A sound field estimation method based on kernel interpolation with an adaptive kernel function is proposed. The kernel-interpolation-based sound field estimation methods enable physics-constrained interpolation from pressure measurements of distributed microphones with a linear estimator, which constrains interpolation functions to satisfy the Helmholtz equation. However, a fixed kernel function would not be capable of adapting to the acoustic environment in which the measurement is performed, limiting their applicability. To make the kernel function adaptive, we represent it with a sum of directed and residual trainable kernel functions. The directed kernel is defined by a weight function composed of a superposition of exponential functions to capture highly directional components. The weight function for the residual kernel is represented by neural networks to capture unpredictable spatial patterns of the residual components. Experimental results using simulated and real data indicate that the proposed method outperforms the current kernel-interpolation-based methods and a method based on physics-informed neural networks.