Beyond Gradient Descent: Learnable Friction Fields and Energy-Aware Geometric Optimization

To address the common problems of low search efficiency, difficult constraint handling, and insufficient algorithm interpretability in high-dimensional continuous optimization problems, this paper proposes a novel continuous optimization method called the Fine-Scale Snake Optimization Algorithm. This method no longer treats the optimization process as a simple stochastic search or population evolution, but rather models it as an anisotropic geometric flow under energy-constrained conditions. The algorithm decomposes decision variables into multi-scale scale substructures, introduces a learnable friction field, a phase propagation mechanism based on topological relationships, and explicit energy budget constraints, enabling the search process to proceed continuously, stably, and structurally awarely on the feasible manifold. Theoretical analysis shows that this method exhibits good properties in terms of local convergence stability, constraint boundary handling, and exploration of non-convex regions. This research provides a new modeling paradigm for continuous optimization algorithms.