A Hybrid Analytic–Fitting Conceptual Paradigm Based on Topological-Dynamic Boundary Tension Heuristic — Taking QiankunNet of USTC as a Reference Case

High-dimensional many-body continuous-field systems have long faced the exponential-complexity barrier of reductionist numerical methods. Traditional purely analytic methods are limited by the explosion of degrees of freedom; purely data-driven machine learning models (e.g., QiankunNet from USTC) can circumvent part of the exponential wall, but are essentially interpolation within the training domain. Their ability to extrapolate across attractor basins is weak, and they lack analytic invariant constraints. Targeting the inherent limitation of neural-network quantum state (NQS) methods in cross–attractor-basin extrapolation, this paper proposes a hybrid analytic–fitting paradigm as a conceptual framework to be validated. For continuous-field systems, a dynamic topological partition is made according to interaction strength: the strongly coupled core region is characterized by topological–physical analytic invariants, while the weakly coupled peripheral region is approximated by attention-based machine learning models. Topological boundaries are no longer generated by human-prescribed meshes; instead, they dynamically emerge from the interaction strengths inside the system. Analytic invariants are used to construct a regularization term that constrains the fitting subsystem, in an attempt to improve the model's extrapolation performance across different steady-state attractor basins. Taking quantum many-electron systems as a demonstration case and referencing the QiankunNet technical route, this paper compares and analyzes its engineering advantages and paradigm-level limitations, and also discusses the formal extensibility of the paradigm to continuous-field problems such as the Navier–Stokes equations and dynamical systems. This work shifts the focus of the computational paradigm from brute-force enumeration of microscopic degrees of freedom to identifying emergent attractors and boundary structures within continuous fields.