Robust discovery of partial differential equations in complex situations

TL;DR

This paper introduces R-DLGA, a robust framework combining physics-informed neural networks and genetic algorithms to accurately discover PDEs in complex, noisy, and high-derivative scenarios, outperforming existing methods.

Contribution

The paper proposes a novel PDE discovery framework that enhances stability and accuracy in complex situations by integrating deep learning-genetic algorithms with PINNs.

Findings

Accurately calculates derivatives in noisy, high-order, and shock wave scenarios.

Demonstrates robustness of the framework in complex PDE discovery tasks.

Outperforms existing methods in stability and accuracy.

Abstract

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 performances of existing methods lack 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 work. In the framework, a preliminary result of potential terms provided by the deep learning-genetic algorithm is added into the loss function of the PINN as physical constraints to improve the accuracy of derivative calculation. It assists to optimize the preliminary result and obtain 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 is able to calculate derivatives accurately with the optimization of PINN and possesses surprising robustness to complex situations, including sparse data with high noise, high-order derivatives, and shock waves.