  The operator learning has received significant attention in recent years,
with the aim of learning a mapping between function spaces. Prior works have
proposed deep neural networks (DNNs) for learning such a mapping, enabling the
learning of solution operators of partial differential equations (PDEs).
However, these works still struggle to learn dynamics that obeys the laws of
physics. This paper proposes Energy-consistent Neural Operators (ENOs), a
general framework for learning solution operators of PDEs that follows the
energy conservation or dissipation law from observed solution trajectories. We
introduce a novel penalty function inspired by the energy-based theory of
physics for training, in which the energy functional is modeled by another DNN,
allowing one to bias the outputs of the DNN-based solution operators to ensure
energetic consistency without explicit PDEs. Experiments on multiple physical
systems show that ENO outperforms existing DNN models in predicting solutions
from data, especially in super-resolution settings.
