  Phylogenetics is now fundamental in life sciences, providing insights into
the earliest branches of life and the origins and spread of epidemics. However,
finding suitable phylogenies from the vast space of possible trees remains
challenging. To address this problem, for the first time, we perform both tree
exploration and inference in a continuous space where the computation of
gradients is possible. This continuous relaxation allows for major leaps across
tree space in both rooted and unrooted trees, and is less susceptible to
convergence to local minima. Our approach outperforms the current best methods
for inference on unrooted trees and, in simulation, accurately infers the tree
and root in ultrametric cases. The approach is effective in cases of empirical
data with negligible amounts of data, which we demonstrate on the phylogeny of
jawed vertebrates. Indeed, only a few genes with an ultrametric signal were
generally sufficient for resolving the major lineages of vertebrates.
Optimisation is possible via automatic differentiation and our method presents
an effective way forwards for exploring the most difficult, data-deficient
phylogenetic questions.
