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Abstract: NISQ devices routinely estimate properties of the noiseless system from measurements at varying noise strengths $γ$. These zero-noise extrapolation (ZNE) schemes fail to estimate transport coefficients of conserved quantities when the noise violates the associated conservation laws. We show how to extrapolate measurements in noisy, one-dimensional, non-integrable spin chains to obtain the spin transport coefficients governing linear response in the zero-noise limit. As the wavevector of the Fourier mode of a conserved quantity goes to zero, its decay rate vanishes. The coefficient of the leading momentum dependence defines a transport coefficient in the noiseless system, and an analogous effective coefficient in the noisy one. Naive ZNE implicitly assumes that this effective coefficient is analytic in $γ$ as $γ\to 0$. Our main result is that universal non-linear corrections to hydrodynamics can produce non-analytic dependence on $γ$. In particular, we argue that the effective spin diffusion coefficient of a dissipative Heisenberg chain and the effective subdiffusion coefficient of a dipole-conserving fracton approach their noiseless value as $γ^{1/2}$. We benchmark these analytical predictions numerically.