Learning Lie Groups Acting on the Manifolds Generated by Linear Differential Equations

Symmetry group is an important construct to understand the behaviour of a pure mathematical or a physical system including system of differential equations. We develop a framework that could learn continuous group symmetries governed by a given set of linear differential equations. The key idea in the proposed method is to build the symmetry group G by learning relevant exp() map, which is a crucial object in the study of Lie groups. exp() for G is learned in an implicit manner in terms of the vector fields spanning the associated Lie algebra g. In our experiments, we validate integrity of these learned vector fields by showing their generalization to various solution domains other than the one in which the model is trained. We also demonstrate the construction of a foreknown canonical vector field associated with G, which should remain in the span of g, from learned ones. These learned symmetries reveal the knowledge regarding global solution for a given set of differential equations as discussed in the article. The framework presents an optimal way to transform one solution for that set of differential equations to its another solution as well. This work is also an important step towards learning continuous group symmetries in other topological spaces as well as finding solutions without solving differential equations each time a new set of initial/boundary conditions are specified.