A graphical neural network force field calculation algorithm for molecular dynamics computer simulation

Abstract

A graph neural network force field calculation algorithm for molecular dynamics computer simulation is provided. A computational method for simulating the motion of elements within a multi-element system using a graph neural network (GNN). The method includes: converting a molecular dynamics snapshot of the elements into a directed graph consisting of nodes and edges. The method further includes the following steps: initially embedding the nodes and edges to obtain initially embedded nodes and edges. The method also includes: updating the initially embedded nodes and edges by passing a first message from a first edge to a first node using a first message function and passing a second message from the first node to the first edge using a second message function to obtain updated embedded nodes and edges, and predicting a force vector for one or more elements based on the updated embedded edges and a unit vector pointing from the first node to the second node or from the second node to the first node.

Description

Technical Field

[0001] The present invention generally relates to a graph neural network (GNN) force field computation 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 methodology used to simulate atomic motion in material systems under realistic operating pressure and temperature conditions. There are methodologies to calculate the fundamental atomic forces used in the simulation of atomic motion. One methodology is the ab-initio quantum mechanics method. This method is very accurate, but is also very expensive due to the extremely large amount of computing resources necessary to apply the method. Although there are other methods that consume less computing resources, these other methods do not provide as much accuracy. Summary of the invention

[0003] According to one embodiment, a computational method for simulating the motion of elements within a multi-element system using a graph neural network (GNN) is disclosed. The method includes: converting a molecular dynamics snapshot of the elements within the multi-element system into a directed graph consisting of nodes and edges. The nodes include first and second nodes, and the edges include first and second edges. The method further includes the following steps: initially embedding the nodes and the edges to obtain initially embedded nodes and edges. The method also includes: updating the initially embedded nodes and edges by passing a first message from a first edge to a first node using a first message function and passing a second message from a first node to a first edge using a second message function to obtain updated embedded nodes and edges. The method also includes: predicting a force vector for one or more elements within the multi-element system based on the updated embedded edges and a unit vector pointing from the first node to the second node or from the second node to the first node.

[0004] According to another embodiment, a computational method for simulating the motion of elements within a multi-element system using a graph neural network (GNN) is disclosed. The method includes: converting a molecular dynamics snapshot of the elements within the multi-element system into a directed graph consisting of nodes and edges. The nodes include first, second, and third nodes, and the edges include first, second, and third edges. The method further includes: initially embedding the nodes and the edges to obtain initially embedded nodes and edges. The method also includes: updating the initially embedded nodes and edges by passing a first message from a first edge to a first node using a first message function and passing a second message from a first node to a first edge using a second message function to obtain updated embedded nodes and edges. The second message function represents the interaction between the first, second, and third nodes. The method also includes: predicting a force vector for one or more elements within the multi-element system based on the updated embedded edges.

[0005] According to yet another embodiment, a computational method for simulating the motion of elements within a multi-element system using a graph neural network (GNN) is disclosed. The method includes: converting a molecular dynamics snapshot of the elements within the multi-element system into a directed graph consisting of nodes and edges. The nodes include first and second nodes, and the edges include first and second edges. The method further includes: initially embedding the nodes and the edges to obtain initially embedded nodes and edges. The method also includes: updating the initially embedded nodes and edges by iteratively passing a first message from a first edge to a first node using a first message function and iteratively passing a second message from a first node to a first edge using a second message function to obtain updated embedded nodes and edges. The method also includes: predicting a force vector for one or more elements within the multi-element system based on the updated embedded edges. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

[0008] Figure 3 is a schematic diagram depicting an MD snapshot as a directed graph according to one embodiment.

[0009] Figure 4 is a schematic diagram of a GNN including an embedding part, a message passing part, and a force vector calculation part according to one embodiment.

[0010] Figure 5A and 5B Depicted are schematic views of a first and a second message function, respectively, according to an embodiment.

[0011] FIG. 6A to FIG. 6E Depicted are graphs showing force prediction results of C, F, H, O, and S atoms, respectively, obtained from a water-perfluorosulfonic acid (Nafion) system using a GNN algorithm of one embodiment.

[0012] 7A to 7D Depicted are graphs showing force prediction results for Al, F, H, and O atoms, respectively, obtained from an Al2O3 system using a GNN algorithm of one embodiment, in which the surface is chemically reacting with HF acid molecules.

