Neural Operators (NOs) are machine learning models designed to solve partial differential equations (PDEs) by learning to map between function spaces. Neural Operators such as the Deep Operator Network (DeepONet) and the Fourier Neural Operator (FNO) have demonstrated excellent generalization properties when mapping between spatial function spaces. However, they struggle in mapping the temporal dynamics of time-dependent PDEs, especially for time steps not explicitly seen during training. This limits their temporal accuracy as they do not leverage these dynamics in the training process. In addition, most NOs tend to be prohibitively costly to train, especially for higher-dimensional PDEs. In this paper, we propose the Temporal Neural Operator (TNO), an efficient neural operator specifically designed for spatio-temporal operator learning for time-dependent PDEs. TNO achieves this by introducing a temporal-branch to the DeepONet framework, leveraging the best architectural design choices from several other NOs, and a combination of training strategies including Markov assumption, teacher forcing, temporal bundling, and the flexibility to condition the output on the current state or past states. Through extensive benchmarking and an ablation study on a diverse set of example problems we demonstrate the TNO long range temporal extrapolation capabilities, robustness to error accumulation, resolution invariance, and flexibility to handle multiple input functions.
Diab et al. (Thu,) studied this question.