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In the recent decade, computational tools have become central in material design, allowing rapid development cycles at reduced costs. Machine learning tools are especially on the rise in photonics. However, the inversion of the Maxwell equations needed for the design is particularly challenging from an optimization standpoint, requiring sophisticated software. We present an innovative, open-source software tool called Neural Inverse Design of Nanostructures (NIDN) that allows designing complex, stacked material nanostructures using a physics-based deep learning approach. Instead of a derivative-free or data-driven optimization or learning method, we perform a gradient-based neural network training where we directly optimize the material and its structure based on its spectral characteristics. NIDN supports two different solvers, rigorous coupled-wave analysis and a finite-difference time-domain method. The utility and validity of NIDN are demonstrated on several synthetic examples as well as the design of a 1550 nm filter and anti-reflection coating. Results match experimental baselines, other simulation tools, and the desired spectral characteristics. Given its full modularity in regard to network architectures and Maxwell solvers as well as open-source, permissive availability, NIDN will be able to support computational material design processes in a broad range of applications.