  The primal approach to physics-informed learning is a residual minimization.
We argue that residual is, at best, an indirect measure of the error of
approximate solution and propose to train with error majorant instead. Since
error majorant provides a direct upper bound on error, one can reliably
estimate how close PiNN is to the exact solution and stop the optimization
process when the desired accuracy is reached. We call loss function associated
with error majorant $\textbf{Astral}$: neur$\textbf{A}$l a
po$\textbf{ST}$erio$\textbf{RI}$ function$\textbf{A}$l Loss. To compare Astral
and residual loss functions, we illustrate how error majorants can be derived
for various PDEs and conduct experiments with diffusion equations (including
anisotropic and in the L-shaped domain), convection-diffusion equation,
temporal discretization of Maxwell's equation, and magnetostatics problem. The
results indicate that Astral loss is competitive to the residual loss,
typically leading to faster convergence and lower error (e.g., for Maxwell's
equations, we observe an order of magnitude better relative error and training
time). We also report that the error estimate obtained with Astral loss is
usually tight enough to be informative, e.g., for a highly anisotropic
equation, on average, Astral overestimates error by a factor of $1.5$, and for
convection-diffusion by a factor of $1.7$.
