Predicting dynamical properties of interacting quantum many-body systems from first principles is a notoriously difficult task. In my talk, I will describe a numerical method named "dissipation-assisted operator evolution" (DAOE) which we proposed to tackle this problem. DAOE builds on recent insights about the spreading of information in closed quantum systems to motivate an approximation scheme, which can be carried out efficiently using tensor networks. I will argue that DAOE can yield quantitatively accurate estimates of transport coefficients at a numerical cost that is essentially exponentially smaller than more brute-force approaches. Then I will go on to discuss various extensions of DAOE, including a formulation tailored to fermionic systems and another aimed at capturing the effect of varying charge density on transport coefficients.