SchrödingerNet: A Universal Neural Network Solver for The Schrödinger Equation

Recent advances in machine learning have facilitated numerically accurate solution of the electronic Schr\"{o}dinger equation (SE) by integrating various neural network (NN)-based wavefunction ansatzes with variational Monte Carlo methods. Nevertheless, such NN-based methods are all based on the Born-Oppenheimer approximation (BOA) and require computationally expensive training for each nuclear configuration. In this work, we propose a novel NN architecture, Schr\"{o}dingerNet, to solve the full electronic-nuclear SE by defining a loss function designed to equalize local energies across the system. This approach is based on a rotationally equivariant total wavefunction ansatz that includes both nuclear and electronic coordinates. This strategy not only allows for the efficient and accurate generation of a continuous potential energy surface at any geometry within the well-sampled nuclear configuration space, but also incorporates non-BOA corrections through a single training process. Comparison with benchmarks of atomic and molecular systems demonstrates its accuracy and efficiency.