DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators

Understand

While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator.

• This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data.
• However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors.
• To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset.