materials. DFT is used to calculate physical properties such as energy, electron density, band gap, stability, reaction activation energy, electrical conductivity, and magnetism. For example, DFT is used in practical material research. An equation used in DFT is, for example, the Kohn-Sham equation shown below.

[0003]

[0004] The various symbols in the Kohn-Sham equation shown in formula (1) are known and will not be explained further. The Kohn-Sham equation shown in formula (1) is a condensed version of the Schrodinger equation, which requires a very large amount of calculation. The following formula (2) is a function of the potential V, which is a functional of the electron density at position r, on the left side of formula (1). ｅｆｆ [n](r). ｅｆｆ As [n](r), a known value can be applied, and therefore a detailed description thereof will be omitted.

[0005]

[0006] In equation (2), the potential V in the Kohn-Sham equation (1) ｅｆｆ [n](r) is the exchange-correlation energy or exchange-correlation functional E, which cannot be written rigorously as shown in the following equation (3), in exchange for the compression of the Schrodinger equation. ＸＣ It contains the functional derivative of the electron density n(r) with respect to a term called [n].

[0007]

[0008] Exchange-correlation energy or exchange-correlation functional E ＸＣ[n] is a functional of the electron density (also called electron density distribution). The electron density n(r) at position r is expressed by the following equation (4). As shown in equation (4), the electron density n(r) is defined by the square of the eigenfunction ψ in the Kohn-Sham equation (1). The eigenfunction ψ in the Kohn-Sham equation (1) is called the Kohn-Sham orbital or wave function.

[0009]

[0010] The right-hand side of equation (4) indicates that the sum of the squares of the absolute values ​​of the eigenfunctions ψ is taken for multiple eigenstates i of electrons. The suffix i∈occ. of the sum symbol in equation (4) is a symbol that identifies multiple eigenstates occupied by multiple electrons. occ. means occupancy.

[0011] Exchange-correlation energy or exchange-correlation functional E ＸＣ The accuracy of the approximation of [n] affects the accuracy of calculation of the physical property values ​​obtained by DFT. ＸＣ [n] is defined by the spatial integral of the exchange-correlation energy density of electrons, for example, as shown in the following equation (5).

[0012]

[0013] As shown in equation (5), the exchange-correlation energy density ε ＸＣ [n](r) is a functional of the electron density and depends on the entire electron density spread over the space to be integrated. However, in practice, an approximation (semi-local approximation) that depends only on the electron density or the gradient of the electron density distribution is adopted for the electron exchange-correlation energy density. For example, an example of a semi-local approximation is the Perdew-Burke-Ernzerhof (PBE) method (hereinafter referred to as conventional method 1).

[0014] FIG. 10 shows the relationship between the electron density n(r) and the electron exchange-correlation energy density ε ＸＣ10 is a schematic diagram showing the relationship between the electron exchange-correlation energy density ε[n](r) and the electron exchange-correlation energy density ε[n](r). ＸＣ [n](r) depends only on the local electron density n(r) and the gradient ∇n(r) of the local electron density n(r). ＸＣ [n](r) has the advantage of being easy to construct because it uses analytical formulas derived from physical theory, but it does not include long-range electron correlations. Therefore, it is difficult to improve the calculation accuracy of band gaps, reaction activation energies, magnetism, long-range interactions, etc. for materials analyzed by DFT using Conventional Method 1.

[0015] As a method for improving the calculation accuracy of band gaps, reaction activation energies, magnetism, long-range interactions, etc., for example, the electron exchange-correlation energy density ε ＸＣ There is a method (hereinafter referred to as conventional method 2) in which [n](r) depends on a descriptor R(r) in a format that convolves the surrounding electron density (non-local format).

[0016]

[0017] FIG. 11 shows the electron density n(r) and the electron exchange-correlation energy density ε ＸＣ [n](r). In conventional method 2, w(|r - r'|) in equation (6) is a convolution function related to the convolution of the electron density n(r') around position r. The convolution function may also be referred to as a convolution kernel or weight function. As shown in FIG. 11, the convolution function w(|r - r'|) describes how much the influence of electrons at a distance |r - r'| is taken into account at position r. Due to physical requirements, the convolution function w(|r - r'|) must satisfy the following conditions: spherical symmetry and attenuation with increasing distance from position r. In this case, the convolution function w(|r - r'|) is given, for example, by a Gaussian function exp(-α|r - r'|^2) using a parameter α set depending on the material.

[0018] In conventional method 2, a non-local descriptor R(r) is calculated by performing the convolution integral shown in equation (6). The descriptor R(r) corresponds to an input to a trained model that outputs the exchange-correlation energy density of electrons. Next, in this method, the non-local descriptor R(r), the electron density n(r), and the gradient ∇n(r) of the electron density n(r) are input to the trained model to calculate the exchange-correlation energy density ε ＸＣ [n](r) = ε ＸＣ [n](n(r), ∇n(r), R(r)) can be found.

[0019] In conventional method 2, the exchange-correlation energy density ε of electrons is not calculated by physical theory alone. ＸＣ Since the shape of [n](r) cannot be determined, we use the nonlocal form of the descriptor R(r) and the trained model to determine the electron exchange-correlation energy density ε ＸＣ [n](r). In this way, in conventional method 2, the long-range electron correlation is calculated as the electron exchange-correlation energy density ε ＸＣ [n](r).

[0020] In the conventional method 2, the calculation of the non-local type descriptor R(r) shown in Equation (6) is performed using, for example, an eigenfunction ψ based on a plurality of basis functions. ｉ This is done using the expansion of (r). The following equation (7) is ｉ This is an equation showing the expansion of (r).

[0021]

[0022] φ on the right side of equation (7) ａ (r) indicates each of the basis functions. ａ The subscript a in (r) is a suffix that identifies the basis function. ｉａ is the basis function φ in eigenstate i ａ The expansion coefficients corresponding to (r) are shown. ａ For example, a Gaussian function exp(-β|r-r'|^2) or a plane wave exp(ikr) is used as (r). In this case, the electron density n(r) is calculated as shown in the following equation (8) using equations (4) and (7).

[0023]

[0024] The calculation of the non-local form descriptor R(r) shown in equation (6) is performed using equation (8) by the following equation (9).

[0025]

[0026] As shown on the right side of equation (9), in conventional method 2, the sum symbol Σ ａｂ In the following, for the sake of simplicity, we will use the eigenfunction ψ ｉ Basis functions φ of ａ (r) and φ ｂ Assume that the total number of (r) is N. In this case, as shown in equation (9), in order to calculate the nonlocal form descriptor R(r), it is necessary to perform sum-object integration for combinations of a = 1, ..., N, b = 1, ..., N. Therefore, in conventional method 2, to obtain the nonlocal form descriptor R(r), a number of calculations (computational amount) of O(N^2) is required. In addition, since the Kohn-Sham equation (1) is a self-consistent equation (which may also be referred to as self-consistent), it is necessary to repeatedly perform the calculation of equation (9) multiple times until the calculation of the Kohn-Sham equation (1) is completed.

[0027] For these reasons, when using conventional method 2 to improve the calculation accuracy of band gaps, reaction activation energies, magnetism, long-range interactions, etc., the calculation of equation (9), which has a high calculation cost, must be repeated. Therefore, conventional method 2 has the problem that it takes a long time to calculate the DFT.

[0028] Ryo Nagai, Ryosuke Akashi & Osamu Sugino, “Completing density functional theory by machine learning hidden messages from molecules” npj Computational Materials volume 6, Article number: 43 (2020), https: / / www. nature. com / articles / s41524-020-0310-0

[0029] The problem to be solved by the present disclosure is to realize processing that can calculate physical property values ​​with high accuracy while reducing calculation costs.

[0030] An information processing device according to an embodiment includes at least one memory and at least one processor, wherein the at least one processor calculates, at each of a plurality of positions in a system to be calculated, a descriptor based on a convolution value obtained by performing a linear operation on an eigenfunction of each of a plurality of electrons in an equation used in density functional theory, inputs at least the descriptor to a trained model at each of the plurality of positions to output an exchange-correlation energy density of each of the plurality of electrons, applies the exchange-correlation energy density at each of the plurality of positions to the equation to calculate an eigenfunction of each of the plurality of electrons and an eigenenergy of each of the plurality of electrons, and calculates an energy of the system using the eigenfunctions and the eigenenergies.

[0031] Fig. 1 is a block diagram illustrating an example of a hardware configuration of an information processing device according to an embodiment. Fig. 2 is a diagram illustrating an example of a functional block in a processor according to an embodiment. Fig. 3 is a diagram illustrating an example of a trained model according to an embodiment. Fig. 4 is a diagram illustrating an eigenfunction ψ in the Kohn-Sham equation according to an embodiment. ｉ (r) and the electron exchange-correlation energy density ε ＸＣFIG. 5 is a schematic diagram showing an overview of the relationship between [n](r). FIG. 5 is a flowchart showing an example of a procedure for solution calculation processing according to an embodiment. FIG. 6 is a diagram showing an example of functional blocks in a processor mounted on a learning device according to an embodiment. FIG. 7 is a flowchart showing an example of a procedure for model generation processing according to an embodiment. FIG. 8 is a diagram showing an example of a trained model generated in an application example of an embodiment. FIG. 9 is a flowchart showing an example of a procedure for model generation processing according to an application example of an embodiment. FIG. 10 is a diagram showing an example of a relationship between electron density n(r) and electron exchange-correlation energy density ε in conventional method 1 (e.g., PBE method). ＸＣ 11 is a schematic diagram showing the relationship between the electron density n(r) and the electron exchange-correlation energy density ε ＸＣ [n](r) is a schematic diagram showing the relationship between [n](r) and [n](r).

[0032] Hereinafter, embodiments of an information processing device, an information processing method, and an information processing program will be described in detail with reference to the drawings.

[0033] (Embodiment) Fig. 1 is a block diagram showing an example of the hardware configuration of an information processing device 1 according to this embodiment. As shown in Fig. 1, the information processing device 1 may be connected to an external device 9A via a communication network 5. The information processing device 1 may also include an external device 9B connected via a device interface 39. The information processing device 1 may allow a user to input various parameters related to a substance composed of multiple atoms. The substance is, for example, a molecule. Note that the substance is not limited to molecules and may be various crystals, etc.

[0034] The information processing device 1 includes a computer 30 and an external device 9B connected to the computer 30 via a device interface 39. The computer 30 includes, for example, a processor 31, a main storage device (memory) 33, an auxiliary storage device (memory) 35, a network interface 37, and a device interface 39. The information processing device 1 may be realized as the computer 30 in which the processor 31, the main storage device 33, the auxiliary storage device 35, the network interface 37, and the device interface 39 are connected via a bus 41.

[0035] Although the computer 30 shown in FIG. 1 includes one of each component, it may also include multiple of the same component. Furthermore, while FIG. 1 shows a single computer 30, the software may be installed on multiple computers, with each of the multiple computers executing the same or different parts of the software. In this case, a distributed computing configuration may be used in which each computer communicates via a network interface 37 or the like to execute the processing. In other words, the information processing device 1 in this embodiment may be configured as a system in which one or more computers execute instructions stored in one or more storage devices to realize various functions described below. Furthermore, information transmitted from a terminal may be processed by one or more computers on a cloud, and the processing results may be transmitted to a terminal such as a display device (display unit) corresponding to the external device 9B.

