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Euplectella aspergillum, as a textbook example presented by nature, is famous for its superior buckling resistance and stiffness subject to compression of arbitrary directions and its double-diagonal reinforced square microstructure has attracted considerable interest. In the present work, a parametric model is developed for a class of sponge-like lattices, and a neural network model is trained to realize accurate and efficient mapping from the geometry parameters to the corresponding effective stiffness and buckling resistance under uniaxial compression along all directions. Then an optimization formulation is proposed for designing a sponge-like lattice with on-demand requirements. Using the same amount of material, in the present design framework, the average buckling resistance of the original sponge-lattice model in literature is increased by about 40% and the design efficiency is improved by about 3–4 orders as compared to the traditional optimization process with exact mechanical modeling. Combining the intelligence of nature, humans and computers, the present AI-enhanced bioinspiration paradigm provides a novel pathway for designing innovative materials and structural systems.