Improved-Physics informed neural networks for solving Kuramoto Sivashinsky euqation

In this abstract, physics-informed neural networks (PINNs) (M. Raissi, 2019) are introduced and implemented to solve Kuramoto Sivashinsky equations. PINNs employ standard feedforward neural networks (NNs) with the basic physical laws encoded into the NN using automatic differentiation to reveal the dynamic behavior of equations directly from collection data. Specifically, here we use this network structure to produce an accurate K-S model from original data. Our approach relies on training deep neural networks that are extended to encode the K-S equation. Based on the above, a 1-dimensional (1D) time-dependent K-S equation is considered and solved using two mesh-less methods, namely Gaussian process (GP) and improved-PINNs. We observe that these methods are trained with small data and can predict the solution of the equation with high accuracy. Numerical results show that the two algorithms can use fewer sample points to reconstruct the exact solution of the K-S equation with a lower level of error, while PINNs method is more flexible to encode the K-S equation into learning process.