[0036] Various computations of the information processing device 1 in this embodiment may be executed in parallel using one or more processors, or using multiple computers via a network. Furthermore, various computations may be distributed to multiple processor cores within a processor and executed in parallel. Furthermore, some or all of the processes, means, etc. disclosed herein may be executed by at least one of a processor and a storage device provided on a cloud that can communicate with the computer 30 via a network. Thus, various functions described below in this embodiment may be implemented in the form of parallel computing using one or more computers.

[0037] The processor 31 may be an electronic circuit (such as a processing circuit, processing circuitry, CPU (Central Processing Unit), GPU (Graphics Processing Unit), FPGA (Field Programmable Gate Array), or ASIC (Application Specific Integrated Circuit)) including a control device and an arithmetic device of the computer 30. The processor 31 may also be a semiconductor device including a dedicated processing circuit. The processor 31 is not limited to an electronic circuit using electronic logic elements, and may be realized by an optical circuit using optical logic elements. The processor 31 may also include an arithmetic function based on quantum computing.

[0038] The processor 31 performs arithmetic processing based on data and software (programs) input from each device, etc., configured internally of the computer 30, and can output the arithmetic results and control signals to each device, etc. The processor 31 may control each component constituting the computer 30 by executing the OS (Operating System) of the computer 30, applications, etc.

[0039] The information processing device 1 in this embodiment may be realized by one or more processors 31. Here, the processor 31 may refer to one or more electronic circuits arranged on one chip, or may refer to one or more electronic circuits arranged on two or more chips or two or more devices. When multiple electronic circuits are used, the electronic circuits may communicate with each other via wire or wirelessly.

[0040] The main memory device 33 is a memory device that stores instructions executed by the processor 31 and various data, and information stored in the main memory device 33 is read by the processor 31. The auxiliary memory device 35 is a memory device other than the main memory device 33. Note that these memory devices refer to any electronic component capable of storing electronic information, and may be semiconductor memory. The semiconductor memory may be either volatile memory or non-volatile memory. The memory device for saving various data used in the information processing device 1 according to this embodiment may be realized by the main memory device 33 or the auxiliary memory device 35, or may be realized by an internal memory built into the processor 31. For example, the memory unit in this embodiment may be realized by the main memory device 33 or the auxiliary memory device 35.

[0041] Multiple processors may be connected (coupled) to one storage device (memory), or a single processor 31 may be connected. Multiple storage devices (memories) may be connected (coupled) to one processor. When the information processing device 1 in this embodiment is configured with at least one storage device (memory) and multiple processors connected (coupled) to this at least one storage device (memory), it may include a configuration in which at least one of the multiple processors is connected (coupled) to at least one storage device (memory). This configuration may also be realized by storage devices (memories) and processors 31 included in multiple computers. Furthermore, it may include a configuration in which the storage device (memory) is integrated with the processor 31 (for example, a cache memory including an L1 cache and an L2 cache).

[0042] The network interface 37 is an interface for connecting to the communication network 5 wirelessly or via a wire. The network interface 37 may be an appropriate interface, such as one that conforms to an existing communication standard. The network interface 37 may exchange information with an external device 9A connected via the communication network 5. The communication network 5 may be any one of a wide area network (WAN), a local area network (LAN), a personal area network (PAN), or a combination thereof, as long as information is exchanged between the computer 30 and the external device 9A. An example of a WAN is the Internet, an example of a LAN is IEEE 802.11 or Ethernet (registered trademark), and an example of a PAN is Bluetooth (registered trademark) or NFC (Near Field Communication), etc.

[0043] The device interface 39 is an interface such as a USB (Universal Serial Bus) that directly connects to an output device such as a display device, an input device, and the external device 9 B. The output device may also have a speaker that outputs sound and the like.

[0044] The external device 9A is a device connected to the computer 30 via the communication network 5. The external device 9B is a device connected directly to the computer 30.

[0045] The external device 9A or the external device 9B may be, for example, an input device (input unit). The input device is, for example, a device such as a camera, a microphone, a motion capture device, various sensors, a keyboard, a mouse, or a touch panel, and provides acquired information to the computer 30. The external device 9A or the external device 9B may also be a device equipped with an input unit, a memory, and a processor, such as a personal computer, a tablet terminal, or a smartphone.

[0046] Furthermore, the external device 9A or the external device 9B may be, for example, an output device (output unit). The output device may be, for example, a display device (display unit) such as an LCD (Liquid Crystal Display), a CRT (Cathode Ray Tube), a PDP (Plasma Display Panel), or an organic EL (Electro Luminescence) panel, or may be a speaker that outputs sound, etc. Furthermore, the external device 9A or the external device 9B may be a device that includes an output device, a memory, and a processor, such as a personal computer, a tablet terminal, or a smartphone.

[0047] The external device 9A or the external device 9B may be a storage device (memory). For example, the external device 9A may be a network storage or the like, and the external device 9B may be a storage device such as an HDD.

[0048] Furthermore, the external device 9A or the external device 9B may be a device having some of the functions of the components of the information processing device 1 in this embodiment. In other words, the computer 30 may transmit or receive some or all of the processing results of the external device 9A or the external device 9B.

[0049] 2 is a diagram illustrating an example of functional blocks in the processor 31. The processor 31 has, as functions realized by the processor 31, for example, a setting unit 311, a nonlocal descriptor calculation unit 313, an exchange-correlation energy density determination unit 315, an equation calculation unit 317, and an evaluation unit 319. The functions realized by the setting unit 311, the nonlocal descriptor calculation unit 313, the exchange-correlation energy density determination unit 315, the equation calculation unit 317, and the evaluation unit 319 are stored as programs in, for example, the main storage device 33 or the auxiliary storage device 35. The processor 31 can realize the processing functions related to the setting unit 311, the nonlocal descriptor calculation unit 313, the exchange-correlation energy density determination unit 315, the equation calculation unit 317, and the evaluation unit 319 by reading and executing the respective programs stored in the main storage device 33 or the auxiliary storage device 35.

[0050] The setting unit 311 sets various parameters related to the substance to be searched for. For example, when the convolution function w(|r - r'|) is a Gaussian function exp(-α|r - r'|^2), the parameters include the value of the parameter α in the Gaussian function, the basis function φ for the expansion of the eigenfunction, ａ When (r) is, for example, a Gaussian function exp(-β|r-r'|^2) or a plane wave exp(ikr), it corresponds to the parameter β or k. Furthermore, the setting unit 311 sets initial values ​​when solving equations used in density functional theory (hereinafter referred to as DFT).

[0051] The equation used in DFT is, for example, the exchange-correlation energy E ＸＣ The Kohn-Sham equation (1) is obtained after functional differentiation of the electron density n(r) with respect to [n]. Note that the equation used in DFT is not limited to the Kohn-Sham equation (1), and the exchange-correlation energy E ＸＣ The energy variation equation may be an energy variation equation having a functional derivative of the density of multiple electrons with respect to [n], or another equation based on the energy variation equation. For example, the equation used in DFT may be an energy variation equation explicitly including a kinetic energy functional derivative with respect to the electron density n(r), or an orbital-free DFT. The energy variation equation corresponds to the eigenvalue equation expressed by formula (1) using formula (2) including formula (3). The eigenfunction corresponds to the function corresponding to the energy eigenvalue in the eigenvalue equation expressed by the Kohn-Sham equation (1) or the energy variation equation.

[0052] The initial value is, for example, the expansion coefficient c ｉａ corresponds to the initial value of the expansion coefficient c ｉａ is updated whenever the Kohn-Sham equation (1) or the energy variation equation is solved. At this time, the setting unit 311 updates the expansion coefficient c ｉａ to the updated value. For the sake of concreteness, the following description will be given assuming that the equation used in the DFT is the Kohn-Sham equation (1).

[0053] The nonlocal descriptor calculation unit 313 calculates multiple nonlocal descriptors (hereinafter referred to as nonlocal descriptors) according to multiple positions r. The nonlocal descriptor calculation unit 313 calculates a descriptor based on a value obtained by convolving the results of a linear operation on the eigenfunctions of each of multiple electrons in the equations used in density functional theory at each of multiple positions in the system to be calculated. The linear operation is a linear operation on the eigenfunction. Examples of the linear operation include constant multiplication, delay, differentiation, and integration. The linear operation includes raising the eigenfunction to the first power (identity transformation), differentiation of the eigenfunction (vector differentiation operation on a scalar eigenfunction), etc. Note that the linear operation is not limited to the above and may be convolution, etc.

[0054] The nonlocal descriptor calculation unit 313 calculates the descriptor based on, for example, a convolution value of the result of a linear operation on the basis functions that serve as the basis of the eigenfunctions of each of the multiple electrons in the system and the expansion coefficients corresponding to the basis functions. Specifically, the nonlocal descriptor calculation unit 313 calculates the descriptor based on the sum of products of the convolution value of the result of a linear operation on the basis functions that serve as the basis of the eigenfunctions and the expansion coefficients when the eigenfunctions are expanded using the basis functions. More specifically, the nonlocal descriptor calculation unit 313 calculates the descriptor by adding the squares of the absolute values ​​of the sum of products across the multiple eigenstates occupied by the multiple electrons. The calculated descriptor corresponds to a nonlocal descriptor. The nonlocal descriptor R(r) is defined by the following equation (10):

[0055]

[0056] As shown in equation (10), the nonlocal descriptor R(r) has a convolution integral obtained by convolving a convolution kernel w(|r - r'|) with the result (L hat ψ) of a linear operation on the eigenfunction ψ in the Kohn-Sham equation (1). That is, the convolution integral shown in equation (10) is obtained by convolving a plurality of eigenfunctions ψ after a linear operation centered on each of the plurality of positions r at each of the plurality of positions r. ｉis an integral obtained by convolving the distribution of |r - r'| with a convolution kernel (also called a weight function). The suffix (subscript) i∈occ. of the sum symbol in formula (10) is a symbol that identifies multiple eigenstates occupied by multiple electrons. For these reasons, the nonlocal descriptor R(r) shown in formula (10) represents the degree of influence of interactions by multiple electrons (electron distribution) from positions away from each of multiple positions r as the center. Formula (10) showing the nonlocal descriptor R(r) is expressed by the multiple basis functions φ ａ Eigenfunction ψ by (r') ｉ Using equation (7) showing the expansion of (r'), it is transformed into equation (11) below.

