Machine learning potential function design method for UO2 and fission products thereof

Abstract

The invention provides a machine learning potential function design method for UO2 and a fission product thereof, and the method comprises the steps: S1, constructing a UO2-Xe / I unit cell with the fission product, and carrying out the optimization of the UO2-Xe / I unit cell; s2, the optimized UO2-Xe / I unit cell structure is disturbed and simulated, and a training data set is generated; s3, establishing a machine learning potential function; s4, performing molecular dynamics simulation on the machine learning potential function; s5, performing density functional theory calculation on the candidate configuration to obtain accurate structure-energy-force data, and repeating the steps S3 to S5 to perform iterative calculation; s6, verifying the trained machine learning potential function to obtain an accurate machine learning potential function; and S7, performing molecular dynamics simulation by using an accurate machine learning potential function. The precise DP model is obtained through machine learning, and a theoretical basis is provided for research of a diffusion mechanism of fission products in UO2 and molecular dynamics simulation of other mechanical properties.

Description

Technical Field

[0001] The present application relates to the field of computational materials science and technology, and in particular to a method for designing a machine learning potential function for UO2 and its fission products. Background Art

[0002] During the operation of the reactor, uranium atoms in the nuclear fuel UO2 absorb neutrons and undergo fission, releasing energy and producing a large amount of fission products. This has a very important impact on the safe and stable operation of the fuel elements. Among them, xenon (Xe) and iodine (I) are typical fission products with high volatility. Xe is a typical inert gas and is basically insoluble in the UO2 matrix. They can migrate to the grain boundaries and form bubbles, causing the fuel to expand and reduce mechanical and thermal properties. Part of Xe is directly produced by fission, and part is produced by the decay of I. The solubility of I in UO2 is also very low, and the presence of I is also an important factor leading to stress corrosion cracking of zirconium alloy cladding materials. Therefore, studying the diffusion behavior of Xe and I in UO2 is of great significance in the basic research of nuclear fuel.

[0003] However, due to the inherent resolution limitations of experimental equipment, it is difficult to characterize the diffusion behavior and diffusion coefficient of fission atoms. Molecular dynamics (MD) simulation is a powerful method in this regard, in which the accuracy of the potential function determines the accuracy of the molecular dynamics simulation. However, at present, many fission products do not have corresponding potential functions. In recent years, as a typical machine learning model, the Deep Potential (DP) model has demonstrated accurate simulation results at the level of Density Functional Theory (DFT) in many different research fields. Summary of the invention

[0004] The present application provides a machine learning potential function design method for UO2 and its fission products to solve the problem that the diffusion behavior and diffusion coefficient of fission products are difficult to characterize and measure experimentally.

[0005] In order to solve the above technical problems, the present application provides a method for designing a machine learning potential function for UO2 and its fission products, comprising the following steps: step S1, constructing a UO2-Xe / I unit cell with fission products, and optimizing the UO2-Xe / I unit cell; step S2, perturbing the optimized UO2-Xe / I unit cell structure, performing ab initio molecular dynamics (AIMD) simulation, and generating a training data set; step S3, performing machine learning force field training on the training data set, and establishing a machine learning potential function; step S4, performing molecular dynamics simulation on the machine learning potential function, and identifying The atomic structure with force deviation within the preset range is taken as the candidate configuration; step S5, the candidate configuration is subjected to density functional theory calculation to obtain accurate structure-energy-force data, and the structure-energy-force data is merged into the training set of subsequent iterations, and steps S3 to S5 are repeated to perform iterative calculations to train the machine learning potential function (DP model); step S6, the trained machine learning potential function is verified, and the calculation results of the machine learning potential function and the density functional theory are compared to verify the accurate machine learning potential function; step S7, molecular dynamics simulation is performed using the accurate machine learning potential function.

[0006] In some exemplary embodiments, in step S1, a UO2-Xe / I unit cell with fission products is constructed, and the UO2-Xe / I unit cell is optimized, including: first, based on the periodicity and atomic coordinates of the fluorite-type face-centered cubic structure, a UO2 perfect unit cell is constructed using VESTA software; then a 2×2×2 cell expansion operation is performed on the UO2 perfect unit cell, that is, 8 UO2 unit cells are stacked in two layers, 4 in each layer; next, starting from the origin, UO2 is numbered in a counterclockwise direction, the UO2 in the lower layer is numbered 1 to 4, and the UO2 in the upper layer is numbered 5 to 8; finally, UO2 numbered 1, 2, and 5 are selected, and fission atoms Xe / I are inserted into the octahedral gap of each UO2, Insert 3 fission atoms in total to construct the first type of UO2-Xe / I structure; select UO2 numbered 1, 2 and 7, insert fission atoms Xe / I in each octahedral gap of UO2, insert 3 fission atoms in total to construct the second type of UO2-Xe / I structure; select UO2 numbered 1, 3 and 5, insert fission atoms Xe / I in each octahedral gap of UO2, insert 3 fission atoms in total to construct the third type of UO2-Xe / I structure; select UO2 numbered 1, 3 and 6, insert fission atoms Xe / I in each octahedral gap of UO2, insert 3 fission atoms in total to construct the fourth type of UO2-Xe / I structure.

