Extended physics-informed neural networks (Ex-PINNs): a machine-learning method for solving partial differential equations in domain with spatially heterogeneous parameters

Abstract In this paper, we propose a machine-learning based method called Extended Physics-informed Neural Networks (Ex-PINNs) to solve the partial differential equations in domain with unknown parameters. In the existing PINNs, the partial differential equations with distributed parameters are often solved by setting several sub-domains, which can be cumbersome when many interfaces and sub-domains must be prescribed. In order to solve such problem, an extra neural network is introduced to the standard PINN architecture for the characterization and reconstruction of the unknown continuous parameters. The loss function of the Ex-PINNs consists of residual part and boundary condition part formulated by not only the neural networks of solution, but also the one characterizing the parameters. By the method, we avoid the need for standard PINNs to introduce a large number of interfaces and sub-domains to approximate solution. We also present the solid mechanics and the heat conduction problems for the validation of the proposed Ex-PINN. Numerical experiments show that the results of the Ex-PINNs are in good agreement with the FEM in relative error of less than 3.0%. The method provides an extension of PINN in latent cases such as solid mechanics.