A Bayesian Hybrid Surrogate Framework for Mechanical Systems with Complex Stiffness Physics-Constrained Inference and Adaptive Solution of Differential Equations

Mechanical systems with heterogeneous materials, frequency-dependent damping, uncertain boundary conditions, and incomplete measurements often requirerepeated solutions of large systems of differential equations. Conventional finiteelement models represent governing mechanics explicitly but may be computationally expensive for calibration, uncertainty propagation, and design exploration.Purely data-driven surrogates can reduce computational cost, but may violate equilibrium, conservation, or constitutive constraints and may give poorly calibrateduncertainty estimates.This paper proposes a Bayesian Physics-Constrained Hybrid Surrogate (BPCHS)framework for mechanical forward and inverse problems involving complex stiffnessmatrices. The framework combines a low-fidelity mechanics solver, a probabilisticdiscrepancy model, and a physics-informed surrogate. Unknown material andstructural parameters are inferred jointly with model discrepancy from noisyobservations. The formulation represents frequency-dependent stiffness through acomplex dynamic operator and incorporates equilibrium and boundary-conditionresiduals into surrogate training. A cost-aware adaptive sampling rule is proposed to select additional high-fidelity simulations or experiments where predictiveuncertainty and mechanical residuals are jointly large.The paper presents a research hypothesis, governing equations, a Bayesianobservation model, a hybrid surrogate construction, and a validation protocol.The proposed principles are testable modeling hypotheses, not established physicallaws. Their validity must be assessed through numerical and experimental studies.No empirical performance claim is made. Potential applications include structuralmodel updating, viscoelastic material characterization, damage-sensitive responseprediction, and uncertainty-aware design.