The Woods-Saxon (WS) potential, a well-established model describing the central mean field potential in atomic nuclei, plays a critical role in studies of nuclear structure and nuclear reactions. Currently, research on the WS potential primarily focuses on the forward problem, solving for wave functions given the potential, whereas the inverse problem of inferring WS potential parameters from wave functions remains underexplored. In this work, three WS central potential parameters, the depth $V_0$, radius $R_0$, and diffuseness $a_0$ are selected. Corresponding wave functions are calculated using the WS potential model, and a Gaussian process emulator is trained, achieving a coefficient of determination $R^2=0.99$, and the mean absolute error is 0.0015 on the test set. The wave functions associated with $V_0=-50$ MeV, $R_0=3.5$ fm, $a_0=0.45$ fm are treated as experimental evidence. Using Bayesian analysis, we obtain posterior distributions for the parameters, $V_0=-50.{30}_{-5.27}^{+5.00}$ MeV, $R_0=3.{56}_{-0.06}^{+0.07}$ fm, $a_0=0.{43}_{-0.04}^{+0.04}$ fm. The central values of the posterior distributions agree closely with the true values, and the full distributions encompass the target WS potential parameters. This demonstrates that Bayesian inference can successfully invert WS parameters from theoretical wave functions, thereby offering a robust new approach for potential inversion and for exploring complex nuclear systems such as exotic nuclei and high-energy scattering processes.

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• Nuclear forces
• Nuclear reactions
• Bayesian methods
• Machine learning