Although neural-network approximations of differential equations have become increasingly effective, they are rarely accompanied by rigorous guarantees. This talk introduces a Learn and Verify framework that constructs certified enclosures of solutions. During learning, physics-informed neural networks produce candidate sub- and super-solutions, guided by a Doubly Smoothed Maximum loss. During verification, interval arithmetic together with adaptive subdivision is used to prove the required differential inequalities over the whole continuous domain, accounting for rounding and discretization. Recent results for parameter-dependent equations will also be presented.