Instant prediction of relaxation in moir\'{e} superlattices using neural networks

The relaxation of moir\'e superlattices in twisted bilayers of transition metal dichalcogenides (TMDs) has been modeled using a set of neural-network-based approaches. We implemented and compared several architectures, including (i) an interpolator combined with an autoencoder, (ii) an interpolator combined with a decoder, (iii) a direct generator mapping input parameters to displacement fields, and (iv) a physics-informed neural network (PINN). Among these, the direct generator architecture demonstrated the best performance, achieving machine-level precision with minimal training data. Remarkably, once trained, this simple fully connected network is able to predict the full displacement field of a moir\'e bilayer within a fraction of a second, whereas conventional continuum simulations require hours or even days. This finding highlights the low-dimensional nature of the relaxation process and establishes neural networks as a practical and efficient alternative to ab initio approaches for rapid modeling and high-throughput screening of 2D twisted heterostructures.