[0013] FIG. 8A to FIG. 8C Graphs are depicted showing force prediction results of Li, O, and P atoms, respectively, obtained from the Li4P2O7 system using a GNN algorithm of one embodiment.

[0014] Fig. 9 can be used to implement the GNN algorithm in one or more embodiments (e.g. Figure 4 Schematic illustration of the computing platform for the GNN algorithm. DETAILED DESCRIPTION

[0015] As required, detailed embodiments of the present invention are disclosed herein. However, it is to be understood that the disclosed embodiments are merely illustrative of the present invention, which may be embodied in various forms and alternative forms. The figures are not necessarily to scale; some features may be enlarged or minimized to show the details of a particular component. Therefore, the specific structural and functional details disclosed herein are not to be construed 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 the disclosed or claimed embodiments. The term "substantially" may modify a value or relative property disclosed or claimed in the present disclosure. In such instances, "substantially" may indicate 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) methodology is 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 cell units, surface coatings, batteries, water desalination, and water filtration. Methodologies exist for calculating the fundamental atomic forces used in simulations of atomic motion. Ab initio quantum mechanical methods are very accurate, but are also very expensive due to the extremely large amount of computing resources necessary to apply the method.

[0018] Neural networks have been used to fit and predict quantum mechanical energies. These methodologies have been referred to as Neural Network Force Fields (NNFF). The derivatives of energy with respect to atomic positions and forces are predicted using quantum mechanical energies. However, these methodologies are also computationally expensive. In view of the foregoing, what is needed is a computational methodology for calculating atomic forces that provides a sufficient 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 of each atom, which are therefore used to modify the velocities of the atoms in the simulation. The resulting trajectories of atoms are utilized to describe physical phenomena such as, but not limited to, ion migration in batteries and fuel cell units, 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 in terms of the number of atoms and the simulated kinetic time). As explained above, an accurate but expensive method uses ab initio quantum mechanics methods, known as ab initio molecular dynamics (AIMD).

[0020] Figure 1 A schematic graph 100 is depicted showing the tradeoff between scale and accuracy for 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 utilized to model anti-stick coating degradation. Force field MD can be utilized to model amorphous electrolytes and proton exchange membrane fuel cell unit (PEMFC) bipolar plate (BPP) oxide coatings through force field fitting 112. Coarse-grained MD 106 can be utilized to model perfluorosulfonic acid polymers for PEMFCs.

[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 with approximately five hundred (500) atoms over a period of five hundred (500) nanoseconds.

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

[0023] One of the drawbacks of classical molecular dynamics models that use predefined functional forms is that they constrain the behavior of the interactions between atoms to assumptions that are still valid under certain conditions. For example, classical force fields that have quadratic functional forms for the bonds between atoms assume that the chemical bonds between atoms are not broken. This assumption is contrary to natural behavior, in which chemical bonds in molecules are formed and broken during chemical reactions. There are relatively more complex predefined functional forms for simulating chemical reactions. For example, reaction force fields (such as ReaxFF) are one such model. These models suffer from the drawback of being difficult to parameterize and require a large number of relatively complex hand-crafted closed function forms.

[0024] As mentioned above, there are NNFF algorithms that require the calculation of quantum mechanical energies. One implementation of NNFF uses 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," Chem. Sci. 2017, 8, 6924; and J. Behler, "Atom-centered Symmetry Functions for Constructing High-Dimensional Neural Network Potentials," J. Chem. 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 Around the central atom i The atoms 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 fact that the rotation of the atoms is constant relative to any point in space (such as the central atom). i ) refers to the energy invariance of rotating the entire Cartesian coordinates of the local atomic environment by the same rotation. Translation invariance refers to the energy invariance of shifting the entire Cartesian coordinates of the local atomic environment by the same displacement. Permutation invariance refers to the energy invariance of changing the order of atoms in the local atomic environment while keeping their orientation fixed. In addition to satisfying these three (3) conditions, the selected fingerprint feature vectors can also meet the following criteria (assuming they are consistent with the physical properties of the system): i The decay contribution of the atoms (for R > Rc decreases to 0); and changes smoothly as the atomic orientation changes smoothly in the local atomic environment.

