Robust discovery of partial differential equations in complex situations

Data-driven discovery of partial differential equations (PDEs) has achieved considerable development in recent years. Several aspects of problems have been resolved by sparse regression-based and neural-network-based methods. However, the performance of existing methods lacks stability when dealing with complex situations, including sparse data with high noise, high-order derivatives, and shock waves, which bring obstacles to calculating derivatives accurately. Therefore, a robust PDE discovery framework, called the robust deep-learning genetic algorithm (R-DLGA), that incorporates the physics-informed neural network (PINN) is proposed in this paper. In the framework, preliminary results of potential terms provided by the DLGA are added into the loss function of the PINN as physical constraints to improve the accuracy of derivative calculation. It assists in optimizing the preliminary result and obtaining the ultimately discovered PDE by eliminating the error compensation terms. The stability and accuracy of the proposed R-DLGA in several complex situations are examined for proof and concept, and the results prove that the proposed framework can calculate derivatives accurately with the optimization of the PINN and possesses surprising robustness for complex situations, including sparse data with high noise, high-order derivatives, and shock waves.

Automated summary

The paper introduces a robust PDE discovery framework, R-DLGA, incorporating the physics-informed neural network (PINN). This framework optimizes preliminary results to improve derivative calculation accuracy and demonstrates remarkable robustness when dealing with complex situations.