Efficient solving of Schrödinger equation using deep convolutional neural network model with an attention mechanism and transfer learning

Abstract We have developed a deep convolutional neural network integrated with attention mechanisms to directly process two-dimensional stochastically constrained electrostatic potentials, and established an end-to-end mapping from electrostatic potentials to ground-state energy. Compared with existing methods, this model achieves superior median absolute error-with only 1/10 of the required data volume-and adopts a lightweight architecture to reduce parameter redundancy. Furthermore, we proposed a transfer learning strategy that uses the pre-trained model as a ‘large model’ and fine-tunes it using three specific potential functions: simple harmonic oscillator (ho), infinite well (iw), and double inverted negative Gaussian (ng). Experimental results demonstrate that the adapted ‘large model’ accurately predicts these specific potential functions, effectively addressing the common generalization limitations in neural network-based partial differential equation solutions. This approach establishes a novel paradigm integrating efficiency and high precision for multi-electron system calculations.