Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space
This addresses the problem of efficiently solving complex differential equations for researchers in computational science and engineering, representing an incremental improvement over existing PINN methods.
The paper tackled the challenge of solving stiff nonlinear multiscale differential equations and inverse problems using Physics-Informed Neural Networks (PINNs), resulting in a framework that combines multi-head training and unimodular regularization to significantly improve efficiency by facilitating transfer learning.
Non-linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well-established method to tackle stiff differential equations. Here we present a machine learning framework to facilitate the solution of nonlinear multiscale differential equations and, especially, inverse problems using Physics-Informed Neural Networks (PINNs). This framework is based on what is called \textit{multi-head} (MH) training, which involves training the network to learn a general space of all solutions for a given set of equations with certain variability, rather than learning a specific solution of the system. This setup is used with a second novel technique that we call Unimodular Regularization (UR) of the latent space of solutions. We show that the multi-head approach, combined with Unimodular Regularization, significantly improves the efficiency of PINNs by facilitating the transfer learning process thereby enabling the finding of solutions for nonlinear, coupled, and multiscale differential equations.