Explainable Deep Operator Networks for Interpretable Solutions of Nonlinear PDEs

Abstract

Deep learning, or more specifically DeepONets, has been a potent tool to solve a variety of nonlinear partial differential equations (PDEs). They are however black-box and thus not easily interpretable which prevents their use in important scientific and engineering fields where a bit of transparency and physical understanding is needed. The presented paper tackles this issue and suggests a framework named Explainable DeepONet (X-DeepONet) that combines explainability-mechanisms with physics-informed learning. The suggested model is built on top of the regular DeepONet network by utilizing feature attribution algorithms and sensitivity analysis in order to give a human-understandable mapping between input functions and PDE solutions. Moreover, physics constraints are also enforced using residual-based loss functions to make sure that the governing equations and boundary conditions are met. The framework is tested on benchmark nonlinear PDEs, where the predictive accuracy of the framework, in terms of mean squared error and relative L2 norms, are shown to be higher, and at the same time provides useful information about the model behavior with interpretable feature importance and response sensitivity. By comparing the results with baseline models, such as standard DeepONet and physics-informed neural networks, it is possible to note that the proposed method is better in terms of accuracy and interpretability. The findings prove the suggested framework as an effective and innovative method of transparent operator learning and there are enormous perspectives of enhancing credible scientific machine learning in intricate Multiphysics systems.