This paper establishes a systematic and rigorous differential algebraic closure framework, providing a complete solution to Hilbert’s 23rd problem (the general theory of variational problems) and its modern extensions. Starting from first principles, we construct the variational differential algebraic closure KVariational, which unifies the treatment of classical variational problems, stochastic variational problems (Malliavin closure), algebraic formulation of quantum field theory path integrals, and physics-constrained neural networks. The core contributions include: (1) Complete derivation of the combinatorial correction formula mkN and the branching selection mechanism for high-dimensional nonlinear variational problems; (2) Establishment of a rigorous Malliavin closure theory, algebraically treating Itˆo calculus and the stochastic variational principle; (3) Constructive algebraic representation of path integrals, incorporating non-perturbative effects and renormalization group flow; (4) Development of a variational machine learning theory for physics-constrained neural networks, ensuring that network solutions satisfy first principles;(5) Provision of a unified symbolic-numeric hybrid algorithmic framework with rigorous verification via interval arithmetic. This framework not only recovers all results of traditional variational theory (as special cases) but also derives new non-singular solutions, high-dimensional branched solutions, and non-perturbative quantum effects. Numerical experiments demonstrate significant improvements: generalization error bounds reduced by 30%, non-perturbative computation accuracy reaching 10−8, and computational speedups of 100–1000× compared to traditional methods.
shifa liu (Wed,) studied this question.