Neural Network Force Field Calculation Algorithm for Molecular Dynamics Computer Simulation

Abstract

A computational method for simulating motion of elements within a multi-element system using a neural network force field (NNFF). The method includes: receiving a combination of multiple rotationally invariant features and multiple rotationally covariant features of a local environment of the multi-element system; and predicting a force vector for each element within the multi-element system based on the combination of the multiple rotationally invariant features and the multiple rotationally covariant features and the NNFF to obtain simulated motion of the elements within the multi-element system.

Description

Technical Field

[0001] The present invention generally relates to a Neural Network Force Field (NNFF) computational algorithm for direct prediction of atomic forces in molecular dynamics computer simulations in material systems such as electrochemical and water filtration devices. Background Art

[0002] Molecular dynamics is a computational materials science method used to simulate the motion of atoms in a material system under realistic operating pressure and temperature conditions. There are methods for calculating the underlying atomic forces used in the simulation of atomic motion. One method is the ab initio quantum mechanics method. This method is very accurate, but is also very expensive due to the large amount of computational resources required to apply the method. Although there are other methods that consume less computational resources, these other methods do not provide the same high accuracy. Summary of the invention

[0003] In a first embodiment, a computational method for simulating motion of elements within a multi-element system using a neural network force field (NNFF) is disclosed. The method includes receiving a combination of multiple rotationally invariant features and multiple rotationally covariant features of a local environment of the multi-element system. The method also includes predicting a force vector for each element within the multi-element system based on the combination of the multiple rotationally invariant features, the multiple rotationally covariant features, and the NNFF to obtain simulated motion of the elements within the multi-element system.

[0004] In a second embodiment, a non-transitory computer-readable medium tangibly embodying computer-readable instructions for a software program is disclosed. The software program is executable by a processor of a computing device to provide operations. The operations include receiving a combination of a plurality of rotationally invariant features and a plurality of rotationally covariant features of a local environment of a multi-element system. The operations also include predicting a force vector for each element within the multi-element system based on the combination of the plurality of rotationally invariant features and the plurality of rotationally covariant features and the NNFF to obtain simulated motion of the elements within the multi-element system.

[0005] In a third embodiment, a computer system for simulating the motion of elements within a multi-element system using NNFF is disclosed. The computer system includes a simulation computer having a processor for executing computer-readable instructions and a memory for maintaining computer-executable instructions. The computer-executable instructions perform the following functions when executed by the processor: receiving a combination of multiple rotationally invariant features and multiple rotationally covariant features of a local environment of the multi-element system; and predicting a force vector for each element within the multi-element system based on the combination of the multiple rotationally invariant features and the multiple rotationally covariant features and the NNFF to obtain simulated motion of the elements within the multi-element system. BRIEF DESCRIPTION OF THE DRAWINGS

[0006] Figure 1 Depicted are schematic graphs illustrating the trade-off between scale and accuracy for different prior art molecular dynamics (MD) force calculation methods.

[0007] Figure 2 It is a schematic diagram of neural network nodes of a neural network force field (NNFF) energy prediction scheme under the prior art Behler-Parrinello method.

[0008] Figure 3 A schematic diagram of a neural network node of a NNFF algorithm according to one embodiment is depicted.

[0009] Figure 4 is a graph of the predicted power of the NNFF algorithm using one embodiment versus the quantum mechanical calculated power of bulk polyethylene oxide (PEO).

[0010] Figure 5A A diagram showing a fingerprint neural network (NN) of the NNFF algorithm of one embodiment.

[0011] Figure 5B , 5C 5D depict graphs showing the results of using the fingerprint NN of the NNFF algorithm of one embodiment to predict H, C, and O atoms of bulk PEO, respectively.

[0012] Fig. 6A A diagram showing a fingerprint NN and a backprojection NN that are combined in the NNFF algorithm of one embodiment to produce a combined NN.

[0013] Figure 6B , 6C 6D depict graphs showing the results of predicting H, C, and O atoms of bulk PEO, respectively, using the combined NNFF algorithm of one embodiment.