[0057]

[0058] As shown in equation (11), the eigenfunction ψ after linear operation ｉ The convolution integral of (r') is the basis function φ ａ (r') is transformed into the convolution integral. ｉ The result of linear operation on (r') (the result of linear operation) is, for example, the result of applying nabla (∇) to the basis function φa(r') (∇φ ａ (r'), etc., and can be set arbitrarily. The result of the linear operation may be referred to as a linear transformation or a linear operation. The basis function φ in Equation (11) ａ (r') is an eigenfunction ψ corresponding to the eigenenergy of each of multiple electrons in the equation (Kohn-Sham equation or energy variation equation) used in density functional theory (hereinafter referred to as DFT). ｉ (r) corresponds to the basis of the basis function φ ａ (r) is expressed, for example, by a Gaussian function exp(-β|r-r'|^2) or a plane wave exp(ikr). As shown in equation (11), the convolution integral is expressed by a plurality of basis functions φ ａ (r) is calculated by multiplying each by a weighting function w(|r−r′|).

[0059] The convolution kernel w(|r-r'|) in Equation (10) and Equation (11) is a basis function φ at a position r' around the position r. ａThe weight function w(|r - r'|) is the convolution function (weight function) for the convolution of (r'). The weight function w(|r - r'|) is the basis function φ of the electrons separated by a distance |r - r'|. ａ The weight function w(|r - r'|) describes how much the influence of (r') is taken into account at position r. That is, the weight function w(|r - r'|) is the basis function φ for an electron at position r', which is away from the center of position r. ａ (r') is the basis function φ located at the center ａ In other words, the half width or full width at half maximum of the convolution kernel w(|r−r′|) indicates the range of influence of the electron distribution.

[0060] The convolution function w(|r-r'|) in equation (11) must satisfy the conditions (hereinafter referred to as physical conditions) that it must be spherically symmetric and attenuate as it moves away from the position r, due to physical requirements. In other words, a weighting function w(|r-r'|) that satisfies the physical conditions is a function that is spherically symmetric with respect to the center corresponding to the position r and decreases as it moves away from the center. In this case, the convolution function w(|r-r'|) is, for example, a Gaussian function exp(-α|r-r'|^2) using a parameter α that is set depending on the material. Note that the specific form of the convolution function w(|r-r'|) is not limited to a Gaussian function, and may be any function that satisfies the physical conditions and has a basis function φ ａ Any function can be set as long as the convolution integral (equation (11)) obtained by multiplying with (r') can be analytically calculated. That is, the weighting function w(|r - r'|) is a function that allows analytical calculation of the convolution integral. The function shape of the convolution function w(|r - r'|) may be set or selected as appropriate by the user or the like using the setting unit 311.

[0061] Note that the expression of the convolution integral is not limited to an expression that can be solved analytically. That is, the convolution integral that expresses the nonlocal descriptor R(r) may be an expression that can be solved numerically. In this embodiment, since the number of times the convolution integral is performed can be reduced, even in a case where, for example, the convolution function is complex and can only be integrated by numerical integration using a numerical grid, and each convolution calculation requires a long time, the number of convolutions themselves can be reduced, thereby realizing the effects of the present application described below.

[0062] From these facts, the non-local descriptor calculation unit 313 calculates the formula (11) (hereinafter referred to as the convolution result) by applying the parameter values ​​set by the setting unit 311 to the analytical calculation result of the convolution integral shown in the formula (11). As a result, the non-local descriptor calculation unit 313 calculates the formula (11) (hereinafter referred to as the convolution result) by applying the parameter values ​​set by the setting unit 311 to the analytical calculation result of the convolution integral shown in the formula (11). ａ A plurality of convolution results according to the total number of (r') are determined.

[0063] Next, the non-local descriptor calculation unit 313 calculates the absolute value in equation 11. Specifically, the non-local descriptor calculation unit 313 first calculates the sum shown in equation 12 below.

[0064]

[0065] Equation (12) is a set of expansion coefficients c ｉａ and a plurality of convolution results over the total number of basis functions. That is, the non-local descriptor calculation unit 313 calculates the sum of products of the expansion coefficients c ｉａ Then, the non-local descriptor calculation unit 313 calculates the sum of products of the convolution result and the vector vectors. Next, the non-local descriptor calculation unit 313 calculates the absolute value of the calculated sum of products as shown in the following equation (13).

[0066]

[0067] The nonlocal descriptor calculation unit 313 calculates the sum of the calculated absolute value over multiple eigenstates i∈opp. of electrons occupied by multiple electrons, as shown in the following equation (14). In this way, the nonlocal descriptor calculation unit 313 calculates the nonlocal descriptor R(r) shown in equation (10). The nonlocal descriptor calculation unit 313 stores the nonlocal descriptor R(r) in the main storage device 33 or the auxiliary storage device 35 according to the position r. As described above and shown in equations (10) to (14), the calculation complexity of the nonlocal descriptor R(r) can be achieved with O(N) calculations.

[0068]

[0069] The exchange-correlation energy density determiner 315 inputs at least a descriptor to the trained model at each of a plurality of positions in the system, and outputs the exchange-correlation energy density of each of a plurality of electrons. For example, the exchange-correlation energy density determiner 315 inputs each of a plurality of nonlocal descriptors R(r), the density of each of a plurality of electrons based on an eigenfunction, the gradient of the density (electron density) of the electron, and the kinetic energy density of the electron to the trained model, and determines the exchange-correlation energy density of the electron, with the density of the electron as a variable, according to a plurality of positions r. Specifically, the exchange-correlation energy density determiner 315 reads the trained model from the main storage device 33 or the auxiliary storage device 35. The trained model inputs the electron density n(r), the gradient ∇n(r) of the electron density n(r), and the kinetic energy density τ(r) of the electron according to the spin direction for each of a plurality of positions r, and outputs the exchange-correlation energy density ε of the electron. ＸＣ The neural network is trained to output [n](r). The electron kinetic energy density τ(r) is calculated by a known method, for example, based on the momentum obtained by applying an operator corresponding to the electron momentum to the electron eigenfunction ψ determined by the expansion coefficients used in calculating the nonlocal descriptor R(r). The calculation of the kinetic energy density is performed, for example, by the exchange-correlation energy density determination unit 315. The generation of the trained model will be described later. The generated trained model is stored in advance in the main storage device 33 or the auxiliary storage device 35.

[0070] 3 is a diagram showing an example of a trained model LM. As shown in FIG. 3, the trained model has an input layer IL, an intermediate layer (hidden layer) ML, and an output layer OL. The input layer IL contains an up-spin electron density n ↑ (r) and the down-spin electron density n ↓ (r), the gradient ∇n(r) of the electron density n(r), the kinetic energy density τ(r), and the non-local descriptor R(r) are input. The input to the trained model is not limited to the above. For example, the up-spin electron density n ↑ (r) or down-spin electron density n ↓ (r), the ratio of electron up spin to down spin, the gradient ∇n(r) of the electron density n(r), the kinetic energy density τ(r), and the nonlocal descriptor R(r) may be input to the input layer IL. Also, the kinetic energy density τ(r) may be omitted in the input to the trained model. As shown in FIG. 3 , the trained model LM outputs the electron exchange-correlation energy density ε ＸＣ The exchange-correlation energy density determination unit 315 outputs the electron exchange-correlation energy density ε[n](r) output from the learned model LM. ＸＣ [n](r) is stored in the main memory 33 or the auxiliary memory 35 depending on the position r.

[0071] Figure 4 shows the eigenfunction ψ in the Kohn-Sham equation (1). ｉ (r) and the electron exchange-correlation energy density ε ＸＣ As shown in FIG. 4 and equations (10) to (14), the electron exchange-correlation energy density ε ＸＣ [n](r) is the eigenfunction ψ ｉ It is determined based on the convolution integral of (r) and the convolution function w(|r-r'|). That is, the electron exchange-correlation energy density ε ＸＣ [n](r) is not a convolution integral of the electron density n(r), but is an eigenfunction ψ that is the basis for calculating the electron density n(r), as shown in equation (4) and FIG. ｉ (r) is calculated based on

[0072] The equation calculation unit 317 calculates the exchange-correlation energy density ε ＸＣ [n](r) is applied to an equation used in density functional theory to calculate the eigenfunctions and eigenenergies of each of the multiple electrons. For example, the equation calculation unit 317 calculates the eigenfunctions and eigenenergies of each of the multiple electrons by applying the determined exchange-correlation energy density ε ＸＣ The exchange-correlation energy based on [n](r) and electron density n(r), electron density n(r), and eigenfunction ψ ｉ (r) and calculates the solution of the Kohn-Sham equation (1). Specifically, the equation calculation unit 317 calculates the solution of the Kohn-Sham equation (1) using the determined energy density ε ＸＣ [n](r), electron density n(r), and expansion coefficient c ｉａ The initial value or expansion coefficient c set in the repeated calculation ｉａ Using the values ​​of the potential V in Eq. (2), ｅｆｆ [n](r). The equation calculation unit 317 calculates the calculated potential V ｅｆｆ [n](r) is used to solve the Kohn-Sham equation (1) shown in Equation (1). As a result, the equation calculation unit 317 calculates the eigenfunction ψ ｉ and the energy eigenvalue ε ｉ It is decided that:

[0073] The equation calculation unit 317 calculates the basis function φ ａ (r) is used to determine the eigenfunction ψ ｉ As a result, the equation calculation unit 317 newly calculates the expansion coefficient c ｉａ Next, the equation calculation unit 317 determines the newly determined expansion coefficients (hereinafter referred to as latest expansion coefficients) c ｉａ and potential V ｅｆｆ The expansion coefficients used to determine [n](r) (hereinafter referred to as the previous expansion coefficients) are compared for each of the subscripts a that identify the basis functions. The previous expansion coefficients are the coefficients used to determine the potential V just before solving the Kohn-Sham equation (1). ｅｆｆ [n](r) are the expansion coefficients used to derive (r). If the difference between the latest expansion coefficients and the immediately preceding expansion coefficients is equal to or less than a predetermined value for the suffix a of all the expansion coefficients, the equation calculation unit 317 determines the latest expansion coefficients and the energy eigenvalues ​​as the solutions to the Kohn-Sham equation (1).

[0074] The convergence condition for the solution of the Kohn-Sham equation (1) is not limited to the determination of the convergence of the expansion coefficients. For example, if the difference between the energy eigenvalues ​​associated with the immediately preceding expansion coefficients and the energy eigenvalues ​​associated with the latest expansion coefficients (hereinafter referred to as the eigenvalue difference) is less than a preset value (hereinafter referred to as the set value), the equation calculation unit 317 may determine the latest expansion coefficients and the energy eigenvalues ​​associated with the latest expansion coefficients as the solution of the Kohn-Sham equation (1). Furthermore, if the changes in both the expansion coefficients and the energy eigenvalues ​​are equal to or less than a preset value, the equation calculation unit 317 may determine the latest expansion coefficients and the energy eigenvalues ​​associated with the latest expansion coefficients as the solution of the Kohn-Sham equation (1).

