Charlot Lab · PINN · verify-first
A physics-informed net solves $u''+\omega^2u=0,\ u(0)=1,\ u'(0)=0$ — whose true answer is $\cos\omega t$. A PINN only softly penalizes the physics, so "the loss went down" is not "the answer is right." Instead we compute an a-posteriori error certificate from the trained net's own residual — a rigorous upper bound on how far it can be from the true solution — and let that decide whether to trust it.
First rung of the soft → structural → proven ladder · bound = |e₀| + |e₁|/ω + (1/ω)∫|residual| (Grönwall / Mishra–Molinaro form) · real Torch-trained nets
The certificate is honest in both directions. Trust: the soft ω=2 net drives its residual down, the bound falls to ~0.02, and the white actual-error tick sits inside the bound — the guarantee is sound. Structure: the hard-constraint net writes $u=1+t^2N(t)$ so the initial conditions hold exactly (e₀=e₁=0) — the bound tightens further with no extra training. Silent failure: at ω=8 the same-size net can't represent the high frequency (spectral bias) — the curve looks like it's trying, but the residual stays large, the bound turns red, and the actual error confirms it is ~80% wrong. The certificate catches what the eye misses. (The ∫|residual| uses fine-grid quadrature; the bound is rigorous up to that quadrature, and holds for this well-posed linear regime — the honest domain of validity.)