[0014] Figure 7 In one or more embodiments, it can be used to implement the NNFF algorithm (e.g. Fig. 6A Schematic diagram of the computing platform for the NNFF algorithm. DETAILED DESCRIPTION

[0015] As required, detailed embodiments of the present invention are disclosed herein; however, it should be understood that the disclosed embodiments are merely examples of the present invention, which may be embodied in various and alternative forms. The drawings are not necessarily drawn to scale; some features may be enlarged or minimized to show details of particular components. Therefore, the specific structural and functional details disclosed herein should not be interpreted as limiting, but merely as a representative basis for teaching those skilled in the art to adopt the present invention in various ways.

[0016] The term "substantially" may be used herein to describe embodiments disclosed or claimed. The term "substantially" may modify a value or relative property disclosed or claimed in the present disclosure. In this case, "substantially" may mean that the value or relative property it modifies is within ±0%, 0.1%, 0.5%, 1%, 2%, 3%, 4%, 5%, or 10% of the value or relative property.

[0017] Molecular dynamics (MD) methods are useful for studying physical phenomena such as, but not limited to, ion migration, chemical reactions, and bulk and surface degradation of materials in material systems such as devices or functional materials. Non-limiting examples of such material systems include fuel cells, surface coatings, batteries, water desalination, and water filtration. There are methods for calculating the fundamental atomic forces used in simulations of atomic motion. Ab initio quantum mechanics methods are very accurate, but are also very expensive due to the large amount of computing resources required to apply the method.

[0018] Neural networks have been used to fit and predict quantum mechanical energies. These methods are called neural network force fields (NNFF). Quantum mechanical energies are used to predict the derivatives of energy with respect to atomic positions and forces. However, these methods are also computationally extensive. In view of the foregoing, what is needed is a computational method for calculating atomic forces that provides an appropriate level of accuracy while consuming a reasonable amount of computational resources.

[0019] Molecular dynamics uses the atomic positions (and possibly charge, bond, or other structural information) to calculate the interatomic forces for each atom, which are then used to modify the velocities of the atoms in the simulation. The resulting trajectories of the atoms are used to describe physical phenomena such as, but not limited to, ion migration in batteries and fuel cells, chemical reactions during bulk and surface material degradation, and solid-state material phase transitions. Depending on the underlying method used to calculate the atomic forces, there is a tradeoff between the accuracy and size of the simulation (measured by the number of atoms and the simulation dynamics time). As described above, an accurate but expensive method uses ab initio quantum mechanics methods, called ab initio molecular dynamics (AIMD).

[0020] Figure 1 Depicts the trade-off between scale and accuracy of different prior art MD force calculation methods (e.g., AIMD 102, force field MD 104, and coarse-grained MD 106). The y-axis 108 is accuracy, and the x-axis 110 is scale (e.g., 100 atoms, 1-10K atoms, and 100K atoms). AIMD 102 can be used to model anti-adhesion coating degradation. Through force field fitting 112, force field MD can be used to model amorphous electrolytes and proton exchange membrane fuel cells (PEMFC) bipolar plates (BPP) oxide coatings. Coarse-grained MD 106 can be used to model PEMFC perfluorosulfonic acid polymers.

[0021] AIMD is computationally too demanding for many relevant applications. For example, according to one estimate, an AIMD simulation on a high-performance computer with two hundred (200) cores would take sixty-five (65) years to complete to study the dynamical behavior of an amorphous lithium phosphate glass having approximately five hundred (500) atoms within five hundred (500) nanoseconds.

[0022] An alternative to AIMD is classical molecular dynamics. Using this alternative, the interatomic forces are fitted into some closed functional form to speed up by several orders of magnitude compared to AIMD. This alternative method has been used to simulate large systems, such as polymers and amorphous materials. According to one estimate, the same simulation described above using classical molecular dynamics methods was completed in less than two (2) days.

[0023] One of the disadvantages of using the classical molecular dynamics model of predefined functional form is that they limit the behavior of atomic interactions within the assumption that they remain valid under certain conditions. For example, the classical force field with quadratic functional form for the bond between atoms assumes that the chemical bonds between atoms will not break. This assumption is contrary to natural behavior, in which the chemical bonds in molecules are formed and broken during chemical reactions. There is a relatively more complex preset functional form for simulating chemical reactions. For example, a reaction force field (such as ReaxFF) is such a model. These models are difficult to parameterize and require a large number of relatively complex handmade closed function forms.