[0026] In one implementation, these five (5) requirements are obtained by surrounding the central atom i List of atomic coordinates of Converted into a fingerprint feature vector described by two-body and three-body functions Some non-limiting examples of fingerprint functions are listed below:

[0027] (1)

[0028] (2)

[0029] (3)

[0030] (4)

[0031] (5)

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

[0033] (6).

[0034] Substitution invariance only holds for atoms of the same element. For example, swapping a C atom at position 1 with a C atom at position 2 describes two systems with the same physical properties, and therefore, substitution invariance holds. As another example, swapping a C atom at position 1 with an O atom at position 2 describes two systems with different physical properties, and substitution invariance does not hold. Therefore, when studying multi-element systems, the above G The summation in the function should be done only for the same elements (for two G ) or pairs of the same elements (for three bodies G ) atoms. Considering that in the MD system of interest, i of m number of elements, computed for each parameterized 2-body function m Different G , and for each parameterized three-body function we calculate m ( m +1) / 2 different G Due to their construction (including rotational invariance), these functions provide a useful representation for learning rotationally invariant properties of quantum mechanics. However, machine learning systems that use these invariant features are limited in use to predict rotationally invariant outputs, such as energy.

[0035] Many quantum mechanical methodologies (e.g., density functional theory) are used to calculate the energy of a system. E However, these methodologies cannot conveniently decompose the electron energy into contributions from individual atoms. Therefore, in one or more embodiments, the calculation for each atom in the system is feature vector, and then use it to train or predict energy E A single scalar such as Figure 2 as shown in . Figure 2 A schematic diagram of the neural network nodes of the NNFF energy prediction scheme under the prior art Behler-Parrinello method is depicted. Atomic coordinate environment R1 to R N 202 are used as inputs to the NNFF algorithm 200. The atomic coordinate environments R1 to R N 202 is used to calculate the symmetric functions G1 to G N 204. Each atomic coordinate environment R1 to R N 202 and each symmetric function G1 to G N 204 related. Using symmetric functions G1 to G N 204 to train the atomic NN 206 to obtain the atomic energy 208, which is used to obtain the total energy E 210.

[0036] The NNFF method for AIMD applications suffers from one or more limitations. For example, during the training cycle, each atomic NN (usually one NN for each element) is trained together to a single energy quantity E , the amount of energy of this single E associated with the system of interest. Therefore, the NNs are interdependent in that each NN needs to be used together to predict a single energy value E , and is optimized only for predicting the exact MD system used in the training set. There is no independent Samples: which may be included for training include Second, during the MD run (force prediction cycle), expensive computations are performed to calculate the atomic i The force vector on . is the total system energy E relative to a single atom The derivative of the displacement of . However, E The prediction requires knowledge of the G , Therefore, the calculation The fingerprint vector needs to be calculated 3N times, as shown in the following equation.

[0037] (7)

[0038] in, Indicates the Cartesian x, y, and z axes, N is the total number of atoms in the MD system, and M i It describes atoms i The total number of fingerprint functions. Therefore, for all N atoms to calculate the force O ( N 2 ) growth. In some implementations, certain algorithm modifications can reduce the number of fingerprint vector calculations to 3P times, among which 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 constraint requires a Predicted 3P Second-rate Calculation (depending on the selected R c , P can range from 100 to 400 atoms), which is similar to the One of the predictions Therefore, there is a direct The predicted NNFF algorithm can enable NNFF-based MD Computational speedups of two (2) to three (3) orders of magnitude.

[0039] In one or more embodiments, a graph neural network (GNN) is utilized because it does not require manual calculation of rotational invariant features of the local atomic environment to predict atomic forces for MD simulations. The GNN model architecture can be configured to automatically extract translational and rotational invariant features from input MD simulation snapshots during the training process.

[0040] The computational algorithm of one or more embodiments is configured to perform a direct prediction. Therefore, Samples can be combined for force field training and development. The computational algorithms of one or more algorithms can be applied practically to relatively large atomic systems of interfaces between materials A and B. A training set can be created by combining isolated smaller structures of material A, material B, and the interface between the two.

