A Physics‐Informed Method for Dynamic Iterations Applied to Mathematical Models Involving Multiple Scales

ABSTRACTWe develop a new physics‐informed method for multiscale problems by exploiting the separation of scales of the physical phenomena being modeled. In particular, we exploit the difference in the orders of magnitude of physical parameters arising in the model equations and present a new mathematical method that adapts to the mathematical model in question to benefit convergence. The proposed method serves not only as a diagnostic tool but also as a means to construct rapidly convergent dynamic iterations. In developing the method, we, for the first time, formulate and prove principles to identify classes of mathematical models for which such dynamic iterations converge rapidly for systems of arbitrary dimensions. To do so, we investigate the propagation of the errors of the iterations applied to ‐dimensional systems of differential equations arising from the linearization of mathematical models involving a separation of scale, both in closed form and in the form of error bounds. We devise a set of new principles, which offer two key benefits: First, they serve as guidelines for constructing rapidly convergent methods, and second, they aid in identifying classes of problems for which such schemes are advantageous. Our theoretical findings and illustrative examples reveal that convergence is model‐dependent, while offering insight into how to design dynamic iterations to benefit convergence, depending on the order of magnitude of parameters inherent to the linearization of a given mathematical model.