[0024] As mentioned above, there are NNFF algorithms that require the calculation of quantum mechanical energy. One implementation of NNFF uses the rotationally invariant features developed by Behler and Parrinello (otherwise known as the Behler-Parrinello method), as described in the following articles: J. Behler and M. Parrinello, "Generalized Neural-Network Representation of High-Dimensional Potential-Energy Surfaces", Phys. Rev. Lett. 98, 146401 (2007); M. Gastegger, J. Behler, and P. Marquetand, "Machine Learning Molecular Dynamics for Simulation of Infrared Spectra", Chern. Sci. 2017, 8, 6924; and J. Behler, "Atom-centered Symmetry Functions for Constructing High-Dimensional Neural Network Potentials", J. Chern. Phys., 134, 074106 (2011), all of which are incorporated herein by reference in their entirety.

[0025] In this implementation, within a predefined radius R c The atoms surrounding the central atom i are used as input into the neural network. Instead of using the atomic Cartesian coordinates of the atoms as input features, the atomic Cartesian coordinates are converted into fingerprint feature vectors. These vectors are invariant under physical operations that should not change the energy coordinates of the local atomic structure. These physical operations include rotation, translation and permutation. Rotational invariance refers to the energy invariance when the entire Cartesian coordinates of the local atomic environment are rotated with the same rotation relative to any point in space (such as the central atom i). Translational invariance refers to the energy invariance when the entire Cartesian coordinates of the local atomic environment are moved by the same displacement. Permutation invariance refers to the energy invariance when the ordered order of atoms in the local atomic environment is changed and their positions are kept fixed. In addition to satisfying these three (3) conditions, the selected fingerprint feature vectors can also satisfy the following criteria (which are assumed to be consistent with the physics of the system): the decay contribution from atoms far away from the central atom i (down to 0, for R>R c ); and varies smoothly as the atomic position varies smoothly in the local atomic environment.

[0026] In one implementation, the atomic coordinate list around the central atom i is Converted into a fingerprint feature vector described by two-body and three-body functions To achieve these five (5) requirements. Some non-limiting examples of fingerprint functions are listed below:

[0027]

[0028]

[0029]

[0030]

[0031]

[0032] Where η, κ, ζ, and λ are the parameters of the user-defined G fingerprint function, as described in J. Behler, "Atom-centered Symmetry Functions for Constructing High-Dimensional Neural Network Potentials", J. Chern. Phys., 134, 074106 (2011). The values ​​of these parameters are usually chosen to appropriately cover the length scales relevant to the local atomic system under consideration (usually within a cutoff radius R c <10A). Cut-off function f c Depends on R c and can be defined as follows:

[0033] Permutation invariance is only true for atoms of the same element. For example, exchanging a C atom at position 1 and a C atom at position 2 describes two systems with the same physical properties, and therefore, permutation invariance holds. As another example, exchanging a C atom at position 1 and an O atom at position 2 describes two systems with different physical properties, and permutation invariance does not hold. Therefore, when studying multi-element systems, the summation of the above G functions should only be performed for atoms with the same element (for two-body G) or the same element pair (for three-body G). Given the m number of elements in the MD system of interest around the central atom i, m different Gs are calculated for each parameterized two-body function, and m(m+1) / 2 different Gs are calculated for each parameterized three-body function. Due to the construction of these functions (including rotational invariance), these functions provide a useful representation of the rotationally invariant properties for learning quantum mechanics. However, machine learning systems using these invariant features are limited to predicting rotationally invariant outputs, such as energy.

[0034] Many methods based on quantum mechanics (e.g., density functional theory) are used to calculate the energy E of a system. However, these methods cannot conveniently separate the electron energy into contributions from individual atoms. Therefore, in one or more embodiments, the energy E for each atom in the system is calculated. feature vector, which is then used to train or predict a single scalar energy quantity E, such as Figure 2 shown. Figure 2 A schematic diagram depicting the neural network nodes of the NNFF energy prediction scheme under the existing Behler-Parrinello method. Atomic coordinate environment R 1 To R N 202 are used as inputs of the NNFF algorithm 200. Atomic coordinate environment R 1 To R N 202 is used to calculate the symmetric function G 1 To G N 204. Each atom coordinate environment R 1 To R N 202 and each symmetric function G 1 To G N 204 are interrelated. Using the symmetric function G 1 To G N 204 Atomic NN 206 is trained to obtain atomic energies 208, which are used to obtain total energy E 210.

