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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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
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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 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.
AbstractList	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 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. 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. 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. 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.
ArticleNumber	335
Author	Protopapas, Pavlos Jimenez, Raul Tarancón-Álvarez, Pedro Tejerina-Pérez, Pablo
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Snippet	Non-linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well-established method to tackle... Abstract Non-linear differential equations are a fundamental tool to describe different phenomena in nature. However, we still lack a well-established method...
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SubjectTerms	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
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Title	Efficient PINNs via multi-head unimodular regularization of the solutions space
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