Charlot Lab · PINN · verify-first

Don't trust a PINN. Compute a certificate.

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

net: training:

PINN uθ(t) true cos(ωt) residual |u″+ω²u| actual error (must stay inside the bound)

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certified error ≤ — actual error — — max residual — IC mismatch e₀=— e₁=—

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The certificate reports 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 domain of validity.)