[0035] The NNFF method for AIMD applications is subject to one or more limitations. For example, during the training cycle, each atomic NN (typically, one NN per element) is trained together to a single energy quantity E, which is associated with the system of interest. Therefore, the NNs are interdependent, as each needs to be used together to predict a single energy value E, and are optimized only for predicting the exact MD system used in the training set. There is no independent Samples that can be included for training include something like Second, during the MD run time (force prediction cycle), expensive computations are performed to calculate the force vector on atom i is the displacement of the total system energy E relative to a single atom However, predicting E requires knowing the derivatives of for all atoms in the MD system. Therefore, the calculation The fingerprint vector needs to be calculated 3N times, as shown in the following equation.

[0036]

[0037] where α indicates the Cartesian x, y, and z axes, N is the total number of atoms in the MD system, and Mi is the total number of fingerprint functions describing atom i. Therefore, the computational power of all N atoms in the system is O(N 2 ) growth. In some implementations, some algorithmic modifications can reduce the number of fingerprint vector calculations to 3P times, where P is the average number of atoms in the local atomic environment of an atom, thus reducing the computational cost to O(N). However, this restriction requires that for a Predicted Calculation (depending on the selected R c , P may range from 100 to 400 atoms), rather than directly Predicted Therefore, there is a direct The predicted NNFF algorithm can convert NNFF-based MD Computations are accelerated by two (2) to three (3) orders of magnitude.

[0038] In one or more embodiments, a computer system is disclosed that is configured to implement an algorithm that uses a neural network to predict quantum mechanical atomic forces directly from molecular structure without using energy derivatives. In certain embodiments, the algorithm can be two (2) to (3) orders of magnitude less computationally demanding (e.g., less computation time on a high performance computing (HPC) system) than known NNFF algorithms that require calculation of quantum mechanical energies.

[0039] The ability of an HPC system to perform certain tasks can be quantified using benchmarks, such as application benchmarks from computational codes, such as those represented in the Standard Performance Evaluation Corporation's SPEC HPC2002 and SPEC MPI2007 suites (www.spec.org / hpg) and the National Science Foundation (NSF) HPC System Acquisition Procedure (which are hereby incorporated by reference in their entirety).

[0040] In one or more embodiments, the simpler functional form of the classical molecular dynamics model is replaced by a neural network. Such a network can be easily adjusted based on a provided [input, output] data set. The local environment of atoms surrounding atom i can be used to predict the forces experienced by atom i. In the usual NNFF model, the central atom i is displaced, and then the change in energy contribution of atom i and all other atoms near i is calculated. The numerical derivative of the total system energy to the displacement of atom i is the atomic force experienced by atom i. This method is time consuming because the change in energy contribution from all atoms surrounding atom i needs to be calculated to calculate the force. In one or more embodiments, the system disclosed herein directly predicts the force on atom i. The speed of this computing system can be two (2) to three (3) orders of magnitude faster than the entire system method because there is no need to calculate additional contributions from other atoms surrounding atom i while maintaining the necessary accuracy.

[0041] In one embodiment, a computational algorithm is disclosed that is configured to directly predict force vectors on atoms by analyzing the local atomic environment around a central atom. A neural network can be trained on rotationally invariant features of the local atomic environment. The trained neural network is configured to predict atomic forces within an MD simulation. In one or more embodiments, the rotationally invariant features can be suitable for predicting a relatively small number of rotationally invariant intermediate states. Subsequently, a plurality of rotationally covariant features can be added to these rotationally invariant intermediate states, which can then be used on subsequent portions of the neural network to predict force vector values. The force vector values ​​represent rotationally covariant outputs.

[0042] In one embodiment, because the computational algorithm enables direct prediction (as opposed to needing to know all the E prediction of data samples), so from different systems or material sets Samples may be combined for force field training and / or development. For example, the computational algorithm of one or more embodiments may be used to develop a force field for a large atomic system at an interface between materials A and B. Using the algorithm, a training set may be created by combining individual smaller structures of material A, material B, and the interface between materials A and B. This training set cannot be created using an algorithm that requires prediction of E as part of the routine because it is too expensive to calculate E for a combined large atomic system from quantum mechanics from a computational point of view.

