Learning accurate dynamical models from data is a central challenge in control systems, robotics, and aerospace engineering, where governing equations are needed for prediction, analysis, and controller design. Classical system identification approaches perform well when model structure is known a priori, but struggle when this structural knowledge is unavailable and trajectory data are corrupted by substantial measurement noise. In the presented work, we address the problem of robust learning of unknown dynamic equations from noise-corrupted data. We build on the Sparse Identification of Nonlinear Dynamics (SINDy) method that selects governing equations by assuming sparsity in a large library of candidate functions and solving a sparse regression problem. Classical SINDy formulations rely on numerical estimates of derivatives, which render the method fragile against measurement noise. Weak-form SINDy mitigates the effect of measurement noise by projecting the equations onto a set of test functions before regression, avoiding numerical differentiation; however, the typically used "bump"-shaped test functions lack physical interpretability, have many hyperparameters that require tuning, and are not mutually orthogonal, therefore introducing redundancy into the regression problem. To address the above robustness and interpretability challenges in sparse system identification, we introduce Fourier Weak SINDy, a minimal noise-robust and interpretable derivative-free equation learning method that combines weak-form sparse equation learning with spectral density estimation for data-driven test function selection. By using orthogonal sinusoidal test functions inspired by their prevalence in Modulating Function-based system identification, the weak-form sparse regression problem reduces to a regression over Fourier coefficients. Dominant frequencies are then selected via multitaper estimation of the frequency spectrum of the data. This formulation unifies weak-form learning and spectral estimation within a compact and flexible framework. We illustrate the effectiveness of this approach in numerical experiments across multiple chaotic and hyperchaotic ODE benchmarks.