We combine the three-dimensional (3D) two-potential approach (3DTPA) with the physics-informed neural network (PINN) to compute the $\alpha$ decay width of deformed nuclei, which is expressed in terms of the 3D bound-state wave function and the Coulomb scattering wave function on the separation surface of 3DTPA. The designed PINN output representation and loss function enable stable and accurate solutions of 3D bound-state wave function whose amplitude is extremely small on the separation surface. The PINN is validated against the exact solutions of anisotropic harmonic-oscillator potential and the widely used Woods-Saxon potential. With a microscopic potential constructed by the quartetting wave-function approach (QWFA), the $\alpha$-cluster preformation and tunneling probabilities of typical deformed $\alpha$ emitters ${}^{256}\mathrm{No}$ and ${}^{274}\mathrm{Ds}$ are self-consistently computed by using the $3\text{DTPA}+\text{PINN}$ method. It is found that the method provides a practical tool for microscopic studies of quasistationary states of deformed $\alpha$ emitters.

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• Alpha decay
• Nuclear structure & decays
• Machine learning