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In the scope of "AI for Science", solving inverse problems is a longstanding challenge in materials and drug discovery, where the goal is to determine the hidden structures given a set of desirable properties. Deep generative models are recently proposed to solve inverse problems, but these currently use expensive forward operators and struggle in precisely localizing the exact solutions and fully exploring the parameter spaces without missing solutions. In this work, we propose a novel approach (called iPage) to accelerate the inverse learning process by leveraging probabilistic inference from deep invertible models and deterministic optimization via fast gradient descent. Given a target property, the learned invertible model provides a posterior over the parameter space; we identify these posterior samples as an intelligent prior initialization which enables us to narrow down the search space. We then perform gradient descent to calibrate the inverse solutions within a local region. Meanwhile, a space-filling sampling is imposed on the latent space to better explore and capture all possible solutions. We evaluate our approach on three benchmark tasks and two created datasets with real-world applications from quantum chemistry and additive manufacturing, and find our method achieves superior performance compared to several state-of-the-art baseline methods. The iPage code is available at https://github.com/jxzhangjhu/MatDesINNe.