[0007] In some exemplary embodiments, the UO2-Xe / I unit cell is optimized, including: performing structural relaxation optimization on the UO2-Xe / I unit cell, using the projected augmented wave method to link the full electron wave function with the pseudo wave function, using the generalized gradient approximation method of the Perdew-Burke-Ernzerhof form to exchange the correlation function, with a cutoff energy of 400 eV and an energy convergence limit of 10 -5 eV, the force convergence criterion parameter was -0.02 eV / Å, and the maximum number of electronic self-consistent cycles was set to 1000 to ensure the convergence of the structure.

[0008] In some exemplary embodiments, in step S2, the optimized UO2-Xe / I unit cell structure is perturbed, including: changing the atomic coordinate positions and the size of the simulation box, generating 30 different perturbed structures for each optimized structure, performing 10 steps of ab initio molecular dynamics simulations in a 50K NVT ensemble with a step length of 2 femtoseconds, obtaining 10 atomic structure-energy corresponding data for each configuration, and integrating the data obtained from all perturbed configurations as the initial training data for the machine learning potential function.

[0009] In some exemplary embodiments, in step S3, the machine learning force field training includes 4×10 5 ~8×10 5 Step 1, 4 independent machine learning potential functions are established for each training, and each independent machine learning potential function has a different initial random number.

[0010] In some exemplary embodiments, in step S4, molecular dynamics simulation is performed on the machine learning potential function, including: using LAMMPS to perform molecular dynamics simulation on the structure with an NPT ensemble under at least 3 different temperature conditions, and during the simulation, atomic structures with a force deviation range of 0.1~0.3 eV / Å are identified as candidate configurations.

[0011] In some exemplary embodiments, in step S5, the calculation method of density functional theory is: setting the maximum ion step to 0, performing static relaxation simulation on the candidate configuration, and obtaining its energy-force-structure data.

[0012] In some exemplary embodiments, in step S5, after iterative calculations are repeated from step S3 to step S5, the final data sets include: under the temperature conditions of 50 K, 100 K, 200 K, 300 K, 400 K, 500 K, 600 K, 700 K, 800 K, 900 K, 1000 K, 1100 K, 1200 K, 1300 K, 1400 K, 1500 K, 1600 K, 1700 K, and 1800 K, the pressure is controlled to be 1 bar, and molecular dynamics simulation of the structure is performed using LAMMPS with an NPT ensemble to obtain the corresponding data sets.

[0013] In some exemplary embodiments, in step S6, the trained machine learning potential function is verified, and the machine learning potential function is compared with the density functional theory calculation results to verify the accurate machine learning potential function, including: comparing the machine learning potential function with the predicted results of density functional theory, analyzing the force and energy errors of the machine learning potential function for the test data set, and verifying the accuracy of the machine learning potential function.

[0014] In some exemplary embodiments, in step S7, a molecular dynamics simulation is performed using an accurate machine learning potential function, including: applying the trained machine learning potential function to the LAMMPS software, calculating the basic physical properties of UO2, and simulating the diffusion behavior and diffusion coefficient of fission atoms in the UO2-Xe and UO2-I systems.

[0015] The technical solution provided by the embodiments of the present application has at least the following advantages:

[0016] The present application provides a method for designing a machine learning potential function for UO2 and its fission products, comprising the following steps: step S1, constructing a UO2-Xe / I unit cell with fission products, and optimizing the UO2-Xe / I unit cell; step S2, perturbing the optimized UO2-Xe / I unit cell structure, performing ab initio molecular dynamics simulation, and generating a training data set; step S3, performing machine learning force field training on the training data set, and establishing a machine learning potential function; step S4, performing molecular dynamics simulation on the machine learning potential function, and identifying atomic structures with force deviations within a preset range as candidate configurations; step S5, performing density functional theory calculations on the candidate configurations to obtain accurate structure-energy-force data, and merging the structure-energy-force data into subsequent iterative training sets, repeating steps S3 to S5 for iterative calculations, and training the machine learning potential function; step S6, verifying the trained machine learning potential function, comparing the calculation results of the machine learning potential function and the density functional theory, and verifying that an accurate machine learning potential function is obtained; step S7, using the accurate machine learning potential function for molecular dynamics simulations.

[0017] This application constructs and perturbs the UO2 structure containing fission products, and then performs machine learning force field training, molecular dynamics simulation and model convergence, and finally realizes accurate molecular dynamics calculation simulation of UO2-base structure based on DP model. The calculated basic properties of UO2 are more accurate than traditional potential functions. For the calibration of diffusion coefficient, traditional ab initio molecular dynamics simulation can generally only simulate 2×2×2 super cells, while deep potential energy accelerated molecular dynamics simulation can easily simulate 12×12×12 super cells, 216 times the size range, which greatly improves the calculation efficiency and reduces the calculation cost while ensuring the calculation accuracy. For the simulation of diffusion behavior, molecular dynamics simulation based on DP model can simulate the process of fission atom diffusion nucleation, which is a phenomenon that is difficult to study both in experiments and on DFT scale. This application provides basic support for high-precision simulation and rational design of UO2 and its fission products, and is expected to play a significant role in related fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] One or more embodiments are exemplarily described by pictures in the corresponding drawings, and these exemplified descriptions do not constitute limitations on the embodiments. Unless otherwise stated, the pictures in the drawings do not constitute proportional limitations.

