  Recent work provides promising evidence that Physics-Informed Neural Networks
(PINN) can efficiently solve partial differential equations (PDE). However,
previous works have failed to provide guarantees on the worst-case residual
error of a PINN across the spatio-temporal domain - a measure akin to the
tolerance of numerical solvers - focusing instead on point-wise comparisons
between their solution and the ones obtained by a solver on a set of inputs. In
real-world applications, one cannot consider tests on a finite set of points to
be sufficient grounds for deployment, as the performance could be substantially
worse on a different set. To alleviate this issue, we establish guaranteed
error-based conditions for PINNs over their continuous applicability domain. To
verify the extent to which they hold, we introduce $\partial$-CROWN: a general,
efficient and scalable post-training framework to bound PINN residual errors.
We demonstrate its effectiveness in obtaining tight certificates by applying it
to two classically studied PINNs - Burgers' and Schr\"odinger's equations -,
and two more challenging ones with real-world applications - the Allan-Cahn and
Diffusion-Sorption equations.
