  Physics-informed neural networks have emerged as a prominent new method for
solving differential equations. While conceptually straightforward, they often
suffer training difficulties that lead to relatively large discretization
errors or the failure to obtain correct solutions. In this paper we introduce
invariant physics-informed neural networks for ordinary differential equations
that admit a finite-dimensional group of Lie point symmetries. Using the method
of equivariant moving frames, a differential equation is invariantized to
obtain a, generally, simpler equation in the space of differential invariants.
A solution to the invariantized equation is then mapped back to a solution of
the original differential equation by solving the reconstruction equations for
the left moving frame. The invariantized differential equation together with
the reconstruction equations are solved using a physics-informed neural
network, and form what we call an invariant physics-informed neural network. We
illustrate the method with several examples, all of which considerably
outperform standard non-invariant physics-informed neural networks.