[0019] Figure 1 A schematic flow chart of a method for designing a machine learning potential function for UO2 and its fission products provided in Example 1 of the present application.

[0020] Figure 2 A schematic diagram of the process framework of a method for designing a machine learning potential function for UO2 and its fission products provided in Example 1 of the present application.

[0021] Figure 3 Comparison of the machine learning potential function (uranium-oxygen-xenon) prediction provided in Example 1 of the present application with the density functional theory calculation results.

[0022] Figure 4 Comparison of the machine learning potential function (uranium-oxygen-iodine) prediction provided in Example 2 of the present application with the density functional theory calculation results.

[0023] Figure 5 The mean square displacement of xenon atoms obtained by the molecular dynamics simulation provided in Example 4 of the present application.

[0024] Figure 6 The mean square displacement of iodine atoms obtained by the molecular dynamics simulation provided in Example 5 of the present application.

[0025] Figure 7 The diffusion, nucleation and growth process of xenon atoms obtained by molecular dynamics simulation provided in Example 6 of the present application.

[0026] Figure 8 The diffusion, nucleation and growth process of iodine atoms obtained by molecular dynamics simulation provided in Example 7 of the present application. DETAILED DESCRIPTION

[0027] As can be seen from the background technology, it is currently difficult to characterize the diffusion behavior and diffusion coefficient of fission atoms due to the inherent resolution limitation of experimental equipment.

[0028] In order to solve the above technical problems, the embodiment of the present application provides a method for designing a machine learning potential function of UO2 and its fission products, comprising the following steps: step S1, constructing a UO2-Xe / I unit cell with fission products, and optimizing the UO2-Xe / I unit cell; step S2, perturbing the optimized UO2-Xe / I unit cell structure, performing ab initio molecular dynamics simulation, and generating a training data set; step S3, performing machine learning force field training on the training data set, and establishing a machine learning potential function; step S4, performing molecular dynamics simulation on the machine learning potential function, and identifying The atomic structure with output deviation within the preset range is taken as a candidate configuration; step S5, the candidate configuration is subjected to density functional theory calculation to obtain accurate structure-energy-force data, and the structure-energy-force data is merged into the training set of subsequent iterations, and steps S3 to S5 are repeated for iterative calculation to train the machine learning potential function; step S6, the trained machine learning potential function is verified, and the calculation results of the machine learning potential function and density functional theory are compared to verify the accurate machine learning potential function; step S7, molecular dynamics simulation is performed using the accurate machine learning potential function. The machine learning potential function design method for UO2 and its fission products provided in this application is suitable for describing the basic physical properties of UO2 and the diffusion behavior and diffusion coefficient of fission atoms in UO2-Xe and UO2-I systems, so as to solve the problem that the diffusion behavior and diffusion coefficient of fission products are difficult to characterize and measure experimentally.

[0029] The various embodiments of the present application will be described in detail below in conjunction with the accompanying drawings so that those skilled in the art can better understand the present application and implement it.

[0030] The following contents are all for the purpose of describing in detail the specific examples of the specific implementation process provided by the technical solutions to be protected in this application. However, this application may also adopt other implementation methods different from this description. Technical personnel in this field can, under the guidance of the ideas of this application, adopt technical solutions with different viewpoints to implement the contents within the scope of this application. Therefore, this application is not strictly limited to the specific implementation cases below.

[0031] See also Figure 1 , the present application provides a method for designing a machine learning potential function of UO2 and its fission products, comprising the following steps:

[0032] Step S1, constructing a UO2-Xe / I unit cell with fission products, and optimizing the UO2-Xe / I unit cell.

[0033] Step S2, perturb the optimized UO2-Xe / I unit cell structure, perform ab initio molecular dynamics simulation, and generate a training data set.

[0034] Step S3: Perform machine learning force field training on the training data set to establish a machine learning potential function.

[0035] Step S4: Perform molecular dynamics simulation on the machine learning potential function, and identify atomic structures whose force deviations are within a preset range as candidate configurations.

[0036] Step S5: Perform density functional theory (DFT) calculations on the candidate configurations to obtain accurate structure-energy-force data, and merge the structure-energy-force data into the training set of subsequent iterations, repeat steps S3 to S5 for iterative calculations, and train the machine learning potential function.

[0037] Step S6: verify the trained machine learning potential function, compare the machine learning potential function with the calculation results of density functional theory, and verify that an accurate machine learning potential function is obtained.

[0038] Step S7: Use accurate machine learning potential function to perform molecular dynamics simulation.

[0039] Since the DP model has demonstrated accurate simulation results at the DFT level in many different research fields, in response to this challenge, this application uses first-principles calculations and classical force field simulations to develop two sets of DP models, denoted as DP Xe and DP I , corresponding to the UO2-Xe and UO2-I systems.

