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.
We investigate a numerical behaviour of robust deterministic optimal control problem governed by a convection diffusion equation with random coefficients by approximating statistical moments of the solution. Stochastic Galerkin approach, turning the original stochastic problem into a system of deterministic problems, is used to handle the stochastic domain, whereas a discontinuous Galerkin method is used to discretize the spatial domain due to its better convergence behaviour for convection dominated optimal control problems. A priori error estimates are derived for the state and adjoint in the energy norm and for the deterministic control in L 2 -norm. To handle the curse of dimensionality of the stochastic Galerkin method, we take advantage of the low-rank variant of GMRES method, which reduces both the storage requirements and the computational complexity by exploiting a Kronecker-product structure of the system matrices. The efficiency of the proposed methodology is illustrated by numerical experiments on the benchmark problems with and without control constraints.
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