  Utilizing physics-informed neural networks (PINN) to solve partial
differential equations (PDEs) becomes a hot issue and also shows its great
powers, but still suffers from the dilemmas of limited predicted accuracy in
the sampling domain and poor prediction ability beyond the sampling domain
which are usually mitigated by adding the physical properties of PDEs into the
loss function or by employing smart techniques to change the form of loss
function for special PDEs. In this paper, we design a symmetry-enhanced deep
neural network (sDNN) which makes the architecture of neural networks invariant
under the finite group through expanding the dimensions of weight matrixes and
bias vectors in each hidden layers by the order of finite group if the group
has matrix representations, otherwise extending the set of input data and the
hidden layers except for the first hidden layer by the order of finite group.
However, the total number of training parameters is only about one over the
order of finite group of the original PINN size due to the symmetric
architecture of sDNN. Furthermore, we give special forms of weight matrixes and
bias vectors of sDNN, and rigorously prove that the architecture itself is
invariant under the finite group and the sDNN has the universal approximation
ability to learn the function keeping the finite group. Numerical results show
that the sDNN has strong predicted abilities in and beyond the sampling domain
and performs far better than the vanilla PINN with fewer training points and
simpler architecture.
