<p>This paper introduces an adaptive gradient descent algorithm based on predictor–corrector mechanism inspired by Heun’s second order numerical integration methods called HEAD (Heun’s enhanced adaptive descent). Traditional deep learning optimization algorithms such as SGD, Adam and AdamW etc. frequently exhibit suboptimal stability and variance susceptibility in high-dimensional non-convex optimization landscapes. In order to overcome these challenges, the proposed HEAD optimizer introduces “gradient surprise” corrector in turn modulated by dynamic trust gate and warmup interpolation so it could effectively approximate second-order trajectory without the need for intractable Hessian computations. The extensive testing was performed against modern baselines such as AdamW on benchmark datasets like MNIST, FMNIST, CIFAR-10 and CIFAR-100, including ResNet-based evaluations of whose empirical and statistical results showed that HEAD optimizer not only achieved significant improvements in convergence speed, variance reduction but also depicted remarkable generalization across various non-convex classification tasks. Furthermore, various other experiments were conducted including those of transfer learning, text-based and MLP tasks which further validated the optimizer’s generalizability. Moreover, integration of HEAD into real-world engineering application such as medical image diagnostics and biometric security extraction could be some avenues to highlight proposed optimizers capacity to stabilize training dynamics in highly adversarial and noisy environments.</p>