[0041] The computational algorithms of one or more embodiments can be integrated into an MD simulation engine. This integration can be used to perform accurate MD simulations without using more expensive AIMD methods. The computational algorithms of one or more embodiments find practical application in modeling the force vectors of one or more elements within a multi-element system, such as atomic / ion dynamics, chemical reactions in fuel cell units, water desalination, catalysis, coatings, and batteries.

[0042] In one or more embodiments, the 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 Atomic / Molecular Massively Parallel Simulator (LAMMPS). LAMMPS is available from Sandia National Laboratories in Albuquerque, New Mexico. LAMMPS uses the Message Passing Interface (MPI) for parallel communication. LAMMPS is open source software that is distributed under the terms of the GNU General Public License.

[0043] The atomic force prediction algorithm of one or more embodiments utilizes a graph neural network (GNN) that can directly operate on graph-structured data. Figure 3 FIG. 3 is a schematic diagram illustrating an MD snapshot 300 as a directed graph 302 according to one embodiment. Figure 3 As shown in , MD snapshot 300 depicts three (3) water molecules. Directed graph 302 can be represented as , which has iThe state of the node , represents atoms i To neighboring atoms j Directed edges with influence , and the direction of the impact , where the unit vector is from the atom i Point to Atom j For simplicity, Figure 3 No unit vectors are shown in . In this embodiment, each node is connected to N its closest neighbors, where N The GNN of this embodiment operates on a directed graph 302, which includes three (3) parts: an embedding part, a message passing part, and a force vector calculation part.

[0044] Figure 4 400, which includes an embedding portion 402, a message passing portion 404, and a force vector calculation portion 406. The GNN 400 of one or more embodiments may take the graph representation 302 of the MD snapshot 300 as input. The input 408 of the GNN 400 includes the node ,side and the unit vector In the embedding portion 402 of the GNN 400, the node and edge Embedded with initial embedding and . can be given as the output of a neural network, where the input is given by A one hot encoded vector representing the atom type. By i and j The distance between is given by applying a Gaussian filter. According to one or more embodiments, the one-hot encoding vector can be replaced by any unique representation of the elements of the atom. The Gaussian filter can be defined by its range and step size. For example, for and Range and step size , the Gaussian filter can be composed of an array Given, where , and When applied to The given atom i and j When the distance between . It is expressed by the following equation (7).

[0045] (7).

[0046] In one embodiment, the embedded portion 402 can be mathematically expressed as the following equations (8) and (9).

[0047] (8)

[0048] (9)

[0049] in f a Indicates the mapping of one-hot encodings of atom types to The neural network, GF The GNN 400 of one embodiment uses a single dense hidden layer for f a .

[0050] The message passing portion 404 of the GNN 400 may be composed of several message passing layers 410, where at each layer, local messages may be passed between nodes and edges. The local messages passed between nodes and edges may be defined by equations (10) and (11):

[0051] Edge to Node: (10)

[0052] Node to edge: (11)

[0053] in, Is a node In layer l , and It is the edge The initial condition is and is determined in the embedding part 402 of the GNN 400. Mark the node connected to The index set of the neighboring nodes of . Message function and are node- and edge-specific neural networks that respectively and Update to and These functions are defined so that after each update (message passing), the embedding and Respectively better represent atoms i The local atomic environment and atoms i and jThe message function can be designed flexibly so that it can be changed to emphasize different interatomic dynamics occurring within the MD simulation.

[0054] In one embodiment, the message function and are defined by equations (12) and (13), respectively.

[0055] (12)

[0056] (13)

[0057] in g Label the hyperbolic tangent function, Indicates the sigmoid function, indicates the element-wise multiplication operator, and Identifies the concatenation operator. and Representation layer l The weights and biases of the dense hidden layers of . The second line of Equation (13) represents the three-body function. The other part of Equation (12) and Equation (13) represents the two-body function.

[0058] Figure 5A and Figure 5B The message functions according to one embodiment are respectively described and Schematic diagrams 500 and 502 of FIG. Figure 5A As shown in Applied to and , where the hyperbolic tangent function g and S-shaped function is applied to the result to obtain two outputs which are fed to the element-wise multiplication operator . Sum these results, as given by Depicted. Add to this result to get .like Figure 5B As shown in Applied to , and , where the hyperbolic tangent function g and S-shaped function is applied to the result to obtain two outputs which are fed to the element-wise multiplication operator .Will Add to this result to get In one embodiment, g Indicates the non-linear activation function.

