An architectural analysis of DeepOnet and a general extension of the physics-informed DeepOnet model on solving nonlinear parametric partial differential equations

The Deep Neural Operator, as proposed by Lu et al. (2021), marks a considerable advancement in solving parametric partial differential equations. This paper examines the DeepOnet model’s neural network design, focusing on the effectiveness of its trunk-branch structure in operator learning tasks. Three key advantages of the trunk-branch structure are identified: the global learning strategy, the independent operation of the trunk and branch networks, and the consistent representation of solutions. These features are especially beneficial for operator learning. Building upon these findings, we have evolved the traditional DeepOnet into a more general form from a network perspective, allowing a nonlinear interfere of the branch net on the trunk net than the linear combination limited by the conventional DeepOnet. The operator model also incorporates physical information for enhanced integration. In a series of experiments tackling partial differential equations, the extended DeepOnet consistently outperforms than the traditional DeepOnet, particularly in complex problems. Notably, the extended DeepOnet model shows substantial advancements in operator learning with nonlinear parametric partial differential equations and exhibits a remarkable capacity for reducing physics loss.