Tensor-AIS is a compact, header-only library that provides an Algebraic Iterative Solver (the AIS part in the name), mainly for solving nonlinear Finite Element problems, but it could be used whenever we need to incorporate powers of the unknowns in the system of equations. The solver relies on the gradient descent iterative scheme. The library also provides a toolset of multidimensional arrays, tensor product and index contraction operations, as well as multidimensional tensor polynomial evaluations.