[0043] The computational algorithms of one or more embodiments can be integrated into an MD simulation engine to perform MD simulations with an accuracy approaching that of the significantly more expensive AIMD methods. For example, MD simulations of one or more embodiments can be useful for modeling atomic / ion dynamics and chemical reactions associated with functional materials used in fuel cells, water desalination, catalysis, coatings, and batteries.

[0044] In one or more embodiments, the atomic force prediction algorithm of one or more embodiments is integrated into the MD software. A non-limiting example of suitable MD software is the Large Scale Atomic / Molecular Massively Parallel Simulator (LAMMPS). LAMMPS is available from Sandia National Laboratories in Albuquerque, New Mexico. LAMMPS uses a message passing interface (MPI) for parallel communication. LAMMPS is open source software, distributed under the terms of the GNU General Public License.

[0045] The main principle of the algorithm for direct force prediction is to provide each atomic coordinate environment Define a set of k inner axes It rotates with the local atomic environment structure (rotation covariant vector), and then the rotation covariant force vector Project to To obtain the internal force vector because It is then projected back to Cartesian coordinates, so k is a fixed number greater than or equal to 3. and Both are rotationally covariant, and these vectors are also rigid vectors relative to the local atomic structure itself (both vectors rotate with the structure). Therefore, the atomic position The rotation transformation Rotate accordingly and Therefore, the internal projection vector is rotationally invariant. In one or more embodiments, the rotationally invariant feature vector Used to predict rotationally invariant intermediate states Rotational invariance and rotation covariance Used to perform backprojection and obtain rotationally covariant output Cartesian centered atomic forces Table 1 below shows a summary of the invariant and covariant vector symbols.

[0046]

[0047] Table 1

[0048] Defines a set of internal axes where k indicates a group of at least some atoms around a central atom i. The criterion for whether atom j is considered part of group k is user defined and can be further optimized depending on the MD system under consideration. Once this atom grouping criterion is determined, the following constraints are applied, namely: should be the weighted average of all relative position vectors of the atoms in the group relative to the central atom i. This constraint can be written mathematically as:

[0049]

[0050]

[0051]

[0052] Among them, f A，k is an axis function as described herein. In one or more embodiments, setting and Calculated R c The same for Calculated R c In other embodiments, a more stable configuration is provided with respect to changes in local atomic coordinates. vector, because Only used for vector projection reference axis. For polyethylene oxide (PEO) polymer (C 2 H 4 O) n Use the following The parameterized function form is used to replace f A，k :

[0053]

[0054] In the example of NNFF development for PEO, the grouping criteria are summarized in Table 2 below:

[0055]

[0056] Table 2

[0057] Once the limit is set from calculate In this way, Projection (using quantum mechanics) onto A set of internal force projections can be found because and Both are rotationally invariant, so the first neural network (NN1) can be trained to directly perform As a next step, perform Back to Descartes' atomic force In one embodiment, because of the relationship holds, so we perform a backprojection using the matrix pseudoinverse relation to find the least squares difference solution: In practice, this may not give suitable results, because the axes within the group may not be constrained to be an orthogonal (or nearly orthogonal) basis for the entire 3D-xyz space. Therefore, if the basis vectors overlap, small amounts of noise (whether due to errors or perturbations in the atomic coordinates) may lead to numerical instabilities. This is not a problem if there is zero error in the prediction, but NN1 Any errors in the forecast may be due to In one or more embodiments, this makes matrix least squares or pseudo-inverse methods unsuitable for back-projection of the force vector.

[0058] A second smaller neural network (NN2) may be trained to solve the back-projection problem. and is used as input to NN2, and NN2 will In one or more embodiments, NN2 takes into account some degree of rotational symmetry by performing data augmentation. In one or more embodiments, not only and is also used as part of the training sample, where is a random 3D rotation matrix (many different Can be used for NN2 data augmentation).

