Taming the expressiveness of neural network wave functions for robust convergence in quantum Monte Carlo simulations

Neural networks are powerful tools for representing complex functions and are widely used as trial wave functions in variational quantum Monte Carlo simulations of interacting many-body systems. A common approach approximates the expected energy of the trial wave function by averaging the local energy E_L over Monte Carlo samples of the trial wave function. Here, we show that this energy minimization approach can be unstable. The main issue is that the expressiveness of neural networks often leads to trial wave functions with sharp features, making the average of E_L a poor approximation of the expected energy. In fact, the averaged E_L can even fall below the ground-state energy for many samples, violating the variational principle. A more robust alternative is the minimization of the variance of E_L, based on the fact that all samples of an exact eigenfunction have the same E_L. Since the minimum value of this variance should be zero when the neural network converges to an eigenstate, minimizing compressed versions of the variance can also be effective. Additionally, adjusting the architectures and weight initializations of neural networks to promote smoothness in the trial wave functions enhances convergence robustness. These concepts are illustrated using systems of interacting fermions in a harmonic trap.