  The discovery of partial differential equations (PDEs) from datasets has
attracted increased attention. However, the discovery of governing equations
from sparse data with high noise is still very challenging due to the
difficulty of derivatives computation and the disturbance of noise. Moreover,
the selection principles for the candidate library to meet physical laws need
to be further studied. The invariance is one of the fundamental laws for
governing equations. In this study, we propose an invariance constrained deep
learning network (ICNet) for the discovery of PDEs. Considering that temporal
and spatial translation invariance (Galilean invariance) is a fundamental
property of physical laws, we filter the candidates that cannot meet the
requirement of the Galilean transformations. Subsequently, we embedded the
fixed and possible terms into the loss function of neural network,
significantly countering the effect of sparse data with high noise. Then, by
filtering out redundant terms without fixing learnable parameters during the
training process, the governing equations discovered by the ICNet method can
effectively approximate the real governing equations. We select the 2D Burgers
equation, the equation of 2D channel flow over an obstacle, and the equation of
3D intracranial aneurysm as examples to verify the superiority of the ICNet for
fluid mechanics. Furthermore, we extend similar invariance methods to the
discovery of wave equation (Lorentz Invariance) and verify it through Single
and Coupled Klein-Gordon equation. The results show that the ICNet method with
physical constraints exhibits excellent performance in governing equations
discovery from sparse and noisy data.
