SKG engine unifies geometric laws with kernel based learning
SKG: A Law-Induced Engine from Inference to Induction
Machine Learning
Summary
This paper introduces the SKG reasoning engine, which combines mathematical laws and data to improve how machines learn from examples and rules. The authors develop a detailed framework that uses geometry and kernels (tools that help machines understand data shapes) to model complex systems with conservation laws and symmetries. They present mathematical proofs ensuring the framework can reconstruct underlying rules correctly and handle noisy or unfamiliar data. The approach is tested on examples like simple oscillators and rotating systems, showing reliable performance.
What this means in practice
- •For control system engineers: Construct and verify models of physical systems that obey conservation laws with enhanced data-driven inference under uncertainty.
- •For robotics software developers: Create learning algorithms that incorporate physical symmetries and constraints for better adaptation to changing environments.
Authors
Jian An, Chaoshi Wu, Jinhui Wu
Abstract
This paper proposes the SKG (Symplectic-Kernel Gradient) reasoning engine for law-induced inference and induction. Laws are a priori; data provide instances. Inference is geometric evolution under law constraints. We organize symplectic geometry, GENERIC, kernel methods, RKHS, group-equivariant networks, and constrained mechanics into a unified, analyzable, implementable, verifiable framework. We propose the SKG octuple S = (M, g, omega, H, S, G, Delta, Pi), whose four components correspond to four levels of laws: conservation laws, constitutive relations, symmetry groups, and axiom systems. Main results: (1) Invariants enter metric construction via the Jacobian: g = J^T Sigma^-1 J + epsilon_G I. For constant cases, we establish inverse reconstruction theory: solvability rank condition (Thm 3.9), consistency criterion (Thm 3.11), and uniqueness theorem (Thm 3.14). (2) The kernel bandwidth matrix is uniquely determined by Fisher information and satisfies task-independence (Thm 3.19). (3) The metric reconstruction error decomposes into four terms (Thm 3.21), with squared condition number kappa(Sigma^*)^2 as the dominant amplification factor (Thm 3.22). (4) For state-dependent cases, we establish function-space reconstruction theory on matrix-valued RKHS (Thms 4.8, 4.9). (5) Three-layer projections handle axiom, symplectic-step, and conservation constraints. (6) The residual equation yields the observable formula, four-dimensional rank condition, and noise propagation, while g_qq is structurally unidentifiable. (7) Kernel parameterization yields the RKHS extrapolation bound (Thm 4.15) and strict OOD gain theorem in group directions (Thm 8.15). (8) The learning-decoupling theorem (Thm 9.1) proves the sole object of induction is the parameters of the task energy. Validation: 1D damped oscillator and 2D rotation-invariant system; 6 unit tests, 10 experiments (paired t-test, p < 0.01).