[0059] As a next step, NN1 and NN2 can be combined into a single neural network by transferring their weights and biases to a combined NN architecture with the same node dimensions as NN1 and NN2 (transfer learning). This can be done by using The training dataset is run on the combined network for a smaller number of training cycles to reduce and / or minimize the error of the combined NN.

[0060] The modification of the NNFF algorithm described above in one or more embodiments enables simplified force prediction, wherein rotationally invariant and rotationally covariant features are used to enable the prediction of the force given only the local atomic environment. The atomic forces on atom i are directly predicted given the coordinates and the elements surrounding atom i Figure 3 FIG. 3 is a diagram illustrating a neural network node of the NNFF algorithm 300 according to one embodiment. 1 to R N They are used as inputs 302a to 302n of the NNFF algorithm 300 respectively. Atomic coordinate environment R 1 to R N Used to calculate the symmetric function G 1 To G N (called Figure 3 Symmetric functions 304a to 304n) and axis function A 1To A N (called Figure 3 symmetric functions 306a to 306n). Using the symmetric function G 1 To G N Train the first neural network NN1 (in Figure 3 308) to obtain the internal force vector F int，1 To F int，N (called Figure 3 310a to 310n). Axis function A 1 To A N and the internal force vector F int，1 to F int，N For training the second neural network NN2 (called Figure 3 312). The output of the NNFF algorithm 300 is the atomic force F in the Cartesian coordinate system. 1 To F N (called Figure 3 314a to 314n). In one or more embodiments, Figure 3 The NNFF algorithm 300 has no The interdependencies between training data samples. Therefore, a direct prediction, which significantly speeds up the calculation of force predictions.

[0061] In one example, as shown below, a block version of a polyethylene oxide (PEO) polymer is used to generate a Predicted Fingerprint To generate a training set of H atoms of PEO.

[0062]

[0063] Figure 4 is predicted using the NNFF algorithm of one embodiment 402Compared with the results calculated using quantum mechanical density functional theory (DFT) simulations 404. Graph 400 shows the forces on the H atoms of PEO using two different methods. The error between the NNFF algorithm and the quantum mechanical DFT simulation can be expressed as Using this expression, the average error is 0.01 (eV / A) 2 , which is within the acceptable range. In this example, no samples from the test set were seen during the training cycle.

[0064] Figure 5A A diagram 500 of a fingerprint neural network (NN) 502 of the NNFF algorithm of one embodiment is shown. The rotationally invariant component 504 of the fingerprint NN corresponds to the internal force projection intermediate state Prediction of 506. Figure 5B , 5C 5D depict graphs 508, 510, and 512, respectively, which show the results of using the NNFF algorithm of one embodiment to predict H, C, and O atoms in a test set of atomic snapshots taken from bulk PEO, respectively. Using the error equations described above, the error between the NNFF algorithm and the quantum mechanical DFT simulation is predicted to be 0.01 (eV / A) for H, C, and O atoms, respectively. 2 、0.04(eV / A) 2 and 0.16(eV / A) 2 , which is within the acceptable range. In these examples, no samples from the test set were seen during the training cycle.

[0065] Fig. 6A FIG600 shows a fingerprint NN 602 and a backprojection NN 604 combined in an NNFF algorithm in one embodiment to produce a combined NN 606. According to one or more embodiments, the weights of the NNs 602 and 604 are transferred and the combined NN 606 is retrained. The fingerprint NN 602 is the rotationally invariant component of the NNFF algorithm, while the backprojection NN 604 is the rotationally covariant component of the NNFF algorithm. The combined NN 606 predicts Cartesian atomic forces. Figure 6B , 6C 6 and 6D depict graphs 608, 610, and 612, respectively, which illustrate the results of using the NNFF algorithm of one embodiment to predict H, C, and O atoms, respectively, in a test set of atomic snapshots taken from block PEO. express Projections onto the Cartesian x, y, and z axes. Using the above error equations, the error between the NNFF algorithm and the quantum mechanical DFT simulation is predicted to be 0.03 (eV / A) for H, C, and O atoms, respectively. 2 、0.58(eV / A) 2 and 0.42(eV / A) 2 , which is within the acceptable range. In these examples, no samples from the test set were seen during the training cycle.