[0075] The evaluation unit 319 calculates the eigenfunctions and eigenenergies for each of the multiple electrons using the latest expansion coefficients. That is, the evaluation unit 319 applies the exchange-correlation energy density at each of the multiple positions to the Kohn-Sham equation (1) to calculate the eigenfunctions for each of the multiple electrons in the calculation target material and the eigenenergies for each of the multiple electrons in the material. Specifically, the evaluation unit 319 determines the eigenfunctions for each of the multiple electrons by calculating Equation (7) using the latest expansion coefficients and basis functions. Furthermore, the evaluation unit 319 calculates the nonlocal descriptor R(r) by performing processing similar to the calculation in the nonlocal descriptor calculation unit 313 using the latest expansion coefficients. Note that the calculation of the nonlocal descriptor R(r) may be performed by the nonlocal descriptor calculation unit 313. The evaluation unit 319 calculates the electron density, the gradient of the electron density, and the kinetic energy density using the latest expansion coefficients. The calculation of the electron density, the gradient of the electron density, and the kinetic energy density using the latest expansion coefficients may be performed by the non-local descriptor calculation unit 313 .

[0076] The evaluation unit 319 determines the exchange-correlation energy density of electrons by inputting the nonlocal descriptors calculated using the latest expansion coefficients, the electron density, the gradient of the electron density, and the kinetic energy density into the trained model. The determination of the exchange-correlation energy density of electrons using the trained model may be performed by the exchange-correlation energy density determination unit 315. The evaluation unit 319 calculates the energy of the system to be calculated using the calculated eigenfunctions and calculated eigenenergies. That is, the evaluation unit 319 calculates the energy related to the material to be calculated using the calculated eigenfunctions and eigenenergies. The energy related to the material includes the Kohn-Sham kinetic energies of multiple electrons, the energy of interaction between multiple electrons and atoms (energy related to ions), the Coulomb energies (Hartree terms) of multiple electrons, and the exchange-correlation energy of electrons. That is, the evaluation unit 319 calculates the total energy using the exchange-correlation energy density of electrons, the electron density, the kinetic energy density, etc. The Kohn-Sham kinetic energy, the electron-atom interaction energy, the Coulomb energy, and the exchange-correlation energy are functionals of the electron density.

[0077] The evaluation unit 319 calculates physical property values ​​related to the substance to be searched for using various energies. The physical property values ​​include, for example, band gap, reaction activation energy, and magnetism. A known method can be applied to calculate the physical property values ​​from the solution of the Kohn-Sham equation, and therefore a description thereof will be omitted. The evaluation unit 319 stores the calculated physical property values ​​in the main storage device 33 and / or the auxiliary storage device 35. The evaluation unit 319 may also display the calculated physical property values ​​on a display or the like.

[0078] The above has described the configuration of the information processing device 1. Below, the procedure of the process for solving the Kohn-Sham equation (hereinafter referred to as the solution calculation process) executed by the information processing device 1 will be described with reference to Fig. 5. Fig. 5 is a flowchart showing an example of the procedure of the solution calculation process.

[0079] (Solution Calculation Process) (Step S501) The setting unit 311 sets various parameters related to the material. For example, the setting unit 311 sets parameter values ​​in the convolution function, parameter values ​​of the basis functions related to the expansion of the eigenfunction, initial values ​​of expansion coefficients related to the expansion of the eigenfunction, etc. These parameters may be set by inputting instructions from a user via an input device, or may be set automatically in response to input of the composition of the material that is the target of material search, using a correspondence table of parameters for the composition.

[0080] (Step S502) The non-local descriptor calculation unit 313 calculates a non-local descriptor using the set parameters, basis functions, and weight functions. Specifically, the non-local descriptor calculation unit 313 calculates the non-local descriptor R(r) by applying the set parameters, basis functions, and weight functions to equations (10) to (14). The non-local descriptor calculation unit 313 stores the calculated non-local descriptor R(r) in the main storage device 33 or the auxiliary storage device 35.

[0081] (Step S503) The exchange-correlation energy density determiner 315 calculates the electron density, the electron density gradient, and the electron kinetic energy density based on the parameters and the basis functions. For example, the exchange-correlation energy density determiner 315 calculates the electron density by applying the parameters and the basis functions to Equation (4). The exchange-correlation energy density determiner 315 also calculates the electron density gradient by applying a gradient (∇: nabla) to the calculated electron density. The exchange-correlation energy density determiner 315 also calculates the electron momentum by applying a momentum operator to an eigenfunction based on the parameters and the basis functions. Next, the exchange-correlation energy density determiner 315 calculates the electron kinetic energy density by applying the electron momentum to an equation for calculating the electron kinetic energy density. The exchange-correlation energy density determiner 315 stores the calculated electron density, the electron density gradient, and the electron kinetic energy density in the main storage device 33 or the auxiliary storage device 35.

[0082] (Step S504) The exchange-correlation energy density determiner 315 reads out the trained model from the main storage device 33 or the auxiliary storage device 35. The exchange-correlation energy density determiner 315 determines the exchange-correlation energy density of electrons by inputting the electron density, the electron density gradient, the electron kinetic energy density, and the nonlocal descriptor into the read trained model. The exchange-correlation energy density determiner 315 stores the determined exchange-correlation energy density of electrons in the main storage device 33 or the auxiliary storage device 35.

[0083] (Step S505) The equation calculation unit 317 calculates the determined exchange-correlation energy density ε ＸＣ [n](r) and the exchange-correlation energy E based on the electron density n(r) ＸＣ [n], electron density n(r), and eigenfunction ψ ｉ (r) and calculates the solution of the Kohn-Sham equation (1). Specifically, the equation calculation unit 317 calculates the solution of the Kohn-Sham equation (1) using the determined energy density ε ＸＣ By calculating equation (5) using [n](r) and electron density n(r), the exchange-correlation energy E ＸＣ Next, the equation calculation unit 317 calculates the exchange-correlation energy E ＸＣ Based on [n] and the electron density n(r), etc., the operator acting on the eigenfunction on the left side of the Kohn-Sham equation (1) is derived (determined). This determines the operators in the Kohn-Sham equation (1), excluding the eigenvalues ​​and eigenfunctions.

[0084] Next, the equation calculation unit 317 solves the Kohn-Sham equation (1) with the operator determined. A known method can be applied to solve the Kohn-Sham equation (1), which is an eigenvalue problem, so a description thereof will be omitted. By solving the Kohn-Sham equation (1) with the operator determined, the equation calculation unit 317 solves the eigenfunction ψ ｉ (r) and eigenvalue ε ｉ The equation calculation unit 317 determines the determined eigenfunction ψ ｉ (r) and eigenvalue ε ｉ and stored in the main storage device 33 or the auxiliary storage device 35.

[0085] (Step S506) The equation calculation unit 317 calculates the determined eigenfunction ψ using equation (7). ｉ (r) is expanded using basis functions. As a result, the equation calculation unit 317 determines a plurality of expansion coefficients as the latest expansion coefficients. Next, just before solving the Kohn-Sham equation, the equation calculation unit 317 calculates the potential V ｅｆｆ The equation calculation unit 317 compares the expansion coefficients (previous expansion coefficients) used to derive [n](r) with the latest expansion coefficients. Note that the equation calculation unit 317 may compare the energy eigenvalues ​​related to the previous expansion coefficients with the energy eigenvalues ​​related to the latest expansion coefficients.

[0086] (Step S507) If the difference between the previous expansion coefficient and the latest expansion coefficient is less than a predetermined value for the suffix a of all expansion coefficients, i.e., if the expansion coefficients have converged (Yes in step S507), the process proceeds to step S509. If the difference between the previous expansion coefficient and the latest expansion coefficient is equal to or greater than a predetermined value for the suffix a of at least one expansion coefficient, i.e., if the expansion coefficients have not converged (No in step S507), the process proceeds to step S508. The convergence condition for the expansion coefficients is not limited to the above difference, and may be determined, for example, by a known statistical method based on the previous expansion coefficients and the latest expansion coefficients.

[0087] Note that the determination in this step is not limited to the above. For example, if the eigenvalue difference is less than a set value (Yes in step S507), the process of step S509 may be executed. At this time, if the eigenvalue difference is equal to or greater than the set value (No in step S507), the process of step S508 is executed. Also, if the changes in both the expansion coefficients and the energy eigenvalues ​​are less than a preset value (Yes in step S507), the process of step S509 may be executed. At this time, if the changes in both the expansion coefficients and the energy eigenvalues ​​are equal to or greater than a preset value (No in step S507), the process of step S508 is executed.

[0088] (Step S508) The equation calculation unit 317 updates the expansion coefficients in the parameters using the latest expansion coefficients as the immediately preceding expansion coefficients. That is, the processing from step S502 onwards is repeated using the latest expansion coefficients as parameters.

[0089] (Step S509) The equation calculation unit 317 determines the latest expansion coefficients and energy eigenvalues ​​as solutions to the Kohn-Sham equation (1). As a result, the equation calculation unit 317 determines the eigenfunctions and eigenvalues ​​of the Kohn-Sham equation (1). Next, the evaluation unit 319 calculates physical property values ​​related to the search target substance based on the determined eigenfunctions and eigenvalues. The evaluation unit 319 outputs the calculated physical property values ​​to the main storage device 33 and / or the auxiliary storage device 35, a display, etc.

[0090] The generation of the trained model used in this embodiment will be described below. The hardware configuration of the information processing device (hereinafter referred to as the learning device) used in this embodiment for generating the trained model is similar to the hardware configuration of the information processing device 1 shown in FIG. 1 , and therefore the description thereof will be omitted.

[0091] 6 is a diagram showing an example of functional blocks in the processor 31 installed in the learning device. The processor 31 has, as functions realized by the processor 31, for example, an acquisition unit 511, an exchange-correlation energy density determination unit 513, an exchange-correlation energy density calculation unit 515, a comparison unit 517, and a learning unit 519. The functions realized by the setting unit 311, the nonlocal descriptor calculation unit 313, the exchange-correlation energy density determination unit 315, the equation calculation unit 317, and the learning unit 519 are each stored as a program in, for example, the main storage device 33 or the auxiliary storage device 35. The processor 31 can realize the processing functions related to the acquisition unit 511, the exchange-correlation energy density determination unit 513, the exchange-correlation energy density calculation unit 515, the comparison unit 517, and the learning unit 519 by reading and executing each program stored in, for example, the main storage device 33 or the auxiliary storage device 35.

[0092] The acquiring unit 511 acquires training data and supervised data (which may also be referred to as teacher data) indicating exchange-correlation energy densities corresponding to the training data. The training data and supervised data are data sets used in generating a trained model. Hereinafter, the training data and supervised data are collectively referred to as training data sets. For example, the acquiring unit 511 acquires multiple training data sets used in the model to be trained from an external device 9A, for example, via the network interface 37 and the communication network 5. In this case, the external device 9A corresponds to a server device and / or a storage device that stores multiple training data sets. The acquiring unit 511 stores the acquired multiple training data sets in the main storage device 33, the auxiliary storage device 35, or the like. Furthermore, the acquiring unit 511 acquires each of the multiple training data sets from the main storage device 33 or the auxiliary storage device 35 in generating a trained model.

