Covariant Neural Network Architecture for Determining Atomic Potentials

A computing device configured for computationally simulating an N-body physical system, wherein the N-body physical system is represented mathematically as a compound object X having N elementary parts E={ei}, i=1, . . . , N, each ei representing one of the N bodies of the N-body physical system, wherein X is hierarchically decomposed into J subsystems, Pj, j=1, . . . , J, each Pj comprising one or more of the elementary parts of E, and wherein each Pj is described by a position vector rj and an internal state vector ψj, the computing device comprising:

• one or more processors;
• and
• memory configured to store computer-executable instructions that, when executed by the one or more processors, cause the computing device to carry out computational operations including:
• constructing a hierarchical artificial neural network (ANN) having J nodes each corresponding to one of the J subsystems, the J nodes including m≥2 leaf nodes as ANN inputs, m=1 non-leaf root node as ANN output, and m≥1 intermediate non-leaf nodes, wherein each node is a neuron of the ANN and is configured to compute an activation corresponding to a different one of the internal state vectors ψj, and wherein:
• for each leaf node, ψj describes the internal state of a respective one of the Pj subsystems having just a single elementary part ei,
• for each given intermediate non-leaf node, ψj describes the internal state of a respective one of the Pj subsystems having 2≤k<N parts ei that are each comprised in a child node of the given intermediate non-leaf node,
• and for the root node, ψj describes the internal state of a subsystem Pj having k=N elementary parts ei that are each comprised in a child node of the root node;
• receiving input data to the leaf nodes specifying respective position vectors and respective internal state vectors of the N elementary parts E;
• for each given non-leaf node, computing ψj from the position vectors and internal states of all the child nodes of the given non-leaf node according to a covariant aggregation rule that represents ψj as a tensor object that is covariant to rotations of the rotation group SO(3);
• applying a Clebsch-Gordan transform to reduce a tensor product of the state vectors of the nodes to irreducible covariant vectors;
• and
• computing ψj of the root node as output of the ANN to determine the internal state of the N-body physical system.

An article of manufacture comprising a non-transitory computer readable media having computer-readable instructions stored thereon for computationally simulating an N-body physical system, wherein the N-body physical system is represented mathematically as a compound object X having N elementary parts E={ei}, i=1, . . . , N, each ei representing one of the N bodies of the N-body physical system, wherein X is hierarchically decomposed into J subsystems, Pj, j=1, . . . , J, each Pj comprising one or more of the elementary parts of E, and wherein each Pj is described by a position vector rj and an internal state vector ψj, and wherein the instructions, when executed by one or more processors of a computing device, cause the computing device to carry out operations including:

• constructing a hierarchical artificial neural network (ANN) having J nodes each corresponding to one of the J subsystems, the J nodes including m≥2 leaf nodes as ANN inputs, m=1 non-leaf root node as ANN output, and m≥1 intermediate non-leaf nodes, wherein each node is a neuron of the ANN and is configured to compute an activation corresponding to a different one of the internal state vectors ψj, and wherein:
• for each leaf node, ψj describes the internal state of a respective one of the Pj subsystems having just a single elementary part ei,
• for each given intermediate non-leaf node, ψj describes the internal state of a respective one of the Pj subsystems having 2≤k<N parts ei that are each comprised in a child node of the given intermediate non-leaf node,
• and for the root node, ψj describes the internal state of a subsystem Pj having k=N elementary parts ei that are each comprised in a child node of the root node;
• receiving input data to the leaf nodes specifying respective position vectors and respective internal state vectors of the N elementary parts E;
• for each given non-leaf node, computing ψj from the position vectors and internal states of all the child nodes of the given non-leaf node according to a covariant aggregation rule that represents ψj as a tensor object that is covariant to rotations of the rotation group SO(3);
• applying a Clebsch-Gordan transform to reduce a tensor product of the state vectors of the nodes to irreducible covariant vectors;
• and
• computing ψj of the root node as output of the ANN to determine the internal state of the N-body physical system.