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Physics Informed Learning of Orthogonal Features With Applications in Solving Partial Differential Equations.

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Title: Physics Informed Learning of Orthogonal Features With Applications in Solving Partial Differential Equations.
Authors: Jia, Qianxing1 (AUTHOR), Wang, Dong1,2,3 (AUTHOR) wangdong@cuhk.edu.cn
Source: Numerical Methods for Partial Differential Equations. Sep2026, Vol. 42 Issue 5, p1-25. 25p.
Subjects: Partial differential equations, Orthogonalization, Helmholtz, Hermann von, 1821-1894, Orthogonal functions
Abstract: The random feature method (RFM) constructs approximation spaces by initializing features from generic distributions, which provides universal approximation properties to solve general partial differential equations. However, such standard initializations lack awareness of the underlying physical laws and geometry, which limits approximation. In this work, we propose the Physics‐Driven Orthogonal Feature Method (PD‐OFM), a framework for constructing feature representations that are explicitly tailored to both the differential operator and the computational domain by pretraining features using physics‐informed objectives together with orthogonality regularization. This pretraining strategy yields nearly orthogonal feature bases. We provide both theoretical and empirical evidence that physics‐informed pretraining improves the approximation capability of the learned feature space. When employed to solve Helmholtz, Poisson, wave, and Navier–Stokes equations, the proposed method achieves residual errors 2–3 orders of magnitude lower than those of comparable methods. Furthermore, the orthogonality regularization improves transferability, enabling pretrained features to generalize effectively across different source terms and domain geometries for the same PDE. [ABSTRACT FROM AUTHOR]
Copyright of Numerical Methods for Partial Differential Equations is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
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DbLabel: Engineering Source
An: 197428257
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  Data: Physics Informed Learning of Orthogonal Features With Applications in Solving Partial Differential Equations.
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  Data: <searchLink fieldCode="JN" term="%22Numerical+Methods+for+Partial+Differential+Equations%22">Numerical Methods for Partial Differential Equations</searchLink>. Sep2026, Vol. 42 Issue 5, p1-25. 25p.
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  Data: <searchLink fieldCode="DE" term="%22Partial+differential+equations%22">Partial differential equations</searchLink><br /><searchLink fieldCode="DE" term="%22Orthogonalization%22">Orthogonalization</searchLink><br /><searchLink fieldCode="DE" term="%22Helmholtz%2C+Hermann+von%2C+1821-1894%22">Helmholtz, Hermann von, 1821-1894</searchLink><br /><searchLink fieldCode="DE" term="%22Orthogonal+functions%22">Orthogonal functions</searchLink>
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  Data: The random feature method (RFM) constructs approximation spaces by initializing features from generic distributions, which provides universal approximation properties to solve general partial differential equations. However, such standard initializations lack awareness of the underlying physical laws and geometry, which limits approximation. In this work, we propose the Physics‐Driven Orthogonal Feature Method (PD‐OFM), a framework for constructing feature representations that are explicitly tailored to both the differential operator and the computational domain by pretraining features using physics‐informed objectives together with orthogonality regularization. This pretraining strategy yields nearly orthogonal feature bases. We provide both theoretical and empirical evidence that physics‐informed pretraining improves the approximation capability of the learned feature space. When employed to solve Helmholtz, Poisson, wave, and Navier–Stokes equations, the proposed method achieves residual errors 2–3 orders of magnitude lower than those of comparable methods. Furthermore, the orthogonality regularization improves transferability, enabling pretrained features to generalize effectively across different source terms and domain geometries for the same PDE. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Data: <i>Copyright of Numerical Methods for Partial Differential Equations is the property of Wiley-Blackwell and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1002/num.70142
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        Text: English
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        PageCount: 25
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      – SubjectFull: Partial differential equations
        Type: general
      – SubjectFull: Orthogonalization
        Type: general
      – SubjectFull: Helmholtz, