  In this paper, we study Physics-Informed Neural Networks (PINN) to
approximate solutions to one-dimensional boundary value problems for linear
elliptic equations and establish robust error estimates of PINN regardless of
the quantities of the coefficients. In particular, we rigorously demonstrate
the existence and uniqueness of solutions using the Sobolev space theory based
on a variational approach. Deriving $L^2$-contraction estimates, we show that
the error, defined as the mean square for differences at the sample points
between the true solution and our trial function, is dominated by the training
loss. Furthermore, we show that as the quantities of the coefficients for the
differential equation increase, the error-to-loss ratio rapidly decreases. Our
theoretical and experimental results confirm the robustness of the error
regardless of the quantities of the coefficients.
