  Tensor networks have been an important concept and technique in many research
areas, such as quantum computation and machine learning. We study the
exponential complexity of contracting tensor networks on two special graph
structures: planar graphs and finite element graphs.
  We prove that any finite element graph has a $O(d\sqrt{\max\{\Delta,d\}N})$
size edge separator. Furthermore, we develop a
$2^{O(d\sqrt{\max\{\Delta,d\}N})}$ time algorithm to contracting a tensor
network consisting of $N$ Boolean tensors, whose underlying graph is a finite
element graph with maximum degree $\Delta$ and has no face with more than $d$
boundary edges in the planar skeleton, based on the $2^{O(\sqrt{\Delta N})}$
time algorithm \cite{fastcounting} for planar Boolean tensor network
contractions.
  We use two methods to accelerate the exponential algorithms by transferring
high-dimensional tensors to low-dimensional tensors.
  We put up a $O(k)$ size planar gadget for any Boolean symmetric tensor of
dimension $k$, where the gadget only consists of Boolean tensors with dimension
no more than $5$.
  Another method is decomposing any tensor into a series of vectors (unary
functions), according to its \emph{CP decomposition} \cite{tensor-rank}.
  We also prove the sub-exponential time lower bound for contracting tensor
networks under the counting \emph{Exponential Time Hypothesis} (\#ETH) holds.
