Abstract: Methods and systems for computationally processing data with a multi-layer convolutional neural network (CNN) having an input and output layer, and one or more intermediate layers are described. Input data represented in a form of evaluations of continuous functions on a sphere may be received at a computing device and input to the input layer. The input layer may compute outputs as covariant Fourier space activations by transforming the continuous functions into spherical harmonic expansions. The output activations from the input layer may be processed sequentially through each of the intermediate layers. Each, intermediate layer may apply Ciebsch-Gordan transforms to compute respective covariant Fourier space activations as input to an immediately next layer, without computing any intermediate inverse Fourier transforms or forward Fourier transforms.

Abstract: Methods and systems for computationally processing data with a multi-layer convolutional neural network (CNN) having an input and output layer, and one or more intermediate layers are described. Input data represented in a form of evaluations of continuous functions on a sphere may be received at a computing device and input to the input layer. The input layer may compute outputs as covariant Fourier space activations by transforming the continuous functions into spherical harmonic expansions. The output activations from the input layer may be processed sequentially through each of the intermediate layers. Each, intermediate layer may apply Ciebsch-Gordan transforms to compute respective covariant Fourier space activations as input to an immediately next layer, without computing any intermediate inverse Fourier transforms or forward Fourier transforms.

Abstract: Methods and systems for computationally simulating an N-body physical system are disclosed. A compound object X having N elementary parts E may be decomposed into J subsystems, each including one or more of the elementary parts and having a position vector rj and state vector ?j. A mural network having J nodes each corresponding to one of the subsystems may be constructed, the nodes including leaf nodes, a non-leaf root node, and intermediate non-leaf nodes, each being configured to compute an activation corresponding to the state of a respective subsystem. Upon receiving input data for the parts E, each node may compute ?j from rj and ?j of its child nodes using a covariant aggregation rule representing ?j as a tensor that is covariant to rotations of the rotation group SO(3). A Clebsch-Gordan transform may be applied to reduce tensor products to irreducible covariant vectors, and ?j of the root node may be computed as output of the ANN.

Abstract: A method and system for modelling atomic or molecular level structures, or the atomic or molecular energies or forces in such structures. The method and system use a Gaussian Process to estimate properties at the atom or molecular level in such structures.

Abstract: A method and system, for modelling atomic or molecular level structures or the atomic or molecular energies or forces in such structures is disclosed. The method and system use a Gaussian Process to estimate properties at the atom or molecular level in such structures.