A Framework Based on Symbolic Regression Coupled with eXtended
  Physics-Informed Neural Networks for Gray-Box Learning of Equations of Motion
  from Data
Elham Kiyani Khemraj Shukla George Em Karniadakis Mikko Karttunen
We propose a framework and an algorithm to uncover the unknown parts of
nonlinear equations directly from data. The framework is based on eXtended
Physics-Informed Neural Networks (X-PINNs), domain decomposition in space-time,
but we augment the original X-PINN method by imposing flux continuity across
the domain interfaces. The well-known Allen-Cahn equation is used to
demonstrate the approach. The Frobenius matrix norm is used to evaluate the
accuracy of the X-PINN predictions and the results show excellent performance.
In addition, symbolic regression is employed to determine the closed form of
the unknown part of the equation from the data, and the results confirm the
accuracy of the X-PINNs based approach. To test the framework in a situation
resembling real-world data, random noise is added to the datasets to mimic
scenarios such as the presence of thermal noise or instrument errors. The
results show that the framework is stable against significant amount of noise.
As the final part, we determine the minimal amount of data required for
training the neural network. The framework is able to predict the correct form
and coefficients of the underlying dynamical equation when at least 50\% data
is used for training.