This paper proposes an Adaptive Basis-inspired Deep Neural Network (ABI-DNN) for solving partial differential equations with localized phenomena such as sharp gradients and singularities. Like the adaptive finite element method, ABI-DNN incorporates an iteration of "solve, estimate, mark, enhancement", which automatically identifies challenging regions and adds new neurons to enhance its capability. A key challenge is to force new neurons to focus on identified regions with limited understanding of their roles in approximation. To address this, we draw inspiration from the finite element basis function and construct the novel Basis-inspired Block (BI-block), to help understand the contribution of each block. With the help of the BI-block and the famous Kolmogorov Superposition Theorem, we first develop a novel fixed network architecture named the Basis-inspired Deep Neural Network (BI-DNN), and then integrate it into the aforementioned adaptive framework to propose the ABI-DNN. Extensive numerical experiments demonstrate that both BI-DNN and ABI-DNN can effectively capture the challenging singularities in target functions. Compared to PINN, BI-DNN attains significantly lower relative errors with a similar number of trainable parameters. When a specified tolerance is set, ABI-DNN can adaptively learn an appropriate architecture that achieves an error comparable to that of BI-DNN with the same structure.