  We build an emulator based on the polynomial chaos expansion (PCE) technique
to efficiently model the non-linear effects associated with the clustering of
the $k$-essence dark energy in the effective field theory (EFT) framework.
These effects can be described through a modification of Poisson's equation,
denoted by the function $\mu(k,z)$, which in general depends on wavenumber $k$
and redshift $z$. To emulate this function, we perform $200$ high-resolution
$N$-body simulations sampled from a seven-dimensional parameter space with the
Latin hypercube method. These simulations are executed using the
$\texttt{k-evolution}$ code on a fixed mesh, containing $1200^3$ dark matter
particles within a box size of $400~\text{Mpc}/ h$. The emulation process has
been carried out within $\texttt{UQLab}$, a $\texttt{MATLAB}$-based software
specifically dedicated to emulation and uncertainty quantification tasks. Apart
from its role in emulation, the PCE method also facilitates the measurement of
Sobol indices, enabling us to assess the relative impact of each cosmological
parameter on the $\mu$ function. Our results show that the PCE-based emulator
efficiently and accurately reflects the behavior of the $k$-essence dark energy
for the cosmological parameter space defined by $w_0 c_s^2 \text{CDM} +\sum
m_{\nu}$. Compared against actual simulations, the emulator achieves
sub-percent accuracy up to the wavenumber $k \approx 9.4 ~h \text{Mpc}^{-1} $
for redshifts $z \leq 3$. Our emulator provides an efficient and reliable tool
for Markov chain Monte Carlo (MCMC) analysis, and its capability to closely
mimic the properties of the $k$-essence dark energy makes it a crucial
component in Bayesian parameter estimations. The code is publicly available at
https://github.com/anourizo/k-emulator .
