  We investigate the inverse problem for Partial Differential Equations (PDEs)
in scenarios where the parameters of the given PDE dynamics may exhibit
changepoints at random time. We employ Physics-Informed Neural Networks (PINNs)
- universal approximators capable of estimating the solution of any physical
law described by a system of PDEs, which serves as a regularization during
neural network training, restricting the space of admissible solutions and
enhancing function approximation accuracy. We demonstrate that when the system
exhibits sudden changes in the PDE dynamics, this regularization is either
insufficient to accurately estimate the true dynamics, or it may result in
model miscalibration and failure. Consequently, we propose a PINNs extension
using a Total-Variation penalty, which allows to accommodate multiple
changepoints in the PDE dynamics and significantly improves function
approximation. These changepoints can occur at random locations over time and
are estimated concurrently with the solutions. Additionally, we introduce an
online learning method for re-weighting loss function terms dynamically.
Through empirical analysis using examples of various equations with parameter
changes, we showcase the advantages of our proposed model. In the absence of
changepoints, the model reverts to the original PINNs model. However, when
changepoints are present, our approach yields superior parameter estimation,
improved model fitting, and reduced training error compared to the original
PINNs model.
