Large Language Models as Universal Function Approximators: A Framework for Physical System Simulation and Autodidactic Learning

We establish Large Language Models (LLMs) as universal function approximators through deterministic scaffolding, achieving both theoretical completeness and practical implementation. Our framework demonstrates universal approximation across R^n while enabling continuous learning at inference time, verified through complex physical simulations including Navier-Stokes dynamics, quantum many-body systems, and Einstein field equations. The system achieves 95% Action Success Rate in physical system modeling versus 15-25% for existing approaches, while maintaining mathematical rigor and physical consistency (CAIDR 12.0922, surpassing current 0.4116 benchmark).

Key implementations span quantum mechanics (wave function approximation, ψ(x,t)), fluid dynamics (∂v/∂t + (v·∇)v = -∇p/ρ + ν∇²v), and number theory (Riemann ζ(s) analytical continuation). The framework generates pin-accurate hardware simulations (DDR2 SDRAM), solves partial differential equations, and explores novel quantum gravitational coupling constants (QGIC = √(h·c)/G = 5.46×10⁻⁸ kg·m). Our AGI-R benchmark demonstrates 100% code generation/execution success across physics domains while maintaining token coherence over 10⁶ sequences.

This work proves LLMs can simultaneously serve as universal approximators and autodidactic engines, enabling rapid theoretical physics exploration and complex systems modeling. The framework bridges computational mathematics, quantum mechanics, and dynamical systems through a unified approximation theory, establishing a new paradigm for scientific computing.