  We give an algorithm for computing exact maximum flows on graphs with $m$
edges and integer capacities in the range $[1, U]$ in
$\widetilde{O}(m^{\frac{3}{2} - \frac{1}{328}} \log U)$ time. For sparse graphs
with polynomially bounded integer capacities, this is the first improvement
over the $\widetilde{O}(m^{1.5} \log U)$ time bound from [Goldberg-Rao JACM
`98].
  Our algorithm revolves around dynamically maintaining the augmenting
electrical flows at the core of the interior point method based algorithm from
[M\k{a}dry JACM `16]. This entails designing data structures that, in limited
settings, return edges with large electric energy in a graph undergoing
resistance updates.
