Clifford neural layers for multivector system modeling
Generally discussed herein are devices, systems, and methods for machine learning (ML) modeling of a system that operates on a multivector object. A method includes receiving, by an ML model, the multivector object as an input that represents a state of the multivector system. The method includes operating, by the ML model and using a Clifford layer that includes neurons that implement a multivector kernel, on the multivector input to generate a multivector output that represents the state of the multivector system responsive to the multivector input.
1 . A method for machine learning (ML) modeling of a multivector system that operates on a multivector object, the method performed by an ML model, the method comprising:
receiving the multivector object that represents a state of the multivector system; and
transforming, using a Clifford layer of the ML model that includes neurons that implement a multivector kernel, the multivector object to a multivector output that represents the state of the multivector system responsive to the multivector object, the Clifford layer is:
(i) a Clifford Fourier Neural Operator (FNO) that includes a Clifford Fourier layer and a first Clifford convolution layer; or
(ii) a second Clifford convolution layer wherein a kernel for the second Clifford convolution layer is constrained such that the second Clifford convolution layer is equivariant.
2 . The method of claim 1 , wherein the Clifford layer is the Clifford FNO that includes the Clifford Fourier layer and the first Clifford convolution layer.
3 . The method of claim 1 , wherein the multivector system is an electromagnetic field or a dynamic fluid.
4 . The method of claim 1 , wherein the multivector object includes two or more of a scalar, a vector, a bi-vector, or a tri-vector.
5 . The method of claim 4 , wherein the multivector object includes the vector.
6 . The method of claim 4 , wherein the multivector object includes the bi-vector.
7 . The method of claim 4 , wherein the multivector object includes the tri-vector.
8 . The method of claim 1 , wherein the Clifford layer is the second Clifford convolution layer and the method further comprises constraining the kernel for the second Clifford convolution layer such that the second Clifford convolution layer is equivariant.
9 . A system comprising:
processing circuitry; and
one or more memories including parameters for neurons trained for machine learning (ML) modeling of a multivector system that operates on a multivector object and instructions that, when executed by the processing circuitry, cause the processing circuitry to perform operations for transforming, by an ML model, a multivector object, the operations comprising:
receiving the multivector object that represents a state of the multivector system; and
transforming, by a Clifford layer of the ML model that includes neurons that implement a multivector kernel, the multivector object to a multivector output that represents the state of the multivector system responsive to the multivector object, the Clifford layer is:
(i) a Clifford Fourier Neural Operator (FNO) that includes a Clifford Fourier layer and a first Clifford convolution layer; or
(ii) a second Clifford convolution layer wherein a kernel for the second Clifford convolution layer is constrained such that the second Clifford convolution layer is equivariant.
10 . The system of claim 9 , wherein the Clifford layer is the Clifford FNO that includes the Clifford Fourier layer and the first Clifford convolution layer.
11 . The system of claim 9 , wherein the multivector system is an electromagnetic field or a dynamic fluid.
12 . The system of claim 9 , wherein the multivector object includes two or more of a scalar, a vector, a bi-vector, or a tri-vector.
13 . The system of claim 9 , wherein the Clifford layer is the second Clifford convolution layer and the operations further include constraining the kernel for the second Clifford convolution layer such that the second Clifford convolution layer is equivariant.
14 . A computer-readable medium including instructions that, when executed by a machine, cause the machine to perform operations for implementing a machine learning (ML) model that models a multivector system that operates on a multivector object, the operations comprising:
receiving the multivector object that represents a state of the multivector system; and
operating, by a Clifford layer of the ML model that includes neurons that implement a multivector kernel, on the multivector object to generate a multivector output that represents the state of the multivector system responsive to the multivector object, the Clifford layer is:
(i) a Clifford Fourier Neural Operator (FNO) that includes a Clifford Fourier layer and a first Clifford convolution layer; or
(ii) a second Clifford convolution layer wherein a kernel for the second Clifford convolution layer is constrained such that the second Clifford convolution layer is equivariant.
15 . The computer-readable medium of claim 14 , wherein the Clifford layer is the Clifford FNO that includes Clifford Fourier layer and the first Clifford convolution layer.
16 . The computer-readable medium of claim 14 , wherein the multivector system is an electromagnetic field or a dynamic fluid.
17 . The computer-readable medium of claim 14 , wherein the multivector object includes two or more of a scalar, a vector, a bi-vector, or a tri-vector.