[0093] The training data includes a nonlocal descriptor, an electron density, a gradient of the electron density, and a kinetic energy density. Note that if kinetic energy density is not required for input to the trained model, kinetic energy density is not required in the training data. The correct answer data corresponds to the exchange-correlation energy density of electrons. The training data and the correct answer data are set in advance and stored in a server device and / or a storage device. The correct answer data corresponds to, for example, the exchange-correlation energy of electrons determined by numerically solving the Schrödinger equation using information about the electron distribution representing the training data (also referred to as training data). Hereinafter, for the sake of concreteness, the model to be trained will be described as a neural network.

[0094] The exchange-correlation energy density determination unit 513 inputs at least information about the electron distribution into the model to be learned and calculates the first exchange-correlation energy density of electrons. The information about the electron distribution includes, for example, at least one of a non-local descriptor obtained by convolving the results of a linear operation on an eigenfunction of each of a plurality of electrons in an equation used in density functional theory (e.g., the Kohn-Sham equation or an energy variational equation having a functional derivative of the electron density with respect to the electron exchange-correlation energy), or a local descriptor based on the eigenfunction. The information about the electron distribution may further include the electron density (electron density distribution), the gradient of the electron density, the electron kinetic energy density, etc. The electron density may include, for example, an up-spin electron density n ↑ (r) and the down-spin electron density n ↓ The electron density is not limited to the above, and for example, the electron density of up spin n ↑ (r) or down-spin electron density n ↓ (r) and the ratio of up spin to down spin of electrons.

[0095] For example, the exchange-correlation energy density determiner 513 inputs the nonlocal descriptor, the electron density, the gradient of the electron density, and the kinetic energy density to the neural network to be trained. Based on the output from the neural network, the exchange-correlation energy density determiner 513 determines a first exchange-correlation energy density of electrons (hereinafter referred to as the first energy density). That is, the exchange-correlation energy density determiner 513 has a function of performing forward propagation (forward propagation processing) on ​​the neural network to be trained. For this reason, the exchange-correlation energy density determiner 513 may also be referred to as a forward propagation processing unit.

[0096] The exchange-correlation energy density calculation unit 515 obtains a second exchange-correlation energy density of electrons calculated by a method different from the input to the model to be trained using information about the electron distribution. The different method is at least one of an analytical method and a numerical method using information about the electron distribution. Furthermore, the different method is, for example, a method of calculating the second exchange-correlation energy density of electrons by inputting information about the electron distribution to another trained model. Note that the different method may be both an analytical and a numerical method. Below, an example of calculating the second exchange-correlation energy density of electrons (hereinafter referred to as the second energy density) will be described.

[0097] The second exchange-correlation energy density of electrons may be calculated by an external device 9A such as a calculation server. In this case, the second energy density is acquired from the external device 9A by the acquiring unit 511 via the communication network 5, together with training data corresponding to the second energy density. The acquiring unit 511 associates ground truth data indicating the second energy density with training data indicating information about the electron distribution, and stores the resulting data as a learning data set in the main storage device 33, the auxiliary storage device 35, or the like. For the sake of concreteness, the following description will be given assuming that the second exchange-correlation energy density of electrons is calculated by the exchange-correlation energy density calculating unit 515.

[0098] The exchange-correlation energy density calculation unit 515 calculates the second energy density using information about the electron distribution in the training data (e.g., the electron density and the gradient of the electron density). In other words, the second energy density is a differentiable and smooth functional. The second energy density is an exchange-correlation energy density that has good convergence when solving the Kohn-Sham equation (1) or the energy variation equation. The second energy density is calculated, for example, by the Perdew-Burke-Ernzerhof (PBE) method. Note that the calculation method for the second energy density is not limited to the PBE method, and other methods such as local density approximation (LDA) may be used for calculation. Note that the second energy density may be output by a pre-constructed neural network capable of outputting the second energy density.

[0099] The comparison unit 517 performs a first comparison between ground truth data associated with information on electron distribution and the first exchange-correlation energy density of electrons. Specifically, the comparison unit 517 multiplies the first energy density by the first electron density and integrates the result of this multiplication over the entire space to calculate the first exchange-correlation energy of electrons. As a result, the comparison unit 517 calculates the absolute value of the difference between the first electron exchange-correlation energy and the ground truth data as the first comparison. The calculation result of the first comparison is used as a loss function in the learning process for the neural network to be trained.

[0100] The following equation (15) shows an example of a term added to the loss function by the first comparison.

[0101]

[0102] The sum symbol Σ in equation (15) indicates the sum over the subscript M that identifies multiple substances (molecules) in the correct data. Also, the first term E in the absolute value of equation (15) ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ ] is the electron density n(M) in the correct data obtained by solving the Schrodinger equation for material M ｅｘａｃｔ The following equation (16) shows the first exchange-correlation energy E of the electron for the material M, where E is the functional. ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ ] is shown as an example.

[0103]

[0104] As shown in equation (16), the first exchange-correlation energy E ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ ] is the electron density n(M) in the ground truth data for material M ｅｘａｃｔ is a functional, and the first energy density (first exchange-correlation energy density of electrons) ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ] (r), the electron density n(M) depending on the position r in the ground truth data for material M ｅｘａｃｔ (r) and integrating over the entire space.

[0105] The second term E in the absolute value of equation (15) ｅｘａｃｔ ＸＣ (M) [n(M) ｅｘａｃｔ ] is the electron density n(M) in the ground truth data for material M ｅｘａｃｔ The second exchange-correlation energy of electrons is expressed as a functional E ｅｘａｃｔ ＸＣ (M) [n(M) ｅｘａｃｔ ] is obtained by solving the Schrodinger equation for material M and is included in the correct data for material M. Note that the second exchange-correlation energy E ｅｘａｃｔ ＸＣ (M) [n(M) ｅｘａｃｔ ] is the first exchange-correlation energy E of the electron ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ ], the exchange-correlation energy density ε of the electron depending on the position r calculated for the material M as the correct data ｅｘａｃｔ ＸＣ [n(M) ｅｘａｃｔ ] (r) may be used to calculate in the same manner as in Equation (16). In addition, the exchange-correlation energy density ε of the electron that depends on the position r calculated for the material M as the correct data exact ＸＣ [n(M) ｅｘａｃｔ ](r), the comparison unit 517 may calculate the absolute value of the difference between the exchange-correlation energy density of the electron and the first energy density as the result of the first comparison.

[0106] The comparison unit 517 also performs a second comparison between the first exchange-correlation energy density of electrons and the second exchange-correlation energy density of electrons. Specifically, the comparison unit 517 calculates a value based on the difference between the first exchange-correlation energy density and the second exchange-correlation energy density as a result of the second comparison. Specifically, the comparison unit 517 multiplies the absolute value of the difference by the electron density and weight, and calculates an integral over a predetermined space as a result of the second comparison. That is, the comparison unit 517 multiplies the absolute value of the difference between the first energy density and the second energy density by the electron density and weight, and then integrates the result of the multiplication over a predetermined space to calculate a penalty term. Thus, the calculation result of the second comparison is used as a loss function in the learning process for the neural network to be trained.

[0107] The comparison unit 517 calculates a penalty term for training the neural network to be trained based on the first energy density and the second energy density. The following equation (17) shows an example of the penalty term added to the loss function by the second comparison.

[0108]

[0109] The sum symbol Σ in equation (17) indicates the sum over the subscript A that identifies the substance included in the correct answer data. The substances included in the correct answer data are not limited to real substances, but may include substances with virtual atomic arrangements and electronic states. In addition, w included in the integrand Ａ (r) indicates the weight for each substance A according to the position r. Furthermore, n(A, r) included in the integrand indicates the electron density according to the substance A and the position r. The weight w Ａ The weight w(r) is set in advance to be larger if the electron density n(A, r) is low. Ａ (r) is set in advance according to (for example, inversely proportional to) the magnitude of the electron density n(A, r), for example.

[0110] The first term ε in the absolute value of equation (17) ＮＮ ＸＣ(A) [n(A)] (r) represents the first energy density at the position (r) with the electron density n(A) for the substance A as a functional. The second term ε in the absolute value of Equation (17) ｒｅｆ ＸＣ (A)[n(A)](r) represents the second energy density at position (r) with the electron density n(A) for substance A as a functional. Also, λ in equation (17) is a parameter indicating the weight of the penalty term (17) in the loss function.

[0111] The penalty term shown in Equation (17) is an example, and is not limited to this. ＮＮ ＸＣ (A) [n(A)] (r) and the second energy density ε ｒｅｆ ＸＣ As long as it includes the difference between (A)[n(A)](r), it can be set arbitrarily. That is, various modifications may be made to the penalty term shown in equation (17). For example, the penalty term may be the electron density n(A, r) and / or the weight w Ａ (r) may be omitted as appropriate. Furthermore, the absolute value shown in equation (17) is not limited to the first power, but may be the second power.

[0112] The comparison unit 517 adds the calculated penalty term to a loss function used when training the neural network to be trained. The loss function has an error function and a penalty term based on the difference between the first energy density and the correct data. The loss function will be described below. Equation (18) shows an example of the loss function.

[0113]

[0114] The comparison unit 517 generates a loss function shown in formula (18) using the first comparison shown in formula (15) and the second comparison shown in formula (17). The loss function shown in formula (18) is expressed as the sum of the first comparison shown in formula (15) and the second comparison shown in formula (17) (hereinafter referred to as the first loss function), but is not limited to this. For example, in a learning process, when a learning target model is trained using the first comparison shown in formula (15) as a loss function (hereinafter referred to as the second loss function) (hereinafter referred to as the first learning), and then the learning target model trained using the first comparison as a loss function is trained using the second comparison shown in formula (17) as a loss function (hereinafter referred to as the third loss function) (hereinafter referred to as the second learning), the comparison unit 517 calculates the first comparison shown in formula (15) as the second loss function during the first learning. Additionally, when the second learning is performed, the comparison unit 517 calculates the second comparison shown in equation (17) as a third loss function.

[0115] The learning unit 519 uses a loss function to learn the neural network to be learned. Specifically, the learning unit 519 determines multiple weights in the neural network by backpropagation using the loss function. That is, the learning unit 519 has a function of performing backpropagation (backpropagation processing) on ​​the neural network to be learned. For this reason, the learning unit 519 may also be referred to as a backpropagation processing unit. Since backpropagation is well known, a description thereof will be omitted.

[0116] For example, the learning unit 519 learns a model to be trained based on the results of the first comparison and the second comparison. Specifically, the learning unit 519 learns the model to be trained using a first loss function based on the first comparison and the second comparison. More specifically, the learning unit 519 learns the model to be trained using a second loss function based on the first comparison and a third loss function based on the second comparison. Note that the learning unit 519 may train the model to be trained using the second loss function using the first comparison, and then further train the model to be trained using the third loss function using the second comparison. The learning unit 519 can arbitrarily set the order of the first learning and the second learning. The learning unit 519 may alternately perform the first learning and the second learning on each piece of training data included in the training dataset. The learning unit 519 may perform the first learning on all of the training data included in the training dataset, and then perform the second learning. Furthermore, the learning unit 519 may perform the second learning on all the learning data included in the learning data set, and then perform the first learning.

