We generalize the pre-orthogonal adaptive Fourier approximation developed by T. Qian et al [11], [9], [5] to functions in the Bergman space on the unit disc and the unit ball $B$ to the Bergman space $A^2(\cD)$ on the irreducible bounded symmetric domain $\cD.$ We show that satisfies the boundary vanishing property, so that the maximum selection principle allows us to give an adaptive expansion of any function $f$ in terms of linear combinations of generalized kernel functions in an optimal way.