A neural network learns a map between finite vectors: $f_\theta:\mathbb{R}^n\to\mathbb{R}^m$, built from layers $x\mapsto\sigma(Wx+b)$. A neural operator instead learns a map between function spaces, $G_\theta:\mathcal{A}\to\mathcal{U}$, so one trained model works at any discretization of the input. Each layer swaps the weight matrix for a kernel integral: $$v_{t+1}(x)=\sigma\!\Big(Wv_t(x)+\int_D \kappa_\theta(x,y)\,v_t(y)\,dy\Big)$$ The FNO computes that integral in Fourier space, which is why one model can solve a whole PDE family — think Navier–Stokes — orders of magnitude faster than a classical solver, at $5 a run instead of $500.