[0040] Specifically, the machine learning potential function design method for UO2 and its fission products provided in this application is as follows: Figure 2As shown, the following steps are included: the data set for potential function model training is prepared by ab initio molecular dynamics (AIMD) calculation; the AIMD data set is trained and tested using a deep potential (DP) model to obtain a fitted molecular dynamics machine learning potential function, namely, a DP model; the physical properties of a perfect UO2 crystal are calculated using the DP model, and compared with the traditional potential function results and experimental values; the DP model is used to simulate the atomic diffusion in the UO2-Xe / I system at a specified fission product concentration and temperature, and the diffusion coefficient of the fission product atoms (Xe and I) is obtained, and the diffusion behavior of the fission atoms (Xe and I) is simulated. The present application obtains an accurate DP model through machine learning, which provides a theoretical basis for the study of the diffusion mechanism of its fission products in UO2 and the molecular dynamics simulation of other mechanical properties. The machine learning potential function design method for UO2 and its fission products provided in the present application can be used for low-cost and high-efficiency preliminary screening, and provide theoretical guidance for experiments. At the same time, the successful establishment of the machine learning potential function of the present application is of great significance for the introduction of machine learning potential functions into MD simulation research on nuclear fuels and their fission products.

[0041] In some embodiments, in step S1, a UO2-Xe / I unit cell with fission products is constructed, and the UO2-Xe / I unit cell is optimized, including: first, based on the periodicity and atomic coordinates of the fluorite-type face-centered cubic structure (space group Fm3m), a UO2 perfect unit cell is constructed using VESTA software; then a 2×2×2 cell expansion operation is performed on the UO2 perfect unit cell, that is, 8 UO2 unit cells are stacked in two layers, 4 in each layer; next, starting from the origin, UO2 is numbered in a counterclockwise direction, the UO2 in the lower layer is numbered 1 to 4, and the UO2 in the upper layer is numbered 5 to 8; finally, UO2 numbered 1, 2, and 5 are selected, and fission atoms Xe / I are inserted into the octahedral gap of each UO2, for a total of 3 fission atoms. , construct the first type of UO2-Xe / I structure; select UO2 numbered 1, 2 and 7, insert fission atoms Xe / I in each octahedral gap of UO2, insert 3 fission atoms in total, and construct the second type of UO2-Xe / I structure; select UO2 numbered 1, 3 and 5, insert fission atoms Xe / I in each octahedral gap of UO2, insert 3 fission atoms in total, and construct the third type of UO2-Xe / I structure; select UO2 numbered 1, 3 and 6, insert fission atoms Xe / I in each octahedral gap of UO2, insert 3 fission atoms in total, and construct the fourth type of UO2-Xe / I structure; a total of 4 types of UO2-Xe / I unit cell structures are constructed.

[0042] In some embodiments, the UO2-Xe / I unit cell is optimized, including: performing structural relaxation optimization on the UO2-Xe / I unit cell, using a projected augmented wave method to link the full electron wave function and the pseudo wave function, using a generalized gradient approximation (GGA) method in the form of Perdew-Burke-Ernzerhof (PBE) to exchange correlation functions, with a cutoff energy of 400 eV and an energy convergence limit of 10 -5 eV, the force convergence criterion parameter was -0.02 eV / Å, and the maximum number of electronic self-consistent cycles was set to 1000 to ensure the convergence of the structure, and preliminary structural relaxation optimization was performed.

[0043] In some embodiments, in step S2, the optimized UO2-Xe / I unit cell structure is perturbed, including: changing the atomic coordinate position and the size of the simulation box, generating 30 different perturbed structures for each optimized structure, and performing 10 steps of ab initio molecular dynamics simulation in a 50K NVT ensemble with a step length of 2 femtoseconds. 10 atomic structure-energy corresponding data are obtained for each configuration, and the data obtained from all perturbed configurations are integrated as the initial training data for the machine learning potential function (DP model).

[0044] In some embodiments, in step S3, the machine learning force field training includes 4×10 5 ~8×10 5 Step 1, 4 independent machine learning potential functions are established for each training, and each independent machine learning potential function has a different initial random number.

[0045] In some embodiments, in step S4, a molecular dynamics simulation is performed on the machine learning potential function, including: using LAMMPS to perform molecular dynamics simulation on the structure with an NPT ensemble under at least 3 different temperature conditions, and during the simulation, atomic structures with a force deviation range of 0.1~0.3 eV / Å are identified as candidate configurations, and a maximum of 200 candidate configurations are selected.

[0046] In some embodiments, in step S5, the density functional theory calculation method is: set the maximum ion step to 0, perform static relaxation simulation on the candidate configuration, obtain its energy-force-structure data, and merge it into a new training set to provide a basis for subsequent iterative calculations. Iterative calculation, repeat steps S3-S5.

[0047] In some embodiments, in step S5, after repeating steps S3 to S5 for iterative calculation, the final data set includes: under the temperature conditions of 50 K, 100 K, 200 K, 300 K, 400 K, 500 K, 600 K, 700 K, 800 K, 900 K, 1000 K, 1100 K, 1200 K, 1300 K, 1400 K, 1500 K, 1600 K, 1700 K, and 1800 K, the pressure is controlled to be 1 bar, and the molecular dynamics simulation of the structure is performed using LAMMPS with the NPT ensemble to obtain the corresponding data set. Finally, the failed configuration accounts for less than 1%. The final data set is simulated under the above-mentioned different temperature conditions.

