Robust loss functions are crucial in deep neural network training for managing outliers and noisy data. This paper presents an adaptive, robust variant of the L2 loss function, leveraging the fractional calculus. The fractional derivative order \(\alpha \) in the proposed Fractional L2 Loss (FL2L) function acts as a bridge, enabling a family of loss functions ranging from L2 to L1 losses. Specifically, as \(\alpha \) increases the FL2L transitions from L2’s strict penalty on large residuals (associated with faster convergence), to L1’s penalizing less (robustness to outliers). Thus, \(\alpha \) acts as an interpretable hyperparameter that adjusts the robustness level of the L2 loss function. Furthermore, we transform \(\alpha \) to a dynamic parameter, allowing the FL2L to automatically adapt its loss landscape during gradient-based optimization, improving error minimization performance. Our experiments in linear regression, drone system identification, and battery cycle life prediction demonstrate the FL2L’s significant improvements.