Whittaker smoother is a widely adopted solution to pre-process satellite image time series. Yet, two key limitations remain: the smoothing parameter must be tuned individually for each pixel, and the standard formulation assumes homoscedastic noise, imposing uniform smoothing across the temporal dimension. This paper addresses both limitations by casting the Whittaker smoother as a differentiable neural layer, in which the smoothing parameter is inferred by a neural network. The framework is further extended to handle heteroscedastic noise through a time-varying regularization, allowing the degree of smoothing to adapt locally along the time series. To enable large-scale processing, a sparse, memory-efficient, and fully differentiable implementation is proposed, exploiting the symmetric banded structure of the underlying linear system via Cholesky factorization. Benchmarks on GPU demonstrate that this implementation substantially outperforms standard dense linear solvers, both in speed and memory consumption. The approach is validated on SITS acquired over the French metropolitan territory between 2016 and 2024. Results confirm the feasibility of large-scale heteroscedastic Whittaker smoothing, though reconstruction differences with the homoscedastic baseline remain limited, suggesting that the transformer architecture used for smoothing parameter estimation may lack the temporal acuity needed to capture abrupt noise variations such as singleday cloud contamination.