A methodology for designing robust, low-order observers for a class of spectral infinite-dimensional nonlinear systems is presented. This approach uses the lowdimensional subspace explicitly in the observer design. Then, robustness to bounded model uncertainties is incorporated using the Lyapunov reconstruction method from robust control theory. Furthermore, the proposed design includes a data-driven learning algorithm that auto-tunes the observer gains to optimize the performance of the state estimation. A numerical study using a model from fluid dynamics -Burgers equation- demonstrates the effectiveness of the proposed observer.