[0059] In force vector calculation section 406, the edge embedding output of message passing section 404 may be used to calculate force vectors for one or more elements within the multi-element system. L The GNN with message passing layers is embedded Edge The final state can be used to calculate , which is the symbol of the atom i Being applied to atoms j A scalar of the magnitude of the force on the atom. i Applied to atoms j The force on Given, and the atom j The force prediction for is the sum of the forces exerted by neighboring atoms. These calculations can be represented mathematically by equations (14) and (15) reproduced below.

[0060] (14)

[0061] (15)

[0062] The function Yes Map to The neural network, and It is an atom j The GNN 400 of one embodiment may apply a series of 3 dense hidden layers to , where the intermediate hidden state is passed through the ReLU activation function.

[0063] During the training process of GNN 400, , , and The weights and biases in are updated by back-propagation to minimize the error between the predicted forces and the forces calculated by MD. In such an embodiment, the force field prediction of GNN400 is translationally invariant, but the input atomic coordinate environment of the MD snapshot is rotationally covariant.

[0064] In one embodiment, in MD simulations for fuel cell units, water desalination, catalysis, coatings, and battery applications, GNN 400 can be used to directly predict atomic force vectors. In addition, the GNN 400 of one or more embodiments can be used as part of MD simulations for drug design and biosciences (including biochemical design). In one or more embodiments, the MD snapshot 300 represented as a directed graph 302 can be composed of nodes, edges, and unit vectors. Nodes can be representations of constituent atoms. Edges can be representations of interatomic influences between adjacent atoms. Unit vectors can be representations of the direction of interatomic influences. For example, a hot encoding of an atom type can be used to initially embed nodes. For example, interatomic distances processed by a Gaussian filter can be used to initially embed edges. Initial node and edge embedding can be iteratively updated by iteratively passing messages from edge to node and from node to edge. The message can be, but is not limited to, a function designed to update node and edge embedding to better represent local atomic environments and interatomic interactions, respectively. The message passing step can be performed more than once. Edge embedding and unit vectors can be used to calculate interatomic forces between a central atom and its neighboring atoms. The values ​​can be summed to predict the force vector for the central atom.

[0065] FIG. 6A to FIG. 6E Graphs 600, 602, 604, 606, and 608 are depicted, respectively, which show the force prediction results of C, F, H, O, and S atoms obtained from the water-nafion system using the GNN algorithm of one embodiment. Graphs 600, 602, 604, 606, and 608 plot the calculated force of C, F, H, O, and S atoms from the quantum mechanical density functional theory (DFT) simulation. The function predicted by GNN . Marking To Descartes x, y and z The following error equation (16) can be used to calculate the average error of the GNN algorithm calculation method:

[0066] (16)

[0067] Using the error equations formulated above, the errors between the GNN algorithm and the quantum mechanical DFT simulations for the predictions of C, F, H, O, and S atoms are 0.273 eV / A, 0.159 eV / A, 0.148 eV / A, 0.221 eV / A, and 0.340 eV / A, respectively, which are within acceptable limits. In these examples, samples from the test set were not seen during the training cycle.

[0068] 7A to 7DGraphs 700, 702, 704, and 706 are depicted, respectively, showing force predictions of Al, F, H, and O atoms, respectively, obtained from an Al2O3 system using a GNN algorithm of one embodiment, in which the surface is chemically reacting with HF acid molecules. Graphs 700, 702, 704, and 706 plot the calculated force of the Al, F, H, and O atoms as calculated from quantum mechanical DFT simulations. The function predicted by GNN Using the error equations formulated above, the errors between the GNN algorithm and the quantum mechanical DFT simulations for the predictions of Al, F, H, and O atoms are 0.297 eV / A, 0.274 eV / A, 0.282 eV / A, and 0.295 eV / A, respectively, which are within acceptable limits. In these examples, samples from the test set were not seen during the training cycle.