[0117] The configuration of an information processing device as a learning device has been described above. Below, the procedure of a process for generating a trained model (hereinafter referred to as a model generation process) executed by the learning device will be described. Additionally, for the sake of specificity, the model generation process will be described assuming that the learning unit 519 trains a model to be trained using a first loss function that uses a first comparison and a second comparison. FIG. 7 is a flowchart showing an example of the procedure of the model generation process.

[0118] (Model Generation Process) (Step S701) The acquisition unit 511 acquires the electron density, the gradient of the electron density, the kinetic energy density, and the non-local descriptor as learning data. The acquisition unit 511 also acquires the electron exchange-correlation energy (third energy) E ｅｘａｃｔ ＸＣ (M) [n(M) ｅｘａｃｔ The acquiring unit 511 stores a training data set including the training data and the supervised data in the main storage device 33 or the auxiliary storage device 35.

[0119] (Step S702) The exchange-correlation energy density determiner 513 inputs the electron density, the gradient of the electron density, the kinetic energy density, and the non-local descriptor to a model (neural network) to be trained. Based on the output from the model (neural network) to be trained, the exchange-correlation energy density determiner 513 determines the electron exchange-correlation energy density (first energy density) ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ The exchange-correlation energy density determiner 513 determines the first energy density) ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ ] (r) Based on the first energy E ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ The exchange-correlation energy density determination unit 513 calculates the first energy density ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ ] (r) and the first energy E ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ ] is stored in the main storage device 33 or the auxiliary storage device 35.

[0120] (Step S703) The exchange-correlation energy density calculation unit 515 calculates the electron exchange-correlation second energy density ε based on the electron density and the gradient of the electron density. ｒｅｆ ＸＣ For example, the exchange-correlation energy density calculation unit 515 calculates the second energy density ε by the PBE method. ｒｅｆ ＸＣ (A) [n(A)](r). The exchange-correlation energy density calculation unit 515 calculates the calculated second energy density ε ｒｅｆ ＸＣ (A)[n(A)](r) is stored in the main memory device 33 or the auxiliary memory device 35.

[0121] (Step S704) The comparison unit 517 performs a first comparison as shown in equation (15). As a result, the comparison unit 517 obtains the first exchange-correlation energy E ＮＮ ＸＣ (M) [n(M)ｅｘａｃｔ ] and correct data E ｅｘａｃｔ ＸＣ (M) [n(M) ｅｘａｃｔ ] are summed for each material M to calculate the first loss function shown in equation (15).

[0122] Furthermore, the comparison unit 517 performs a second comparison as shown in equation (17). As a result, the comparison unit 517 obtains the first energy density ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ ] (r) and the second energy density ε ｒｅｆ ＸＣ Based on (A) [n(A)](r), the comparison unit 517 calculates a second loss function corresponding to the penalty term as shown in equation (17). The comparison unit 517 generates a loss function by adding the first loss function and the second loss function as shown in equation (18). From the above, the comparison unit 517 calculates the first energy density ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ ] (r) and the second energy density ε ｒｅｆ ＸＣ (A) [n(A)] (r) is used to calculate the penalty term shown in Equation (17), and the first energy E ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ ] and correct data E ｅｘａｃｔ ＸＣ (M) [n(M) ｅｘａｃｔ ] (Equation (15)) and the penalty term (Equation (17)) are added to generate a loss function for training the model (neural network) to be trained.

[0123] (Step S705) The learning unit 519 uses the loss function to learn the model (neural network) to be learned. Specifically, the learning unit 519 determines multiple weights in the neural network by backpropagation using the loss function.

[0124] (Step S706) The learning unit 519 determines whether learning for the model (neural network) to be learned is complete. A known method can be applied to determine whether learning is complete, so a description thereof will be omitted. If learning is complete (Yes in step S706), the process of step S707 is executed. If learning is not complete (No in step S706), the process from step S701 onward is repeated. Note that if multiple learning datasets for learning have been acquired in step S701, the process from step S702 onward is repeated after No in step S706.

[0125] (Step S707) The learning unit 519 stores the model (neural network) to be learned as a trained model in the main storage device 33 or the auxiliary storage device 35. At this time, the learning unit 519 may transmit the trained model to the information processing device 1 on which the solution calculation process is executed.

[0126] Based on the above, the information processing device 1 according to this embodiment calculates, at each of multiple positions in a system to be calculated, a descriptor (nonlocal descriptor) based on a convolution value of the results of a linear operation on the eigenfunctions of each of multiple electrons in an equation used in density functional theory (DFT), inputs at least the descriptor to a trained model at each of the multiple positions, outputs the exchange-correlation energy density of each of the multiple electrons, applies the exchange-correlation energy density at each of the multiple positions to the equation, calculates the eigenfunctions and eigenenergies of each of the multiple electrons in the system to be calculated, and calculates the energy of the system to be calculated using the calculated eigenfunctions and eigenenergies. The information processing device 1 according to this embodiment calculates the descriptor based on a convolution value of the results of a linear operation on the basis functions that form the basis of the eigenfunctions and expansion coefficients corresponding to the basis functions. The information processing device 1 according to this embodiment calculates the descriptor based on a product sum of the expansion coefficients and the convolution value of the results of a linear operation on the basis functions that form the basis of the eigenfunctions. The information processing device 1 according to the present embodiment calculates the descriptor by adding the squares of the absolute values ​​of the sums of products across multiple eigenstates occupied by multiple electrons. Furthermore, the non-local descriptor in the information processing device 1 according to the present embodiment represents the degree of influence of interactions by multiple electrons from positions away from each of multiple positions as the center.

[0127] The equation in the information processing device 1 according to this embodiment is the Kohn-Sham equation. Furthermore, the equation in the information processing device 1 according to this embodiment may be an energy variational equation having a functional derivative of the electron density with respect to the electron exchange-correlation energy, or an equation based on the energy variational equation. Furthermore, in the information processing device 1 according to this embodiment, the energy of the system to be calculated includes the Kohn-Sham kinetic energies of the multiple electrons, the energy of the interaction between the multiple electrons and atoms, the Coulomb energy of the multiple electrons, and the exchange-correlation energy of the electrons, and the Kohn-Sham kinetic energy, the interaction energy, the Coulomb energy, and the exchange-correlation energy are functionals of the electron density.

[0128] Furthermore, the trained model in the information processing device 1 according to this embodiment is generated by inputting at least information about electron distribution to a model to be trained, calculating a first exchange-correlation energy density of electrons, obtaining a second exchange-correlation energy density of electrons calculated using a method different from that input to the model to be trained using the information about the electron distribution, performing a first comparison between ground truth data associated with the information about the electron distribution and the first exchange-correlation energy density of the electrons, performing a second comparison between the first exchange-correlation energy density of the electrons and the second exchange-correlation energy density of the electrons, and training the model to be trained based on the results of the first and second comparisons. The information processing device 1 according to this embodiment trains the model to be trained, for example, using a first loss function based on the first comparison and a second comparison. The information processing device 1 according to this embodiment trains the model to be trained, for example, using a second loss function based on the first comparison and a third loss function based on the second comparison.

[0129] In generating a trained model in the information processing device 1 according to this embodiment, the information on the electron distribution includes at least one of a nonlocal descriptor obtained by convolving the results of linear calculations on eigenfunctions for each of multiple electrons in equations used in density functional theory, or a local descriptor based on the eigenfunctions. Furthermore, in generating a trained model in the information processing device 1 according to this embodiment, the different method is at least either an analytical method or a numerical method that uses information on the electron distribution. Furthermore, in generating a trained model in the information processing device 1 according to this embodiment, the different method calculates a second exchange-correlation energy density of the electron by inputting information on the electron distribution into another trained model.

[0130] In generating a trained model in the information processing device 1 according to this embodiment, the information processing device 1 calculates, as the second comparison, a value based on the difference between the first exchange-correlation energy density of the electron and the second exchange-correlation energy density of the electron. For example, as a result of the second comparison, the information processing device 1 according to this embodiment multiplies the absolute value of the difference by the density and weight of the electron to calculate an integral over a predetermined space. Furthermore, in generating a trained model in the information processing device 1 according to this embodiment, the correct answer data corresponds to the exchange-correlation energy of the electron determined by numerically solving the Schrödinger equation using information about the electron distribution.

[0131] As a result, the trained model in the information processing device 1 according to this embodiment is trained using a loss function having, as a penalty term, a constraint condition that does not deviate significantly from the existing exchange-correlation energy density with good convergence used in DFT. In addition, during training of the model to be trained, the weight w Ａ Since (r) is used as the penalty term, the penalty can be made stronger in regions where the electron density n(A, r) is low. Therefore, even in regions where the electron density n(A, r) is low, the exchange-correlation energy density ε ＸＣ [n](r) can be made more stable and smooth.

[0132] Therefore, according to the information processing device 1 of this embodiment, by using a trained model that can improve stability, suppress divergence of the solution, and output the exchange-correlation energy density, it is possible to solve the Kohn-Sham equation with high convergence and reduced computational effort, similar to known methods.

[0133] For these reasons, the information processing device 1 according to this embodiment can calculate the nonlocal descriptor R(r) while incorporating long-range electron correlations and reducing the calculation cost compared to conventional methods, specifically by dramatically reducing the amount of calculation from O(N^2) to O(N). Furthermore, the information processing device 1 according to this embodiment can determine the exchange-correlation energy density of electrons using a trained model trained using a penalty term that improves convergence when solving the Kohn-Sham equation, and the nonlocal descriptor R(r) that incorporates long-range electron correlations and reduces the calculation cost.

[0134] From the above, according to the information processing device 1 of this embodiment, it is possible to solve the Kohn-Sham equation using the electron exchange correlation energy density in which the calculation amount is reduced compared to the conventional method while incorporating long-range electron correlation and the numerical stability of the functional indicating the electron exchange correlation energy density is improved. From the above, according to the information processing device 1 of this embodiment, it is possible to realize a process of calculating physical property values ​​with high accuracy while reducing calculation costs.

[0135] (Application Example) This application example involves generating a trained model that receives inputs of the density of each of a plurality of electrons based on an eigenfunction, the gradient of the density, and the kinetic energy density of the electron, and outputs the exchange-correlation energy density of the electron, with the density of the electron as a variable. The hardware configuration of the information processing device (learning device) used to generate the trained model in this application example is similar to the hardware configuration of the information processing device 1 shown in FIG. 1 , and therefore a description thereof will be omitted.

[0136] 8 is a diagram showing an example of a trained model LMA generated in this application example. As shown in FIG. 8, the trained model has an input layer ILA, an intermediate layer (hidden layer) MLA, and an output layer OLA. The input layer ILA contains the up-spin electron density n ↑ (r) and the down-spin electron density n ↓ (r), the gradient ∇n(r) of the electron density n(r), and the kinetic energy density τ(r) are input. The input to the trained model is not limited to the above. For example, the electron density n ↑(r), the ratio of electron up spin to down spin, the gradient ∇n(r) of electron density n(r), and the kinetic energy density τ(r) may be input to the input layer ILA. Also, the kinetic energy density τ(r) may be omitted in the input to the trained model. Also, the gradient ∇n(r) of electron density n(r) may be omitted in the input to the trained model. As shown in FIG. 8 , the trained model LMA outputs the electron exchange-correlation energy density ε by the OLA in the output layer. ＸＣ [n](r) is output.

