Previous acoustic transfer methods rely on extensive precomputation and storage of data to enable real-time interaction and auditory feedback. However, these methods struggle with complex scenes, especially when dynamic changes in object position, material, and size significantly alter sound effects. These continuous variations lead to fluctuating acoustic transfer distributions, making it challenging to represent with basic data structures and render efficiently in real time. To address this challenge, we present Neural Acoustic Transfer, a novel approach that utilizes an implicit neural representation to encode precomputed acoustic transfer and its variations, allowing for real-time prediction of sound fields under varying conditions. To efficiently generate the training data required for the neural acoustic field, we developed a fast Monte-Carlo-based boundary element method (BEM) approximation for general scenarios with smooth Neumann conditions. Additionally, we implemented a GPU-accelerated version of standard BEM for scenarios requiring higher precision. These methods provide the necessary training data, enabling our neural network to accurately model the sound radiation space. We demonstrate our method's numerical accuracy and runtime efficiency (within several milliseconds for 30s audio) through comprehensive validation and comparisons in diverse acoustic transfer scenarios. Our approach allows for efficient and accurate modeling of sound behavior in dynamically changing environments, which can benefit a wide range of interactive applications such as virtual reality, augmented reality, and advanced audio production.