  Neural networks are universal approximators and are studied for their use in
solving differential equations. However, a major criticism is the lack of error
bounds for obtained solutions. This paper proposes a technique to rigorously
evaluate the error bound of Physics-Informed Neural Networks (PINNs) on most
linear ordinary differential equations (ODEs), certain nonlinear ODEs, and
first-order linear partial differential equations (PDEs). The error bound is
based purely on equation structure and residual information and does not depend
on assumptions of how well the networks are trained. We propose algorithms that
bound the error efficiently. Some proposed algorithms provide tighter bounds
than others at the cost of longer run time.
