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White dwarfs (WDs) in active galactic nucleus (AGNs) discs might migrate tothe inner radii of the discs and form restricted three-body systems with twoWDs moving around the central supermassive black hole (SMBH) in close orbits.These systems could be dynamical unstable, which can lead to very closeencounters or direct collisions. In this work, we use N-body simulations tostudy the evolution of such systems with the different initial orbitalseparation $p$, relative orbital inclination $\Delta{i}$ and SMBH mass $M$. Itis found that the close encounters of WDs mainly occur at $1.1R_{\rm H}\lesssim p \lesssim 2\sqrt{3}R_{\rm H}$, where $R_{\rm H}$ is the mutual Hillradius. For $p<1.1R_{\rm H}$, the majority of WDs move in horseshoe or tadpoleorbits, and only few of them with small initial orbital phase differenceundergo close encounters. For $p=3.0R_{\rm H}$, WD-WD collisions occur in mostof the samples within a time of $10^5P_1$, and considerable collisions occurwithin a time of $t<62P_1$ for small orbital radii, where $P_1$ is the orbitalperiod. The peak of the closest separation distribution increase and the WD-WDcollision fraction decreases with an increase of the relative inclination. Theclosest separation distribution is similar in cases with the different SMBHmass, but the WD-WD collision fraction decreases as the mass of SMBHsincreases. According to our estimation, the event rate of the cosmic WD-WDcollision in AGN discs is about $300{\rm Gpc^{-3}yr^{-1}}$, roughly $1\%$ ofthe one of the observed type Ia supernova. The corresponding electromagneticemission signals can be observed by large surveys of AGNs.

White Dwarf--White Dwarf collisions in AGN discs via close encounters