[0048] In some embodiments, in step S6, the trained machine learning potential function is verified, and the machine learning potential function is compared with the density functional theory calculation results to verify the accurate machine learning potential function, including: comparing the machine learning potential function with the prediction results of density functional theory, analyzing the force and energy errors of the machine learning potential function for the test data set, and verifying the accuracy of the machine learning potential function.

[0049] Specifically, step S6 mainly analyzes the force and energy errors of the DP model for the test data set, compares the prediction results of DP and DFT, and verifies the accuracy of the DP model.

[0050] In some embodiments, in step S7, a molecular dynamics simulation is performed using an accurate machine learning potential function, including: applying the trained machine learning potential function to the LAMMPS software, calculating the basic physical properties of UO2, and simulating the diffusion behavior and diffusion coefficient of fission atoms in the UO2-Xe and UO2-I systems.

[0051] This application proposes a method for designing machine learning potential functions for UO2 and its fission products (Xe and I), which is suitable for describing the basic physical properties of UO2 and the diffusion behavior and diffusion coefficient of fission atoms in UO2-Xe and UO2-I systems. This application constructs and perturbs the structure of UO2 containing fission products, then conducts machine learning force field training, molecular dynamics simulation and model convergence, and finally realizes accurate molecular dynamics calculation simulation of UO2-base structure based on DP model. The calculated basic properties of UO2 are more accurate than traditional potential functions. For the calibration of diffusion coefficient, traditional ab initio molecular dynamics simulation can generally only simulate 2×2×2 super cells, while deep potential energy accelerated molecular dynamics simulation can easily simulate up to 12×12×12, 216 times the size range, which greatly improves the calculation efficiency and reduces the calculation cost while ensuring the calculation accuracy. For the simulation of diffusion behavior, molecular dynamics simulation based on DP model can simulate the process of fission atom diffusion nucleation, which is a phenomenon that is difficult to study both in experiments and on DFT scale. This application provides basic support for the high-precision simulation and rational design of UO2 and its fission products, and is expected to play a significant role in related fields.

[0052] The following is a detailed introduction to the machine learning potential function design method for UO2 and its fission products provided in the present application through specific examples.

[0053] Example 1

[0054] A method for designing a machine learning potential function of UO2 and its fission product Xe, the design method comprising the following steps:

[0055] Step 1: Establish and optimize the structure.

[0056] The method for constructing the UO2-Xe unit cell with fission products is as follows: Based on the periodicity and atomic coordinates of the fluorite-type face-centered cubic structure (space group Fm3m), the VESTA software is used to construct a UO2 perfect unit cell, and then a 2×2×2 expansion operation is performed on it, which is equivalent to 8 UO2 unit cells stacked in two layers, 4 in each layer. Starting from the origin, in a counterclockwise direction, the UO2 in the lower layer is numbered 1-4, and the UO2 in the upper layer is numbered 5-8. Subsequently, one fission atom Xe is inserted into each UO2 octahedral gap in the combination of 1-2-5, 1-2-7, 1-3-5, and 1-3-6, and a total of 3 Xe atoms are inserted to construct 4 types of structures.

[0057] After constructing the UO2-Xe unit cell of the fission product, the structure was optimized by using the projected augmented wave (PAW) method to link the full electron wave function with the pseudo wave function, using the generalized gradient approximation of the PBE form to approximate the exchange correlation function of the GGA, setting the cutoff energy to 400 eV, and the energy convergence criterion parameter to 10 -5 eV, the force convergence criterion parameter was -0.02 eV / Å, and the maximum number of electronic self-consistent cycles was set to 1000 to ensure the convergence of the structure, and preliminary structural relaxation optimization was performed.

[0058] Step 2: Generate a dataset.

[0059] The optimized structure was perturbed, and the perturbation included changing the atomic coordinate position and the size of the simulation box, where the box perturbation limit was 0.03 and the atomic perturbation limit was 0.01. For each optimized structure, 30 different perturbed structures were generated. The original structure and each perturbed structure were simulated by ab initio molecular dynamics (AIMD) for 10 steps in the NVT ensemble at 50K, with a step length of 2fs. Each configuration obtained 10 atomic structure-energy corresponding data, and the data obtained from all perturbed configurations were integrated to generate a data set of 4×31×10=1240 frames as the initial training data for the DP model.

[0060] Step 3: DP model training.

[0061] Start training the potential function. The specific details of the machine learning force field training are as follows: The number of training steps is 4×10 5 -8×10 5 Step 4, each training builds 4 independent machine learning models, each independent machine learning model has a different initial random number.

