Lie-Group Induced Dynamics on PDEs

Diffusion models are frequently used in generative modeling and are applied in the generation of solutions for partial differential equations (PDEs). The data used are solutions of a specific PDE; the diffusion model learns the distribution of these solutions and then samples from this distribution. The solutions of the PDE lie in the Euclidean space R𝑛. In some cases, there is a low-dimensional manifold M ⊆ R𝑛 on which the data lies; however, this is not always Euclidean. To enable diffusion models to be applied to non-Euclidean spaces, adaptations of the classical diffusion model already exist, although these are often computationally intensive. If the data on the manifold exhibit symmetries, these can be described by a Lie group that operates on the data manifold. By using the corresponding Lie algebra, which is a Euclidean space, this operation can be used to compute in Euclidean spaces. The use of symmetries has already been used for protein datasets, but so far, it has not been exploited that all PDEs possess a symmetry group that describes the invariances of their solutions. This bachelor's thesis investigates whether these symmetric invariances can be exploited by inserting the symmetry group of the PDE into a Lie group-based diffusion model to solve the PDE.