Efficient PINNs via multi-head unimodular regularization of the solutions space

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 eq...

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Bibliographic Details
Published in:Communications physics Vol. 8; no. 1; pp. 335 - 14
Main Authors: Tarancón-Álvarez, Pedro, Tejerina-Pérez, Pablo, Jimenez, Raul, Protopapas, Pavlos
Format: Journal Article
Language:English
Published: London Nature Publishing Group UK 15.08.2025
Nature Publishing Group
Nature Portfolio
Subjects:
639/705/1042
639/766/259
Accuracy
Algorithms
Boundary conditions
Einstein equations
Geometry
Inverse problems
Linear equations
Machine learning
Neural networks
Nonlinear differential equations
Numerical analysis
Ordinary differential equations
Physics
Physics and Astronomy
Regularization
ISSN:2399-3650, 2399-3650
Online Access:Get full text
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Summary: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 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. Physics-Informed Neural Networks (PINNs) face challenges in generalizing solutions for nonlinear multiscale differential equations and inverse problems. Here, the authors employ a framework of multihead training with unimodular regularization, to enhance PINN efficiency and enable effective transfer learning in case of systems, including the flame equation, van der Pol oscillator, and Einstein Field Equations.
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ISSN:2399-3650
2399-3650
DOI:10.1038/s42005-025-02248-1