This paper proposes a novel, physics-regularized sound field reconstruction framework based on adaptive Gaussian kernels. Unlike traditional kernel methods that rigidly anchor basis functions to fixed microphone positions, our model parameterizes the unknown sound field using Gaussian functions with fully trainable center positions. This allows the kernels to adaptively converge to spatial regions with severe acoustic gradients, providing extreme flexibility to accurately track complex acoustic modes. Leveraging this explicit architecture, we directly derive an analytical expression for the spatial Laplacian operator. This circumvents computationally expensive automatic differentiation (AD), enabling highly efficient physical regularization and significantly accelerating the overall training process. Simulations under sparse sampling show that the proposed method achieves higher accuracy than traditional kernel methods and physics-informed neural networks (PINNs), while the analytical PDE loss enables significantly faster training.