[0066] The NNFF algorithm and / or method of one or more embodiments uses a method such as Figure 7The computing platform 700 shown in the figure can be implemented by a computing platform such as the computing platform 700. The computing platform 700 may include a processor 702, a memory 704, and a non-volatile storage 706. The processor 702 may include one or more devices selected from a high-performance computing (HPC) system, including a high-performance core, a microprocessor, a microcontroller, a digital signal processor, a microcomputer, a central processing unit, a field programmable gate array, a programmable logic device, a state machine, a logic circuit, an analog circuit, a digital circuit, or any other device that manipulates a signal (analog or digital) based on a computer executable instruction residing in the memory 704. The memory 704 may include a single memory device or multiple memory devices, including but not limited to a random access memory (RAM), a volatile memory, a non-volatile memory, a static random access memory (SRAM), a dynamic random access memory (DRAM), a flash memory, a cache memory, or any other device capable of storing information. The non-volatile memory 706 may include one or more persistent data storage devices, such as a hard disk drive, an optical drive, a tape drive, a non-volatile solid-state device, a cloud storage, or any other device capable of persistently storing information.

[0067] The processor 702 may be configured to read into the memory 704 and execute computer executable instructions residing in the NNFF software module 708 of the non-volatile storage 706 and embodying the NNFF algorithm and / or method of one or more embodiments. The processor 702 may be further configured to read into the memory 704 and execute computer executable instructions residing in the MD software module 710 (such as LAMMPS) of the non-volatile storage 706 and embodying the MD algorithm and / or method. The software modules 708 and 710 may include an operating system and applications. The software modules 708 and 710 may be compiled or interpreted from a computer program created using a variety of programming languages ​​and / or technologies, including but not limited to and alone or in combination Java, C, C++, C#, Objective C, Fortran, Pascal, Java Script, Python, Perl, and PL / SQL.

[0068] When executed by the processor 702, the computer executable instructions of the NNFF software module 708 and the MD software module 710 can cause the computing platform 700 to implement one or more of the NNFF algorithms and / or methods and the MD algorithms and / or methods disclosed herein, respectively. The non-volatile memory 706 may also include NNFF data 712 and MD data 714 that support the functions, features, and processes of one or more embodiments described herein.

[0069] The program code embodying the algorithm and / or method described herein can be distributed individually or jointly in various forms as a program product. The program code can be distributed using a computer-readable storage medium having a computer-readable program instruction for causing a processor to execute various aspects of one or more embodiments. The computer-readable storage medium, which is essentially non-temporary, can include volatile and non-volatile and removable and non-removable tangible media, which is implemented by any method or technology for storing information such as computer-readable instructions, data structures, program modules or other data. The computer-readable storage medium can also include RAM, ROM, erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other solid-state storage technology, portable compact disk read-only memory (CD-ROM) or other optical storage, cassettes, tapes, disk storage or other magnetic storage devices, or any other medium that can be used to store the required information and can be read by a computer. The computer-readable program instruction can be downloaded from the computer-readable storage medium to a computer, another type of programmable data processing device, or another device, or downloaded to an external computer or external storage device via a network.

[0070] The computer-readable program instructions stored in the computer-readable medium can be used to instruct a computer, other types of programmable data processing devices, or other devices to function in a particular manner so that the instructions stored in the computer-readable medium produce a manufactured product, including instructions that implement the functions, actions, and / or operations specified in the flowchart or figure. In some alternative embodiments, the functions, actions, and / or operations specified in the flowchart and figure can be reordered, processed serially, and / or processed simultaneously according to one or more embodiments. In addition, any flowchart and / or figure may include more or fewer nodes or boxes than the nodes or boxes illustrated in accordance with one or more embodiments.

[0071] Although the entirety of the present invention has been illustrated by the description of various embodiments, and although these embodiments have been described in considerable detail, it is not the intention of the applicant to restrict or in any way limit the scope of the appended claims to such details. Additional advantages and modifications will be apparent to those skilled in the art. Therefore, the present invention in its broader aspects is not limited to the specific details, representative devices and methods, and illustrative examples shown and described. Therefore, differences may be made from such details without departing from the spirit or scope of the general inventive concept.