[0062] In the DP model, the potential energy E of the atomic configuration is expressed as the sum of the atomic energies Ei of each atom i, where Ei is obtained from the descriptor by fitting the network. In this work, the descriptor describes the local environment of atom i within the cutoff radius Rc, which is set to 11 Å here. In order to maintain the translational, rotational and permutation symmetries of the descriptor, the embedding network is an essential component of the method. In addition, a smoothed version of the DP model is used to eliminate the discontinuity caused by the cutoff radius, and the inner cutoff radius is set to 10 Å. The sizes of the embedded and fitted networks are set to (25, 50, 100) and (240, 240, 240), respectively. During the training process, pε, pf and pξ are the weight coefficients corresponding to the energy, force and virial tensors, respectively. pε increases from 0.02 to 2, pf decreases from 1000 to 1, and pξ increases from 0.02 to 1.

[0063] Step 4: Short-term MD simulation.

[0064] Then, molecular dynamics simulation was performed, and the accuracy of the potential function was preliminarily verified. The method of molecular dynamics simulation was as follows: the MD calculation in each cycle was performed under at least 3 different temperature conditions, with a control pressure of 1 bar, a step size of 1 fs, and a step number of 4000-10000. LAMMPS was used to perform molecular dynamics simulation of the structure with the NPT ensemble. During the simulation, the force deviations calculated by the 4 models were statistically analyzed. Atomic structures with a force deviation range of 0.1 eV / Å were identified as accurate configurations, and atomic structures with a force deviation of more than 0.3 eV / Å were identified as failed configurations. Atomic structures with a force deviation range of 0.1-0.3 eV / Å were identified as candidate configurations, and up to 200 candidate configurations were selected.

[0065] Step 5: DFT static relaxation.

[0066] The DFT method is used to perform static relaxation simulation on the candidate configurations obtained. The specific settings are: the interaction between ions and electrons uses the plane wave pseudopotential method (PAW), the exchange correlation potential of electron-electron interaction uses the generalized gradient approximation GGA, and the exchange and correction energy between electrons is processed by PBE. The maximum ion step is set to 0. After calculating the energy-force-structure data, it is merged into a new training set to provide a basis for subsequent iterative calculations.

[0067] Step 6: Iterate the calculation.

[0068] Step 3, step 4, and step 5 are repeated repeatedly until the obtained DP model meets the accuracy requirements.

[0069] In the end, the proportion of failed configurations was less than 1%. The final data set was simulated at several different temperature conditions: 50 K, 100 K, 200 K, 300 K, 400 K, 500 K, 600 K, 700 K, 800 K, 900 K, 1000 K, 1100 K, 1200 K, 1300 K, 1400 K, 1500 K, 1600 K, 1700 K, and 1800 K.

[0070] Step 7: DP model verification.

[0071] Analyze the force and energy errors of the DP model for the test data set, and the prediction effect is as follows Figure 3 As shown, by comparing the MD simulation results based on the machine learning potential function model and the DFT calculation results, the two showed good consistency, with an accuracy rate of more than 98%, which verified the rationality of the trained model.

[0072] Step 8: DP model application.

[0073] The trained DP model (denoted as DPXe ) was applied to LAMMPS software for subsequent calculations.

[0074] Example 2

[0075] A method for designing a machine learning potential function for UO2 and its fission product I. The design method is the same as that in Example 1, except that the corresponding Xe atoms are replaced by I atoms. The DP model (denoted as DP I ) The predicted effect is as follows Figure 4 As shown in the figure, the results of DP calculation and DFT fitting accuracy are also as high as 98%, which verifies the rationality of the trained model.

[0076] Example 3

[0077] The DP model obtained in Example 1 and Example 2 is applied to the calculation of the basic properties of UO2, and the specific calculation method is:

[0078] 1. Establish a perfect crystal model of UO2, use the DP model, and use lammps to minimize relaxation to obtain the cohesive energy of UO2.

[0079] 2. A perfect crystal model of UO2 was established. After initial relaxation, deformation was applied to the simulated box in six directions, with the maximum normal strain set to 2% and the maximum shear strain set to 5%. The elastic constants were calculated by measuring the changes in the stress tensor.

[0080] As shown in Table 1, the calculated results are compared with the experimental values ​​and the traditional potential function results.

[0081] Table 1 Comparison of calculated results with experimental values ​​and traditional potential function results

[0082] nature <![CDATA[DP Xe ]]> <![CDATA[DP I ]]> experiment Basak Morelon Cooper <![CDATA[Cohesive energy (eV / UO2)]]> -31.84 -31.18 -22.32 -43.2 -65.9 \ <![CDATA[Elastic constant C 11 (GPa)]]> 393.81 378.49 389.3 408.1 216.9 406.3 <![CDATA[Elastic constant C 12 (GPa)]]> 129.4 120.83 118.7 61.2 79.1 124.7 <![CDATA[Elastic constant C 44 (GPa)]]> 67.35 76.87 59.7 59.5 78.5 63.89

[0083] From Table 1, it can be found that the accuracy of the traditional potential function in predicting cohesive energy needs to be improved. The DP model developed in this application has a more accurate prediction result for cohesive energy, indicating that the DP potential function model has made significant progress compared with the traditional potential function. For the prediction of elastic constants, except for a slight overestimation of the value of C44 (about 10% error), the DP model is more accurate in overall performance.