[0137] The functional block configuration of the processor 31 installed in the learning device in this application example is the same as the functional block configuration of the embodiment shown in Fig. 6. Therefore, the following flowchart will explain functions that differ from the processing content in the functional block configuration of the embodiment shown in Fig. 6. Fig. 9 is a flowchart showing an example of the procedure for model generation processing according to this application example.

[0138] (Model Generation Process) (Step S901) The acquisition unit 511 acquires the electron density, the electron density gradient, and the kinetic energy density as training data. The acquisition unit 511 also acquires the correct electron exchange-correlation energy density as supervised data, which serves as a teacher when training the neural network. The acquisition unit 511 stores a training data set including the training data and the supervised data in the main storage device 33 or the auxiliary storage device 35.

[0139] (Step S902) The exchange-correlation energy density determiner 513 inputs the electron density, the gradient of the electron density, and the kinetic energy density into a learning model (neural network). Based on the output from the learning model (neural network), the exchange-correlation energy density determiner 513 determines the electron exchange-correlation energy density ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ ](r). The exchange-correlation energy density determining unit 513 determines the exchange-correlation energy density ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ ] (r) Based on the first energy E ＮＮ ＸＣ(M) [n(M) ｅｘａｃｔ The exchange-correlation energy density determining unit 513 calculates the exchange-correlation energy density ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ ] (r) and the first energy E ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ ] is stored in the main storage device 33 or the auxiliary storage device 35.

[0140] (Step S903) The exchange-correlation energy density calculation unit 515 calculates the exchange-correlation energy density ε of electrons based on the electron density and the gradient of the electron density. ｒｅｆ ＸＣ (A) [n(A)] (r) is calculated. The calculation of the exchange correlation energy density in this step is similar to that in step S702, and therefore a description thereof will be omitted.

[0141] (Step S904) The comparison unit 517 performs a first comparison to obtain the first exchange-correlation energy E ＮＮ ＸＣ (M) [n(M) ｅｘａｃｔ ] and correct data E ｅｘａｃｔ ＸＣ (M) [n(M) ｅｘａｃｔ ] for the material M, the absolute values ​​of the differences between the material M and the first energy density ε ＮＮ ＸＣ [n(M) ｅｘａｃｔ ] (r) and the second energy density ε ｒｅｆ ＸＣ Based on (A)[n(A)](r), the comparison unit 517 calculates a second loss function corresponding to the penalty term as shown in equation (17). The comparison unit 517 generates a loss function by adding the first loss function and the second loss function as shown in equation (18).

[0142] (Step S905) The learning unit 519 uses a loss function to learn a model (neural network) to be learned. For example, the learning unit 519 determines multiple weights in the neural network by backpropagation using the loss function.

[0143] (Step S906) The learning unit 519 determines whether learning for the model (neural network) to be learned is complete. A known method can be applied to determine whether learning is complete, so a description thereof will be omitted. If learning is complete (Yes in step S906), the process of step S907 is executed. If learning is not complete (No in step S906), the process from step S901 onward is repeated. Note that if multiple learning datasets for learning have been acquired in step S901, the process from step S902 onward is repeated after No in step S906.

[0144] (Step S907) The learning unit 519 stores the model (neural network) to be learned in the main storage device 33 or the auxiliary storage device 35 as a trained model.

[0145] Based on the above, an information processing device (learning device) 1 according to an application example of this embodiment inputs the density of each of multiple electrons in an equation used in density functional theory and the gradient of the density into a model to be trained, determines the exchange-correlation energy density of the electron output from the model to be trained according to multiple positions corresponding to the multiple electrons, calculates the exchange-correlation energy density of the electron by a predetermined calculation using the density and the gradient, adds the spatial integral of the absolute value of the difference between the exchange-correlation energy density output from the model to be trained and the calculated exchange-correlation energy density as a penalty term to a loss function during training, and trains the model to be trained using the loss function. As a result, the trained model in the information processing device 1 according to this embodiment is trained using a loss function (loss function) having, as a penalty term, a constraint condition that prevents a significant deviation from existing exchange-correlation energy densities with good convergence used in DFT. Therefore, the information processing device 1 according to an application example of this embodiment can improve stability, suppress divergence of solutions, and generate a trained model capable of outputting exchange-correlation energy density. As a result, by using the trained model generated by the information processing device 1 according to the application example of this embodiment, it is possible to solve the Kohn-Sham equation with high convergence and reduced computational effort, similar to known methods.

[0146] When the technical idea of ​​the embodiments is realized as an information processing method, the information processing method includes, by at least one processor, calculating, at each of a plurality of positions in a system to be calculated, a descriptor based on a value obtained by convolving the results of linear operations on the eigenfunctions of each of a plurality of electrons in an equation used in density functional theory, inputting at least the descriptor to a trained model at each of the plurality of positions to output an exchange-correlation energy density for each of the electrons, applying the exchange-correlation energy density at each of the plurality of positions to the equation to calculate an eigenfunction for each of the plurality of electrons in the system to be calculated and an eigenenergy for each of the plurality of electrons in the system, and calculating the energy of the system to be calculated using the calculated eigenfunctions and eigenenergies. The procedures and effects of the solution calculation process and model generation process related to the information processing method are similar to those described in the embodiments, and therefore will not be described again.

[0147] When the technical idea of ​​the embodiments is realized as an information processing method, the information processing method includes, by at least one processor, inputting at least information regarding electron distribution into a model to be trained, calculating a first exchange-correlation energy density of electrons, obtaining a second exchange-correlation energy density of electrons calculated using a method different from the input to the model using the information regarding the electron distribution, performing a first comparison between ground truth data associated with the information regarding the electron distribution and the first exchange-correlation energy density, performing a second comparison between the first exchange-correlation energy density and the second exchange-correlation energy density, and training the model to be trained based on the results of the first comparison and the second comparison. The procedure and effects of the model generation process related to the information processing method are similar to those described in the embodiments, and therefore will not be described again. Note that when the technical idea of ​​the embodiments is realized as an information processing program, the information processing program may cause at least one processor to execute any of the above information processing methods.

[0148] When the technical idea of ​​the application example of the embodiment is realized as an information processing method, the information processing method inputs the density of each of multiple electrons in an equation used in density functional theory and the gradient of the density into a model to be trained, determines the exchange-correlation energy density of the electron output from the model to be trained according to multiple positions corresponding to the multiple electrons, calculates the exchange-correlation energy density of the electron by a predetermined calculation using the density and the gradient, adds the spatial integral of the absolute value of the difference between the exchange-correlation energy density output from the model to be trained and the calculated exchange-correlation energy density as a penalty term to a loss function during training, and trains the model to be trained using the loss function. In this way, the information processing method functions as a model generation method. The procedure and effects of the model generation process related to the information processing method are similar to those described in the embodiment, so description thereof will be omitted.

[0149] When the technical idea in the embodiment is realized by an information processing program, the information processing program causes a computer to calculate, at each of a plurality of positions in a system to be calculated, a descriptor based on a value obtained by convolving the results of a linear operation on the eigenfunctions of each of a plurality of electrons in an equation used in density functional theory, input at least the descriptor to a trained model at each of the plurality of positions to output the exchange-correlation energy density of each of the electrons, apply the exchange-correlation energy density at each of the plurality of positions to the equation to calculate the eigenfunctions of each of the plurality of electrons in the system to be calculated and the eigenenergies of each of the plurality of electrons in the system, and calculate the energy of the system to be calculated using the calculated eigenfunctions and eigenenergies.

[0150] For example, the information processing program can be realized by installing the information processing program in a computer in a simulation device or simulation server that calculates the state of electrons in a substance and expanding the program in memory. In this case, the program that can cause a computer to execute the solution calculation process and model generation process can also be stored and distributed on a storage medium such as a magnetic disk (such as a hard disk), an optical disk (such as a CD-ROM or DVD), or a semiconductor memory. The procedure and effects of the solution calculation process and model generation process using the information processing program are the same as those in the embodiment, so a description thereof will be omitted.

[0151] When the technical concept of the application example of the embodiment is realized by an information processing program, the information processing program causes a computer to input the density of each of a plurality of electrons in an equation used in density functional theory and the gradient of the density into a model to be trained, determine the exchange-correlation energy density of the electron output from the model to be trained according to a plurality of positions corresponding to the plurality of electrons, calculate the exchange-correlation energy density of the electron by a predetermined calculation using the density and the gradient, add the spatial integral of the absolute value of the difference between the exchange-correlation energy density output from the model to be trained and the calculated exchange-correlation energy density as a penalty term to a loss function during training, and train the model to be trained using the loss function. For example, the information processing program can also be realized by installing the information processing program on a computer such as a learning device or learning server that trains a neural network and expanding it in memory. In this case, the program that can cause a computer to execute the model generation process can also be stored and distributed on a storage medium such as a magnetic disk (e.g., hard disk), optical disk (e.g., CD-ROM, DVD), or semiconductor memory. The procedure and effect of the model generation process by the information processing program are similar to those of the application example of the embodiment, and therefore a description thereof will be omitted.

[0152] Some or all of the devices in the above-described embodiments may be configured as hardware, or may be configured as information processing software (programs) executed by a CPU, GPU, or the like. In the case of software information processing, software that realizes at least some of the functions of each device in the above-described embodiments may be stored on a non-transitory storage medium (non-transitory computer-readable medium) such as a flexible disk, CD-ROM (Compact Disc-Read Only Memory), or USB memory, and the software information processing may be executed by loading the software into the computer 30. The software may also be downloaded via the communication network 5. Furthermore, the software may be implemented in a circuit such as an ASIC or FPGA, so that the information processing is executed by hardware.

[0153] The type of storage medium that stores the software is not limited. The storage medium is not limited to removable media such as magnetic disks or optical disks, but may be fixed storage media such as hard disks or memory. The storage medium may be provided inside the computer or outside the computer.

[0154] In this specification (including the claims), when the expression "at least one of a, b, and c" or "at least one of a, b, or c" (including similar expressions) is used, it includes any of a, b, c, a-b, a-c, bc, or a-bc. It may also include multiple instances of any element, such as a-a, a-bb-b, a-a-bb-cc-c, etc. Furthermore, it also includes the addition of elements other than the listed elements (a, b, and c), such as having d, as in a-bc-d.

[0155] In this specification (including the claims), when expressions such as "using data as input / based on / according to / in response to" (including similar expressions) are used, unless otherwise specified, this includes cases where various data itself is used as input, or where various data that has been processed in some way (e.g., noise-added, normalized, intermediate representation of various data, etc.) is used as input. Furthermore, when it is stated that a result is obtained "based on / according to / in response to data," this includes cases where the result is obtained based solely on the data in question, as well as cases where the result is obtained in response to other data, factors, conditions, and / or states other than the data in question. Furthermore, when it is stated that "data is output," unless otherwise specified, this includes cases where various data itself is used as output, or where various data that has been processed in some way (e.g., noise-added, normalized, intermediate representation of various data, etc.) is output.

