Diffusion models have achieved remarkable success in content generation but often incur prohibitive computational costs due to iterative sampling. Recent feature caching methods accelerate inference via temporal extrapolation, yet can suffer quality degradation from inaccurate modeling of the complex dynamics of feature evolution. We propose HiCache (Hermite Polynomial-based Feature Cache), a training-free acceleration framework that improves feature prediction by aligning mathematical tools with empirical properties. Our key insight is that feature-derivative approximations in diffusion Transformers exhibit multivariate Gaussian characteristics, motivating the use of Hermite polynomials as a potentially optimal basis for Gaussian-correlated processes. We further introduce a dual-scaling mechanism that ensures numerical stability while preserving predictive accuracy, and is also effective when applied standalone or integrated with TaylorSeer. Extensive experiments demonstrate HiCache's superiority, achieving 5.55x speedup on FLUX.1-dev while matching or exceeding baseline quality, and maintaining strong performance across text-to-image, video generation, and super-resolution tasks. Moreover, HiCache can be naturally added to previous caching methods to enhance their performance, e.g., improving ClusCa from 0.9480 to 0.9840 in terms of image rewards. Code: https://github.com/fenglang918/HiCache