Physics informed neural networks solve fast changing problems better
Equation informed time reparametrization for physics informed neural networks
Computational Engineering, Finance, and Science
Summary
Physics informed neural networks (PINNs) help solve equations that change over time, but they struggle with times when things change very quickly. The authors found a way to change the way time is measured during training, zooming into these fast changes. First, they run a rough solution to see where fast changes happen, then they adjust the timeline so the neural network pays more attention there. This method makes the solutions much more accurate when tested on challenging problems.
What this means in practice
- •For engineers simulating material stress: Improve accuracy in simulating material damage over time where sudden changes occur, enabling better predictions near failure points.
- •For chemical process modelers: Enhance modeling of chemical reactions that exhibit rapid changes or blow-up behavior, supporting safer and more reliable design.
Authors
Orestis Belokas, Christoforos Rekatsinas
Abstract
Physics Informed Neural Networks offer a mesh-free approach to solving differential equations, but their application to time-dependent problems remains challenging, particularly when the governing dynamics involve widely different time scales. With collocation points spread uniformly in time, short intervals of rapid change are under resolved. This work introduces an equation informed time reparameterization framework that modifies the temporal coordinate used during PINN training based on the local dynamics of the governing equation. The proposed method follows a two stage procedure. First, a pilot PINN training provides an approximate solution, which is used to linearize the governing differential operator and estimate its local response to small perturbations. These responses are combined into a time-dependent rate function, large where the dynamics are fast, that defines a strictly increasing transformation from physical time to a new computational time coordinate. A second PINN is then trained on the transformed governing equation, with uniformly sampled collocation points in the new coordinate producing a nonuniform distribution in physical time. Different combination rules and spatial reduction strategies are considered. With the cost of the pilot training included, the method reduces the error by a factor of 4 to 29 relative to the best alternative method on the Kachanov Rabotnov creep-damage model near rupture, and by a factor of 5 to 16 relative to every alternative method on the Frank-Kamenetskii equation near blow up.