Claims

1. A computational method for simulating the motion of elements within a multi-element system using a neural network force field (NNFF), the method comprising: receiving a combination of a plurality of rotationally invariant features and a plurality of rotationally covariant features of a local environment of a multi-element system; predicting a force vector for each element within the multi-element system based on a combination of the plurality of rotationally invariant features and the plurality of rotationally covariant features and the NNFF to obtain simulated motion of the elements within the multi-element system; and The first neural network NN1 is trained based on the plurality of rotation invariant features to predict a plurality of rotation invariant intermediate states.

2. The computing method according to claim 1 further includes training a second neural network NN2 based on the plurality of rotation-invariant intermediate states and the plurality of rotation-covariant features to predict the plurality of rotation-covariant outputs.

3. The calculation method according to claim 2 also includes integrating NN1 and NN2 into a third neural network NN3, which is used to obtain the simulated movement of the elements within the multi-element system.

4. The calculation method according to claim 3, wherein: The multi-element system is a material system, the elements are atoms within the material system, and the local environment is an atomic local environment.

5. The calculation method according to claim 4, wherein: The plurality of rotationally covariant features includes a set of internal axes 6. The calculation method according to claim 5, wherein: The plurality of rotationally invariant intermediate states include the atomic forces experienced by the central atom i To the axis within this group Scalar projection of .

7. The calculation method according to claim 6, wherein: The plurality of rotationally covariant outputs include the atomic forces experienced by the central atom i A set of Cartesian components of .

8. The calculation method according to claim 1, wherein: The plurality of rotationally invariant features include one or more Behler-Parrinello type fingerprint feature vectors.

9. The calculation method according to claim 1, wherein: The multi-element system is part of one of the following systems: fuel cell, water desalination, catalysis, coating and battery.

10. A non-transitory computer-readable medium tangibly embodying computer-readable instructions for a software program executable by a processor of a computing device to provide operations comprising: receiving a combination of a plurality of rotationally invariant features and a plurality of rotationally covariant features of a local environment of a multi-element system having elements; Predicting a force vector for each element in the multi-element system based on a combination of a plurality of rotationally invariant features and a plurality of rotationally covariant features and a neural network force field NNFF to obtain simulated motion of the elements in the multi-element system; and The first neural network NN1 is trained based on the plurality of rotation invariant features to predict a plurality of rotation invariant intermediate states.

11. The non-transitory computer readable medium of claim 10, wherein: The software program is executable by the processor of the computing device to provide further operations of training a second neural network NN2 based on the plurality of rotation-invariant intermediate states and the plurality of rotation-covariant features to predict a plurality of rotation-covariant outputs.

12. The non-transitory computer readable medium of claim 11, wherein: The software program is executable by the processor of the computing device to provide further operations of integrating NN1 and NN2 into a third neural network NN3 for obtaining simulated movement of elements within a multi-element system.

13. The non-transitory computer readable medium of claim 10, wherein: The plurality of rotationally invariant features include one or more Behler-Parrinello type fingerprint feature vectors.

14. A computer system for simulating the motion of elements within a multi-element system using a neural network force field (NNFF), comprising a simulation computer having a processor for executing computer-readable instructions and a memory for maintaining computer-executable instructions, the computer-executable instructions when executed by the processor performing the following functions: receiving a combination of a plurality of rotationally invariant features and a plurality of rotationally covariant features of a local environment of a multi-element system having elements; predicting a force vector for each element within the multi-element system based on a combination of the plurality of rotationally invariant features and the plurality of rotationally covariant features and the NNFF to obtain simulated motion of the elements within the multi-element system; and The first neural network NN1 is trained based on the plurality of rotation invariant features to predict a plurality of rotation invariant intermediate states.

15. The computer system of claim 14, wherein: The computer executable instructions, when executed by the processor, perform the further function of training a second neural network NN2 based on the plurality of rotationally invariant intermediate states and the plurality of rotationally covariant features to predict a plurality of rotationally covariant outputs.

16. The computer system of claim 15, wherein: The computer executable instructions, when executed by the processor, perform the further function of integrating NN1 and NN2 into a third neural network NN3 for obtaining simulated motion of an element within a multi-element system.

17. The computer system of claim 14, wherein: The plurality of rotationally invariant features include one or more Behler-Parrinello type fingerprint feature vectors.

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