[0069] FIG. 8A to FIG. 8C Graphs 800, 802, and 804 are depicted, respectively, which show the force prediction results of Li, O, and P atoms obtained from the Li4P2O7 system using the GNN algorithm of one embodiment. Graphs 800, 802, and 804 plot the force of Li, O, and P atoms calculated from quantum mechanical DFT simulations. The function predicted by GNN Using the error equations formulated above, the errors between the GNN algorithm and the quantum mechanical DFT simulations for the predictions of Li, O, and P atoms are 0.238 eV / A, 0.419 eV / A, and 0.683 eV / A, respectively, which are within acceptable limits. In these examples, samples from the test set were not seen during the training cycle.

[0070] The GNN algorithm and / or methodology of one or more embodiments uses a Fig. 9. The computing platform 900 may include a memory 902, a processor 904, and a non-volatile storage device 906. The processor 904 may include one or more devices selected from a high-performance computing (HPC) system, which includes 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 signals (analog or digital) based on computer executable instructions residing in the memory 902. The memory 902 may include: a single memory device or multiple memory devices, including but not limited to random access memory (RAM), volatile memory, non-volatile memory, static random access memory (SRAM), dynamic random access memory (DRAM), flash memory, cache memory, or any other device capable of storing information. Non-volatile storage 906 may include one or more persistent data storage devices such as a hard drive, an optical drive, a tape drive, a non-volatile solid-state device, cloud storage, or any other device capable of persistently storing information.

[0071] The processor 904 may be configured to read into the memory 902 and execute computer executable instructions residing in the GNN software module 908 of the non-volatile storage device 906 and embodying the GNN algorithm and / or methodology of one or more embodiments. The processor 904 may be further configured to read into the memory 902 and execute computer executable instructions residing in the MD software module 910 (such as LAMMPS) of the non-volatile storage device 906 and embodying the MD algorithm and / or methodology. The software modules 908 and 910 may include operating systems and applications. The software modules 908 and 910 may be compiled or interpreted from computer programs created using various programming languages ​​and / or technologies, which include, without limitation and individually or in combination, Java, C, C++, C#, Objective C, Fortran, Pascal, Java Script, Python, Perl, and PL / SQL. In one embodiment, PyTorch, which is a software package for the Python programming language, may be used to implement code for the GNN of one or more embodiments. The code framework may be based on the Crystalline Graph Convolutional Neural Network (CGCNN) code, which is available under license from the Massachusetts Institute of Technology, Cambridge, MA.

[0072] After execution by the processor 904, the computer executable instructions of the GNN software module 908 and the MD software module 910 can enable the computing platform 900 to implement one or more of the GNN algorithms and / or methodologies and MD algorithms and / or methodologies disclosed herein, respectively. The non-volatile storage device 906 may also include GNN data 912 and MD data 914 that support the functions, features, and processes of one or more embodiments described herein.

[0073] The program code embodying the algorithms and / or methodologies described herein can be distributed individually or collectively as a program product in a variety of different forms. The program code can be distributed using a computer-readable storage medium having computer-readable program instructions for causing a processor to perform various aspects of one or more embodiments. Computer-readable storage media, which are inherently non-transitory, may include volatile and non-volatile, and removable and non-removable tangible media implemented in any method or technology for storing information (such as computer-readable instructions, data structures, program modules, or other data). Computer-readable storage media may further include RAM, ROM, erasable programmable read-only memory (EPROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other solid-state memory technology, portable compact disk read-only memory (CD-ROM) or other optical storage device, cassette, tape, magnetic disk storage device or other magnetic storage device, or any other medium that can be used to store desired information and can be read by a computer. The computer-readable program instructions may be downloaded from the computer-readable storage medium to a computer, another type of programmable data processing apparatus, or another device, or downloaded to an external computer or an external storage device via a network.

[0074] Computer-readable program instructions stored in a computer-readable medium can be used to guide 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, which includes instructions for implementing the functions, actions, and / or operations specified in the flowchart or diagram. In certain alternative embodiments, the functions, actions, and / or operations specified in the flowchart and diagram can be reordered, processed serially, and / or processed concurrently in accordance with one or more embodiments. In addition, any flowchart and / or diagram may include more or fewer nodes or boxes than those illustrated in accordance with one or more embodiments.

[0075] 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, the applicant does not intend that the scope of the appended claims be constrained or in any way limited to such details. Additional advantages and modifications will readily appear 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, deviations from such details may be made without departing from the spirit or scope of the overall inventive concept.

