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This paper builds an infinity shaped ("∞"-shaped) laser scanning welding test platform based on a self-developed motion controller and galvanometer scanner control gateway, takes the autogenous bead-on-plate welding of 304SS with 3 mm thick specimens as the experimental objects, designs the experimental parameters by the Latin hypercube sampling method for obtaining different penetration depth welded joints, and presents a methodology based on the neuroevolution of augmenting topologies for predicting the penetration depth of "∞"-shaped laser scanning welding. Laser power, welding speed, scanning frequency, and scanning amplitude are set as the input parameters of the model, and welding depth (WD) as the output parameter of the model. The model can accurately reflect the nonlinear relationship between the main welding parameters and WD by validation. Moreover, the normalized root mean square error (NRMSE) of the welding depth is about 6.2%. On the whole, the proposed methodology and model can be employed for guiding the actual work in the main process parameters' preliminary selection and lay the foundation for the study of penetration morphology control of "∞"-shaped laser scanning welding.