Physics-informed machine learning method with space-time Karhunen-Loève expansions for forward and inverse partial differential equations

We propose a physics-informed machine-learning method based on space-time-dependent Karhunen-Loève expansions (KLEs) of the state variables and the residual least-square formulation of the solution of partial differential equations. This method, which we name dPICKLE, results in a reduced-order model for solving forward and inverse time-dependent partial differential equations. By conditioning KLEs on data, dPICKLE seamlessly assimilates data in forward and inverse solutions. KLEs are linear in unknown parameters. Because of this, and unlike physics-informed deep-learning methods based on the residual least-square formulation, for well-posed partial differential equation (PDE) problems, dPICKLE leads to linear least-square problems (directly for linear PDEs and after linearization for nonlinear PDEs), which guarantees a unique solution. The efficiency and accuracy of dPICKLE are demonstrated for linear and nonlinear forward and inverse problems via comparison with analytical, finite difference, and physics-informed neural network (PINN) solutions. • We introduce space-time-dependent conditional Karhunen-Loève expansion (CKLE) for dynamic systems. • We demonstrate the accuracy of the proposed method for linear and non-linear differential equations. • CKLEs improve solutions for both forward and inverse problems.