Claims

1. A computational method for simulating element motion within a multi-element system using a graph neural network (GNN), the method comprising: converting a molecular dynamics snapshot of an element within the multi-element system into a directed graph consisting of nodes and edges, the nodes comprising first and second nodes, and the edges comprising first and second edges, wherein the first and second nodes represent first and second neighboring atoms, and the first and second edges represent interatomic influences between the first and second neighboring atoms; Initially embedding the edges using a Gaussian filter to obtain initially embedded edges, and mapping the nodes using a unique representation of elements of atoms to obtain initially embedded nodes; updating the initial embedded nodes and edges by passing a first message from the first edge to the first node using a first message function that is a node-specific neural network and passing a second message from the first node to the first edge using a second message function that is an edge-specific neural network to obtain updated embedded nodes and edges; as well as A force vector for one or more elements within the multi-element system is predicted based on the updated embedded edges and a unit vector directed from the first node to the second node or from the second node to the first node.

2. The calculation method according to claim 1, wherein: The first message function uses the first message and the initially embedded nodes as input, and the second message function uses the second message and the initially embedded edges as input.

3. The calculation method according to claim 1, wherein: The computational method was integrated into the molecular dynamics (MD) software.

4. The calculation method according to claim 1, wherein: The unit vector is representative of the direction of the interatomic influence between the first and second neighboring atoms.

5. The calculation method according to claim 4, wherein: A single dense hidden layer is used for the unique representation of the atomic elements.

6. A computational method for simulating element motion within a multi-element system using a graph neural network (GNN), the method comprising: converting a molecular dynamics snapshot of an element within the multi-element system into a directed graph consisting of nodes and edges, the nodes comprising first, second, and third nodes, and the edges comprising first, second, and third edges, wherein the first and second nodes represent first and second neighboring atoms, and the first and second edges represent interatomic influences between the first and second neighboring atoms; Initially embedding the edges using a Gaussian filter to obtain initially embedded edges, and mapping the nodes using a unique representation of elements of atoms to obtain initially embedded nodes; updating the initial embedded nodes and edges by passing a first message from the first edge to the first node using a first message function that is a node-specific neural network and passing a second message from the first node to the first edge using a second message function that is an edge-specific neural network to obtain updated embedded nodes and edges, the second message function representing interactions between the first, second, and third nodes; as well as Based on the updated embedded edges, force vectors are predicted for one or more elements within the multi-element system.

7. The calculation method according to claim 6, wherein: The first message function uses the first message and the initially embedded nodes as input, and the second message function uses the second message and the initially embedded edges as input.

8. The calculation method according to claim 6, wherein: The computational method was integrated into the molecular dynamics (MD) software.

9. The calculation method according to claim 6, wherein: A single dense hidden layer is used for the unique representation of the atomic elements.

10. A computational method for simulating element motion within a multi-element system using a graph neural network (GNN), the method comprising: converting a molecular dynamics snapshot of an element within the multi-element system into a directed graph consisting of nodes and edges, the nodes comprising first and second nodes, and the edges comprising first and second edges, wherein the first and second nodes represent first and second neighboring atoms, and the first and second edges represent interatomic influences between the first and second neighboring atoms; Initially embedding the edges using a Gaussian filter to obtain initially embedded edges, and mapping the nodes using a unique representation of elements of atoms to obtain initially embedded nodes; updating the initial embedded nodes and edges by iteratively passing a first message from a first edge to a first node using a first message function that is a node-specific neural network and iteratively passing a second message from the first node to the first edge using a second message function that is an edge-specific neural network to obtain updated embedded nodes and edges; as well as Based on the updated embedded edges, force vectors are predicted for one or more elements within the multi-element system.

11. The calculation method according to claim 10, wherein: The iterative passing step of the update step is performed more than once.

12. The calculation method according to claim 10, wherein: The first message function uses the first message and the initially embedded nodes as input, and the second message function uses the second message and the initially embedded edges as input.

13. The calculation method according to claim 10, wherein: The computational method was integrated into the molecular dynamics (MD) software.

14. The calculation method according to claim 10, wherein: A single dense hidden layer is used for the unique representation of the atomic elements.

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