[0156] When the terms "connected" and "coupled" are used in this specification (including the claims), they are intended as open-ended terms that include any of direct connection / coupling, indirect connection / coupling, electrically connection / coupling, communicatively connection / coupling, functionally connection / coupling, and physically connection / coupling. These terms should be interpreted appropriately depending on the context in which they are used, but any connection / coupling form that is not intentionally or naturally excluded should be interpreted as being included in these terms without any restriction.

[0157] In this specification (including the claims), when the expression "A configured to B" is used, it may include a situation in which the physical structure of element A has a configuration capable of performing operation B, and a permanent or temporary setting / configuration of element A is configured / set to actually perform operation B. For example, when element A is a general-purpose processor, it is sufficient that the processor has a hardware configuration capable of performing operation B, and is configured to actually perform operation B by setting a permanent or temporary program (instruction). Furthermore, if element A is a dedicated processor or dedicated arithmetic circuit, etc., it is sufficient that the circuit structure of the processor is implemented to actually execute operation B, regardless of whether control instructions and data are actually attached.

[0158] When used in this specification (including the claims), terms implying containing or possessing (e.g., "comprising," "including," "having," etc.) are intended to be open-ended terms that include cases in which something other than the object indicated by the object of the term is contained or possessed. When the object of such a term implies no quantity or a singular number (e.g., an article such as "a" or "an"), the expression should be construed as not being limited to a specific number.

[0159] In this specification (including the claims), although expressions such as "one or more" or "at least one" are used in some places and expressions that do not specify a quantity or that imply a singular number (expressions using the articles "a" or "an") are used in other places, the latter expressions are not intended to mean "one." In general, expressions that do not specify a quantity or that imply a singular number (expressions using the articles "a" or "an") should be interpreted as not necessarily being limited to a specific number.

[0160] In this specification, when a particular advantage / result is described as being obtained with respect to a particular configuration of a certain embodiment, it should be understood that the same advantage / result can also be obtained with one or more other embodiments having the same configuration, unless otherwise stated. However, it should be understood that the presence or absence of the effect generally depends on various factors, conditions, and / or states, etc., and that the effect is not necessarily obtained with the configuration. The effect is merely obtained by the configuration described in the embodiment when various factors, conditions, and / or states, etc. are satisfied, and the effect does not necessarily occur in a claimed invention that defines the same configuration or a similar configuration.

[0161] When terms such as "maximize" are used in this specification (including the claims), they include finding a global maximum, finding an approximation of a global maximum, finding a local maximum, and finding an approximation of a local maximum, and should be interpreted appropriately according to the context in which the term is used. They also include probabilistic or heuristic approximations of these maxima. Similarly, when terms such as "minimize" are used, they include finding a global minimum, finding an approximation of a global minimum, finding a local minimum, and finding an approximation of a local minimum, and should be interpreted appropriately according to the context in which the term is used. They also include probabilistic or heuristic approximations of these minima. Similarly, when terms such as "optimize" are used, they include finding a global optimum, finding an approximation of a global optimum, finding a local optimum, and finding an approximation of a local optimum, and should be interpreted appropriately according to the context in which the term is used. It also includes finding approximations of these optimum values ​​probabilistically or heuristically.

[0162] In this specification (including claims), when multiple pieces of hardware perform a predetermined process, the pieces of hardware may cooperate to perform the predetermined process, or some of the hardware may perform all of the predetermined process. Furthermore, some of the hardware may perform part of the predetermined process, and other hardware may perform the rest of the predetermined process. In this specification (including claims), when an expression such as "one or more pieces of hardware perform a first process, and the one or more pieces of hardware perform a second process" is used, the hardware performing the first process and the hardware performing the second process may be the same or different. In other words, it is sufficient that the hardware performing the first process and the hardware performing the second process are included in the one or more pieces of hardware. Note that the hardware may include an electronic circuit or a device including an electronic circuit.

[0163] In this specification (including the claims), when multiple storage devices (memories) store data, each of the multiple storage devices (memories) may store only a portion of the data, or may store the entire data.

[0164] Although the embodiments of the present disclosure have been described in detail above, the present disclosure is not limited to the individual embodiments described above. Various additions, modifications, substitutions, partial deletions, etc. are possible within the scope of the conceptual idea and spirit of the present invention derived from the content defined in the claims and their equivalents. For example, in all of the above-described embodiments, when numerical values ​​or formulas are used in the explanation, they are shown as examples and are not limited to these. Furthermore, the order of each operation in the embodiments is shown as an example and is not limited to these.

[0165] REFERENCE SIGNS LIST 1 Information processing device 5 Communication network 9A External device 9B External device 30 Computer 31 Processor 33 Main memory device 35 Auxiliary memory device 37 Network interface 39 Device interface 41 Bus 311 Setting unit 313 Non-local descriptor calculation unit 315 Exchange-correlation energy density determination unit 317 Equation calculation unit 319 Evaluation unit 511 Acquisition unit 513 Exchange-correlation energy density determination unit 515 Exchange-correlation energy density calculation unit 517 Comparison unit 519 Learning unit

Claims

1. An information processing device comprising at least one memory and at least one processor, wherein the at least one processor calculates, at each of a plurality of positions in a system to be calculated, a descriptor based on a value obtained by convolving the results of linear operations on the eigenfunctions of each of a plurality of electrons in an equation used in density functional theory; inputs at least the descriptor into a trained model at each of the plurality of positions and outputs the exchange-correlation energy density of each of the plurality of electrons; applies the exchange-correlation energy density at each of the plurality of positions to the equation to calculate the eigenfunctions and eigenenergies of each of the plurality of electrons; and calculates the energy of the system using the eigenfunctions and the eigenenergies.

2. The information processing device according to claim 1, wherein the at least one processor calculates the descriptor based on a value obtained by convolving a result of the linear operation on a basis function that forms a basis of the eigenfunction and an expansion coefficient corresponding to the basis function.

3. The information processing device according to claim 2, wherein the at least one processor calculates the descriptor based on the sum of products of the expansion coefficients and a value obtained by convolving a result of the linear operation on a basis function that forms a basis of the eigenfunction.

4. The information processing device according to claim 3, wherein the at least one processor calculates the descriptor by summing the squares of the absolute values ​​of the sums of products over a plurality of eigenstates occupied by the plurality of electrons.

5. The information processing device according to claim 1, wherein the descriptor represents the degree of influence of interactions by the plurality of electrons from positions distant from each of the plurality of positions, with each of the plurality of positions being the center.

6. The information processing device according to claim 1, wherein the equation is an energy variation equation having a functional derivative of the density of the plurality of electrons with respect to the exchange-correlation energy of the plurality of electrons, or an equation based on the energy variation equation.

7. The information processing device according to claim 1, wherein the equation is a Kohn-Sham equation.

8. The information processing device according to claim 7, wherein the energy comprises Kohn-Sham kinetic energy of the plurality of electrons, an interaction energy between the plurality of electrons and an atom, a Coulomb energy of the plurality of electrons, and an exchange-correlation energy of the plurality of electrons, and the Kohn-Sham kinetic energy, the interaction energy, the Coulomb energy, and the exchange-correlation energy are functionals of electron density.

9. An information processing device according to any one of claims 1 to 8, wherein the trained model is generated by: inputting at least information regarding electron distribution into a model to be trained to calculate a first exchange-correlation energy density of electrons; obtaining a second exchange-correlation energy density of the electrons calculated using the information regarding the electron distribution by a method different from the input to the model; performing a first comparison between ground truth data associated with the information regarding the electron distribution and the first exchange-correlation energy density; performing a second comparison between the first exchange-correlation energy density and the second exchange-correlation energy density; and training the model to be trained based on the results of the first comparison and the results of the second comparison.

10. An information processing device comprising at least one memory and at least one processor, wherein the at least one processor inputs at least information regarding electron distribution into a model to be learned to calculate a first exchange-correlation energy density of electrons, obtains a second exchange-correlation energy density of the electrons calculated using a method different from that input to the model using the information regarding the electron distribution, performs a first comparison between ground truth data associated with the information regarding the electron distribution and the first exchange-correlation energy density, performs a second comparison between the first exchange-correlation energy density and the second exchange-correlation energy density, and learns the model to be learned based on the results of the first comparison and the results of the second comparison.

11. The information processing device according to claim 10, wherein the at least one processor trains the model to be trained using a first loss function based on the first comparison and the second comparison.

12. The information processing device according to claim 10, wherein the at least one processor trains the model to be trained using a second loss function based on the first comparison and a third loss function based on the second comparison.

13. The information processing device according to claim 10, wherein the information relating to the electron distribution includes at least one of a non-local descriptor obtained by convolving the results of a linear operation on the eigenfunctions of each of a plurality of electrons in an equation used in density functional theory, or a local descriptor based on the eigenfunctions.

14. The information processing device according to claim 10, wherein the different methods are at least either an analytical method or a numerical method that uses information about the electron distribution.

15. The information processing device according to claim 10, wherein the different method is a method of calculating the second exchange-correlation energy density by inputting information about the electron distribution into another trained model.

16. An information processing device according to any one of claims 10 to 15, wherein the at least one processor calculates a value based on the difference between the first exchange-correlation energy density and the second exchange-correlation energy density as a result of the second comparison.

17. The information processing device according to claim 16, wherein the at least one processor multiplies the absolute value of the difference as a result of the second comparison by the density and weight of the electron to calculate an integral over a predetermined space.

18. An information processing device according to any one of claims 10 to 15, wherein the correct data corresponds to the exchange-correlation energy of the electrons determined by numerically solving the Schrodinger equation using information about the electron distribution.

19. An information processing method comprising: calculating, by at least one processor, at each of a plurality of positions in a system to be calculated, a descriptor based on a value obtained by convolving the results of linear operations on the eigenfunctions of each of a plurality of electrons in an equation used in density functional theory; inputting at least the descriptor into a trained model at each of the plurality of positions and outputting the exchange-correlation energy density of each of the plurality of electrons; applying the exchange-correlation energy density at each of the plurality of positions to the equation to calculate the eigenfunctions and eigenenergies of each of the plurality of electrons; and calculating the energy of the system using the eigenfunctions and the eigenenergies.

20. An information processing method comprising: inputting at least information regarding electron distribution into a model to be learned, calculating a first exchange-correlation energy density of electrons, using the information regarding the electron distribution to obtain a second exchange-correlation energy density of the electrons calculated by a method different from the input to the model, performing a first comparison between ground truth data associated with the information regarding the electron distribution and the first exchange-correlation energy density, performing a second comparison between the first exchange-correlation energy density and the second exchange-correlation energy density, and learning the model to be learned based on the results of the first comparison and the results of the second comparison.

21. An information processing program causing at least one processor to execute the information processing method according to claim 19 or 20.

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