This paper presents a novel approach to symbol grounding in neuro-symbolic systems, aiming to bridge the gap between neural network training and symbolic constraint solving. The proposed framework features three key components: (1) modeling symbol solution states as a Boltzmann distribution to avoid expensive state searching and facilitate interactions between network training and symbolic reasoning; (2) leveraging projection and SMT solvers in a Markov Chain Monte Carlo (MCMC) technique to efficiently sample from disconnected symbol solution spaces; and (3) an annealing mechanism to help the system escape from sub-optimal symbol groundings. Experimental results on three tasks— handwritten formula evaluation, visual Sudoku classification, and shortest path search—demonstrate the effectiveness of the proposed method, showing superior performance over existing approaches. The framework's advantages include improved interaction between neural perception and symbolic reasoning, efficient sampling from sparse and disconnected solution spaces, and better generalization capabilities.This paper presents a novel approach to symbol grounding in neuro-symbolic systems, aiming to bridge the gap between neural network training and symbolic constraint solving. The proposed framework features three key components: (1) modeling symbol solution states as a Boltzmann distribution to avoid expensive state searching and facilitate interactions between network training and symbolic reasoning; (2) leveraging projection and SMT solvers in a Markov Chain Monte Carlo (MCMC) technique to efficiently sample from disconnected symbol solution spaces; and (3) an annealing mechanism to help the system escape from sub-optimal symbol groundings. Experimental results on three tasks— handwritten formula evaluation, visual Sudoku classification, and shortest path search—demonstrate the effectiveness of the proposed method, showing superior performance over existing approaches. The framework's advantages include improved interaction between neural perception and symbolic reasoning, efficient sampling from sparse and disconnected solution spaces, and better generalization capabilities.