Geometric Machine Learning

GeometricMachineLearning.jl implements various scientific machine learning models that aim at learning dynamical systems with geometric structure, such as Hamiltonian (symplectic) or Lagrangian (variational) systems.

Installation

GeometricMachineLearning.jl and all of its dependencies can be installed via the Julia REPL by typing

]add GeometricMachineLearning

Architectures

There are several architectures tailored towards problems in scientific machine learning implemented in GeometricMachineLearning.

• SympNet Architecture
• • Principle
  • $LA$-SympNet
  • $G$-SympNets
  • Universal approximation theorem
  • Loss function
  • How are SympNets Implemented?
  • Library Functions
  • References

Manifolds

GeometricMachineLearning supports putting neural network weights on manifolds. These include:

Special Neural Network Layer

Many layers have been adapted in order to be used for problems in scientific machine learning. Including:

• The Attention Layer
• • Reweighting of the input sequence
  • Volume-Preserving Attention
  • How is structure preserved?
  • Historical Note
  • References

Tutorials

Tutorials for using GeometricMachineLearning are:

• SympNets with GeometricMachineLearning
• • Specifying the architecture
  • Examples
• MNIST tutorial

Reduced Order Modeling

A short description of the key concepts in reduced order modeling (where GeometricMachineLearning can be used) are in:

• Symplectic Autoencoder
• • Hamiltonian Model Order Reduction
  • Symplectic Solution Manifold
  • Workflow for Symplectic ROM
  • Proper Symplectic Decomposition
  • Symplectic Autoencoders
  • References
• Kolmogorov $n$-width
• • References