  We propose a new approach for sampling and Bayesian computation that uses the
score of the target distribution to construct a transport from a given
reference distribution to the target. Our approach is an infinite-dimensional
Newton method, involving a linear PDE, for finding a zero of a
``score-residual'' operator. We prove sufficient conditions for convergence to
a valid transport map. Our Newton iterates can be computed by exploiting fast
solvers for elliptic PDEs, resulting in new algorithms for Bayesian inference
and other sampling tasks. We identify elementary settings where score-operator
Newton transport achieves fast convergence while avoiding mode collapse.
