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\paragraph*{\bf Acknowledgments} P. W., Q. F., R. P. and F. Y. acknowledge support from the NSFC (No. 91950120), Scientific funding of Shanghai (No.9ZR1424400), and  Shanghai Outstanding Academic Leaders Plan (No.20XD1402000).
Y.V.K. and L.T. acknowledge support from the Severo Ochoa Excellence Programme (CEX2019-000910-S), Fundacio Privada Cellex, Fundacio Privada Mir-Puig, and CERCA/Generalitat de Catalunya. V.V.K. acknowledges financial support from the Portuguese Foundation for Science and Technology (FCT) under Contracts PTDC/FIS-OUT/3882/2020 and UIDB/00618/2020.

\paragraph*{\bf Author contributions} P. W. and Q. F. contributed equally to this work. All authors (P. W., Q. F., R.P.,Y.K.,L.T., V. K. and F. Y.)contributed significantly to the paper.

\paragraph*{\bf Competing interests} The authors declare no competing interests.


 
\vspace{12 pt}
{\bf \large Figures}

\begin{figure}[htb]
\centering
\includegraphics[width=0.63\textwidth]{figure1.pdf}
\caption{
{\bf Tilted moir\'e lattices.} 
({\bf a}) Schematics of the tilted moir\'e lattice based on a reference square lattice $V(\bm r)$ with primitive translation vectors $a\bf i$ and $a\bf j$, where $a$ is the lattice constant (here set to 1) (i). One of the constitutive sublattices is  obtained by twisting the reference lattice by the angle $\theta$ around the $z-$axis, yielding the potential $V({R}(\theta){\bm r})$ (the red-coloured lattice  in (ii)). The second sublattice is obtained by rotating the reference lattice by an angle $\alpha$ around the $x-$axis (the olive-coloured lattice in (iii)), thus producing the potential 
$V({\bm r}-\alpha z \bf j)$. Superimposing $V({\bm r}-\alpha z \bf j)$ and $V({R}(\theta){\bm r})$  results in the moir\'e potential $U$ shown in (iii). The tilted moir\'e lattice has a period $Z$ along the direction of  the incident beam (shown by a dark-red arrow).  ({\bf b}) Cross-sections of the optically induced  moir\'e lattice $U({\bm r},z)$ with $p=0.3$ and $E_0=7$ (upper row) and the instantaneous bandgap structures (lower row) at distances $z=0$, $Z/4$, $Z/2$, and $3Z/4$. The red arrows indicate the primitive translation vectors ${\bm e}_1$ and ${{\bm e}}_2$ of the tilted moir\'e lattice. Blue and red lines in lower row of (b) show the first and second bands in the instantaneous lattice spectrum for the same distances. ({\bf c}) Blue and red lines show the evolution of the bottom edge of the first band and top edge of the second band with distance $z$ on one lattice period $Z$. In (b)  and (c), $\alpha=0.015$.
}
\label{fig:one}
\end{figure}


\begin{figure}[hbt]
\centering
\includegraphics[width=0.75\textwidth]{figure2.pdf}
\caption{
{\bf Numerical simulation of Thouless pumping in a titled moir\'e lattices} with $p=0.3$, Pythagorean angle $\theta=\arctan(4/3)$ ($m=2$, $n=1$) and tilt angles $\alpha=0.002$ (\textbf{a}, \textbf{b}, \textbf{e}) and $\alpha=0.015$ (\textbf{c}, \textbf{d}, \textbf{f},
\textbf{g}).  (\textbf{a}) and (\textbf{c}) Displacements of the light beam COM, $|{\bm r}_c|$, with distance $z$. (\textbf{b}) and (\textbf{d}) trajectories of the COM ${\bm r}_c=(x_c, y_c)$ in the transverse plane. The asterisks correspond to the points $z=Z$, i.e., to one longitudinal lattice period. Lines of different colour correspond to different amplitudes $E_0$ of the moir\'e lattice. Dashed diagonal lines show the evolution of ${\bm R}_{\rm av}(z)$ in (b) and (d), and of its modulus in (a) and (c) upon propagation. The results for classical motion in the potential $U$ are shown by the dark-blue solid lines. The arrows along dotted lines in (b) and (d) show the direction of tilting of one sub-lattice with respect to another one. (\textbf{e}) Snapshots of beam propagation for $\alpha=0.002$ and $E_0=10.0$ (zoom of the initial intensity distribution is shown within white square). (\textbf{f}) and (\textbf{g}) Snapshots of beam propagation for $\alpha=0.015$  and $E_0=7.5$  and $E_0=9.0$, respectively. (\textbf{h}) Evolution of the mean radius of the beam propagating inside tilted moir\'e lattice (red line), a tilted lattice obtained without twisting of two sublattices (green line) with $p=0.3$, and in a single reference lattice (blue line) with depth $p=1.3$ for $\alpha=0.015$. In all cases, $a=\pi$.}
\label{fig:two}
\end{figure}
 

 
\begin{figure}[htb]
\centering
\includegraphics[width=1.0\textwidth]{figure3.pdf}
\caption{{\bf Experimental observation of two-dimensional Thouless pumping.} ({\bf a}) Optically induced moir\'e lattices with $I_2/I_1=1.0$ (left column) and $I_2/I_1=0.1$ (right column) at different propagation distances inside the sample. Panels b, c and d correspond to the right column. ({\bf b}) Measured COM displacement $|{\bm r}_c|$ with $z$. ({\bf c}) The measured trajectory ${\bm r}_c=(x_c, y_c)$ in the transverse plane for various voltages applied to the crystal. The arrow along dotted lines shows the direction of the sliding of one sub-lattice with respect to another one. ({\bf d}) Surface and contour plots showing signal beam intensity distributions at different distances insider the sample at $\text{U}_\text{bias}=600$ V. The $x$ and $y$ axes in these distributions are determined by the directions of the primitive translation vectors ${\bm e}_1$ and ${\bm e}_2$. }
\label{fig:three}
 \end{figure}



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