In one of the previous posts we have talked about brand new Laplace Neural Operator, this time we focus on extention of well-known Deep Operator Network (DeepONet). DeepONet is a deep learning framework for solving forward and inverse problems involving nonlinear operators. It can learn to map a function to another function by using pairs of input-output functions as training data. DeepONet can handle complex and high-dimensional problems without requiring any knowledge of the underlying equations or boundary conditions.

A recent paper by Liu et al. named "A novel deeponet model for learning moving-solution operators with applications to earthquake hypocenter localization" proposes a novel method for solving inverse problems using deep neural networks. The method is called Extended DeepONet (X-DeepONet).

The Extended DeepONet method enhances the DeepONet framework by incorporating two key changes to the architecture of the DeepONet. While the DeepONet uses only the dot product operator for the output, X-DeepONet incorporates also subtraction and sum operations. The results of all of these operations are fed to a single neural network named Root Net, which produces output. This extended architecture is better at generalization of more complex tasks, such as variable source position determination, as in earthquake hypocenter localization.

The paper demonstrates the effectiveness of the Extended DeepONet method on several velocity models, namely Marmousi velocity model, smooth velocity model and OpenFWI velocity model. While with the first two models the architecture proved to predict highly accurate positions of the source, the third model is order of magnitude more complex, and the error reached with this model opens space for further improvements.

The Extended DeepONet method is a promising technique for solving inverse problems using deep learning. It can potentially be applied to various engineering domains, such as fluid mechanics, solid mechanics, electromagnetics, and heat transfer. The paper opens up new possibilities for combining physics-based modeling and data-driven learning for solving challenging engineering problems.

For further reading we refer to here.