Nonlinearly preconditioned inexact Newton methods have been applied successfully for some difficult nonlinear systems of algebraic equations arising from the discretization of partial differential equations. The preconditioning step involves identifying and balancing the nonlinearities in the system. One of the challenging tasks when applying the methods is to accurately and efficiently identify the unbalanced nonlinearities. In this work, we propose an unsupervised learning strategy based on the classical principal component analysis that learns the bad behavior of a Newton solver in the nonlinear residual subspace of a training problem. A new initial guess is produced by the nonlinear preconditioner where a projected low dimensional Jacobian system corresponding to the slow subspace of the current residuals is solved for the Newton correction vector. Numerical experiments for high Reynolds number incompressible flow problems show that the proposed method is more robust and efficient compared with existing nonlinear solvers.