Efficient PINNs via Multi-Head Unimodular Regularization of the Solutions Space

TL;DR

This paper introduces a novel multi-head training approach combined with Unimodular Regularization to enhance the efficiency of Physics-Informed Neural Networks in solving complex nonlinear, multiscale differential equations and inverse problems.

Contribution

The paper proposes a new multi-head training framework with Unimodular Regularization to improve PINNs' ability to solve diverse and complex differential equations more efficiently.

Findings

Multi-head training captures a general solution space.

Unimodular Regularization improves transfer learning in PINNs.

Enhanced PINNs effectively solve nonlinear, coupled, and multiscale equations.

Abstract

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.