[0084] Example 4

[0085] The DP model obtained in Example 1 is applied to the calculation of the diffusion coefficient of the fission product Xe in UO2. The specific calculation method is as follows: establish a UO2 unit cell, perform a 12×12×12 expansion operation on it, replace 10% of U with Xe atoms, delete 10% of O atoms, maintain electrical neutrality, and perform a simulation for 80ps under the NPT ensemble at a temperature of 998K and 1atm. Statistic the mean square displacement (MSD) of the Xe atoms, as shown in Figure 5 As shown, the fitting straight line area shows that the diffusion coefficient of Xe at this concentration and temperature is 6.22E-12, which is less than 8% compared with 6.66E-12 calculated by DFT.

[0086] Example 5

[0087] The DP model obtained in Example 2 is applied to the calculation of the diffusion coefficient of fission product I in UO2. The specific calculation method is as follows: establish a UO2 unit cell, perform a 12×12×12 cell expansion operation on it, replace 10% of U with I atoms, and delete 10% of O atoms to maintain electrical neutrality. Under the NPT ensemble, control the temperature at 998K and 1atm to perform a simulation for 80ps. Statistic the mean square displacement (MSD) of I atoms, as shown in Figure 6 As shown, the fitting straight line area shows that the diffusion coefficient of I at this concentration and temperature is 1.62E-11, which is less than 10% of the 1.78E-11 calculated by DFT.

[0088] Example 6

[0089] The DP model obtained in Example 1 is applied to the simulation of the diffusion behavior of the fission product Xe in UO2. The specific calculation method is as follows: establish a UO2 unit cell, perform a 10×10×10 cell expansion operation on it, delete 10% of the UO2 atoms, and add 5% of the Xe atoms in the UO2 gap. Under the NPT ensemble, the temperature is controlled at 1400K and 1atm for 2000ps. At the same time, the Monte Carlo (MC) algorithm is combined every 0.1ps to perform atomic exchange.

[0090] Figure 7 The diffusion, nucleation and growth behavior of Xe atoms in the first 700ps are shown. Figure 7 As can be seen in the figure, as time evolves, the Xe atoms diffuse and nucleate into a huge cluster.

[0091] Example 7

[0092] The DP model obtained in Example 2 is applied to the simulation of the diffusion behavior of fission product I in UO2. The specific calculation method is the same as that in Example 6, except that the corresponding Xe atoms are replaced by I atoms.

[0093] Figure 8 The diffusion, nucleation and growth behaviors of I atoms in the first 700ps are shown. Figure 8 As can be seen in the figure, over time, the I atoms diffuse and nucleate into five clusters of almost the same size.

[0094] Based on the above technical scheme, the present application provides a method for designing a machine learning potential function for UO2 and its fission products, comprising the following steps: step S1, constructing a UO2-Xe / I unit cell with fission products, and optimizing the UO2-Xe / I unit cell; step S2, perturbing the optimized UO2-Xe / I unit cell structure, performing ab initio molecular dynamics simulation, and generating a training data set; step S3, performing machine learning force field training on the training data set, and establishing a machine learning potential function; step S4, performing molecular dynamics simulation on the machine learning potential function, and identifying the force deviation. The atomic structure with a difference within a preset range is taken as a candidate configuration; step S5, the candidate configuration is subjected to density functional theory calculation to obtain accurate structure-energy-force data, and the structure-energy-force data is merged into the training set of subsequent iterations, and steps S3 to S5 are repeated to perform iterative calculations to train the machine learning potential function; step S6, the trained machine learning potential function is verified, and the calculation results of the machine learning potential function and the density functional theory are compared to verify that an accurate machine learning potential function is obtained; step S7, molecular dynamics simulation is performed using the accurate machine learning potential function.

[0095] Those skilled in the art can understand that the above-mentioned embodiments are specific examples for implementing the present application, and in practical applications, various changes can be made to them in form and detail without departing from the spirit and scope of the present application. Any person skilled in the art can make their own changes and modifications without departing from the spirit and scope of the present application, so the scope of protection of the present application shall be based on the scope defined in the claims.

Claims

1. A machine learning potential function design method for UO2 and its fission products, characterized in that: The following steps are involved: Step S1, constructing a UO2-Xe / I unit cell with fission products, and optimizing the UO2-Xe / I unit cell; Step S2, perturbing the optimized UO2-Xe / I unit cell structure, performing ab initio molecular dynamics simulation, and generating a training data set; Step S3, performing machine learning force field training on the training data set to establish a machine learning potential function; Step S4, performing molecular dynamics simulation on the machine learning potential function, and identifying atomic structures with force deviations within a preset range as candidate configurations; Step S5, performing density functional theory calculation on the candidate configuration to obtain accurate structure-energy-force data, and merging the structure-energy-force data into the training set of subsequent iterations, repeating steps S3 to S5 for iterative calculation, and training the machine learning potential function; Step S6, verifying the trained machine learning potential function, comparing the machine learning potential function with the calculation results of density functional theory, and verifying that an accurate machine learning potential function is obtained; Step S7: Use accurate machine learning potential function to perform molecular dynamics simulation.

