4DIP Framework: A Symbolic Geometric Approach to Solving Differential Systems

Classical numerical solvers, such as Runge-Kutta (RK45) and Backward Differentiation Formulas (BDF), excel for smooth ordinary differential equations (ODEs) but falter in stiff, chaotic, or high-dimensional systems due to adaptive step-size control and numerical derivative approximations. These limitations hinder applications in complex physics, such as plasma dynamics or tensorial field equations in cosmology. The Four-Dimensional Iterative Prediction (4DIP) framework offers a transformative alternative, employing a symbolic, geometry-based method to contract a residual vector between the target solution Fn and an evolving guess Gn. Like a spring settling into equilibrium, 4DIP iteratively refines solutions without time discretization, preserving system structure across domains like electromagnetism and relativistic hydrodynamics. This revised paper addresses prior feedback by enhancing theoretical foundations, expanding comparisons to state-of-the-art solvers, adding test cases, and improving accessibility for a broad audience. We provide rigorous convergence analysis, detailed failure mode discussions, and opensource code to ensure reproducibility, positioning 4DIP as a versatile tool for computational science.