  In this work, we are concerned with inverse problems involving
poly-fractional operators, where the poly-fractional operator is of the form
  \[P( (-\Delta_g)^s)u := \sum_{i=1}^M \alpha_i(-\Delta_{g_i})^{s_i}u\]
  for $s=(s_1,\dots,s_M)$, $0<s_1<\cdots<s_M<\infty$,
$s_M\in\mathbb{R}_+\backslash\mathbb{Z}$, $g=(g_1,\dots,g_M)$. There are three
major contributions in this work that are new to the literature. First, we
propose equations involving such poly-fractional operators $P$, which have not
been previously considered in the general setting. Such equations arise
naturally from the superposition of multiple stochastic processes with
different scales, including classical random walks and L\'evy flights.
Secondly, we give novel results for the unique continuation properties for
fractional polyharmonic $u$, in the sense that $u$ satisfies
$\tilde{P}((-\Delta_{\tilde{g}})^{\tilde{s}})=0$ in a bounded Lipschitz domain
$\Omega$ for some $\tilde{P}$. With these results in hand, we consider the
inverse problems for $P$, and proved the uniqueness in recovering the
potential, the source function in the semilinear case, and the coefficients
associated to the non-isotropy of the fractional operator.