2. The method for designing a machine learning potential function for UO2 and its fission products according to claim 1, characterized in that: In step S1, a UO2-Xe / I unit cell with fission products is constructed, including: Based on the periodicity and atomic coordinates of the fluorite-type face-centered cubic structure, the perfect unit cell of UO2 was constructed using VESTA software; Then the UO2 perfect unit cell is expanded by 2×2×2, that is, 8 UO2 units are stacked in two layers, 4 in each layer; Starting from the origin, UO2 is numbered in a counterclockwise direction. The UO2 in the lower layer is numbered 1 to 4, and the UO2 in the upper layer is numbered 5 to 8. Select UO2 with numbers 1, 2 and 5, insert fissionable atoms Xe / I into each octahedral gap of UO2, insert three fissionable atoms in total, and construct the first type of UO2-Xe / I structure; Select UO2 with numbers 1, 2 and 7, insert fissionable atoms Xe / I into each octahedral gap of UO2, insert three fissionable atoms in total, and construct the second type of UO2-Xe / I structure; Select UO2 with numbers 1, 3 and 5, insert fissionable atoms Xe / I into each octahedral gap of UO2, insert three fissionable atoms in total, and construct the third type of UO2-Xe / I structure; Select UO2 numbered 1, 3 and 6, insert fissionable atoms Xe / I into each octahedral gap of UO2, insert 3 fissionable atoms in total, and construct the fourth type of UO2-Xe / I structure.

3. The method for designing a machine learning potential function for UO2 and its fission products according to claim 1, characterized in that: Optimization of the UO2-Xe / I unit cell, including: The UO2-Xe / I unit cell was optimized by structural relaxation. The full electron wave function and pseudo wave function were linked by the projected augmented wave method. The generalized gradient approximation method of the Perdew-Burke-Ernzerhof form was used to exchange the correlation function with a cutoff energy of 400 eV and an energy convergence limit of 10 -5 eV, the force convergence criterion parameter was -0.02 eV / Å, and the maximum number of electronic self-consistent cycles was set to 1000 to ensure the convergence of the structure.

4. The method for designing a machine learning potential function for UO2 and its fission products according to claim 1, characterized in that: In step S2, the optimized UO2-Xe / I unit cell structure is perturbed and ab initio molecular dynamics simulation is performed to generate a training data set, including: The atomic coordinate position and the size of the simulation box were changed. For each optimized structure, 30 different perturbation structures were generated. Each perturbation structure was subjected to 10 steps of ab initio molecular dynamics simulation in the NVT ensemble at 50K with a step length of 2 femtoseconds. Ten atomic structure-energy corresponding data were obtained for each configuration. The data obtained from all perturbation configurations were integrated as the initial training data for the machine learning potential function.

5. The method for designing a machine learning potential function for UO2 and its fission products according to claim 1, characterized in that: In step S3, the machine learning force field training includes 4×10 5 ~8×10 5 Step 1, 4 independent machine learning potential functions are established for each training, and each independent machine learning potential function has a different initial random number.

6. The method for designing a machine learning potential function for UO2 and its fission products according to claim 1, characterized in that: In step S4, molecular dynamics simulation is performed on the machine learning potential function, including: using LAMMPS to perform molecular dynamics simulation on the structure with an NPT ensemble under at least 3 different temperature conditions, and during the simulation, atomic structures with a force deviation range of 0.1~0.3 eV / Å are identified as candidate configurations.

7. The method for designing a machine learning potential function for UO2 and its fission products according to claim 1, characterized in that: In step S5, the density functional theory calculation method is: setting the maximum ion step to 0, performing static relaxation simulation on the candidate configuration, and obtaining its energy-force-structure data.

8. The method for designing a machine learning potential function for UO2 and its fission products according to claim 1, characterized in that: In step S5, after iterative calculations are performed by repeating steps S3 to S5, the final data sets include: under the temperature conditions of 50 K, 100 K, 200 K, 300 K, 400 K, 500 K, 600 K, 700 K, 800 K, 900 K, 1000 K, 1100 K, 1200 K, 1300 K, 1400 K, 1500 K, 1600 K, 1700 K, and 1800 K, the pressure is controlled to be 1 bar, and molecular dynamics simulation of the structure is performed using LAMMPS with an NPT ensemble to obtain the corresponding data sets.

9. The method for designing a machine learning potential function for UO2 and its fission products according to claim 1, characterized in that: In step S6, the trained machine learning potential function is verified, and the machine learning potential function is compared with the density functional theory calculation results to verify that an accurate machine learning potential function is obtained, including: By comparing the prediction results of the machine learning potential function and density functional theory, the force and energy errors of the machine learning potential function for the test data set are analyzed to verify the accuracy of the machine learning potential function.

10. The method for designing a machine learning potential function for UO2 and its fission products according to any one of claims 1 to 9, characterized in that: In step S7, a molecular dynamics simulation is performed using an accurate machine learning potential function, including: applying the trained machine learning potential function to the LAMMPS software, calculating the basic physical properties of UO2, and simulating the diffusion behavior and diffusion coefficient of fission atoms in the UO2-Xe and UO2-I systems.

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