Robust discovery of partial differential equations in complex situations. (arXiv:2106.00008v1 [cs.LG])

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.

Source: https://arxiv.org/abs/2106.00008

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