New work on Geometric Clifford Algebra Networks (GCANs). We propose geometric templates for modeling dynamical systems. A 🧵on geometric / Clifford algebras, and symmetry group transformations in neural networks. 📜

(2/12) This work was done during an internship @MSFTResearch #AI4Science #Amsterdam in collaboration with the amazing @jo_brandstetter , @rejuvyesh , @enkimute , and @wellingmax .

(3/12) GCANs build on Clifford neural layers by @jo_brandstetter, @vdbergrianne, @rejuvyesh, @wellingmax, but with strong emphasis on geometry which we model via plane-based geometric algebra.

(4/12) Plane-based geometric algebra constructs isometries, such as rotations or translations, using the Pin(p, q, r) group, whose elements are compositions of reflections. The Pin(p, q, r) group is naturally represented using a Clifford algebra.

(5/12) Elements of the Clifford algebra are (in general) multivectors. In addition to Pin(p, q, r) group actions, their components can also be used to represent the invariant subspaces of (Euclidean) transformations.

(6/12) In this way, we can use the algebra to represent various data types: vectors, planes, points, lines, and so on.

(7/12) We create an optimizable geometric template by linearly combining group actions. Neural networks constructed from these layers are excellent in representing and manipulating geometric transformations, often found in dynamical systems.

(8/12) In this way, we build GCA-MLPs, GCA-GNNs, and GCA-CNNs networks.

(9/12) We evaluate GCA-MLPs and GCA-GNNs on a rigid body motion prediction task. In this case, geometric algebra allows us to couple positions and velocities by representing them in a single multivector!

(10/12) We evaluate the GCA-CNNs on large-scale fluid dynamics and weather forecasting tasks, where scalar (pressure) and vector (wind) fields transform together. In these tasks, we propose GCA-UNets, a straightforward extension of UNets to the geometric algebra regime.

(11/12) For all these datasets, we experiment with different numbers of training trajectories. GCAN layers consistently improve generalization capabilities of the tested architectures.

(12/12) In conclusion, GCA networks are a promising direction for representing and manipulating geometric transformations. We are going to release our combined Clifford neural layers codebase very soon, stay tuned!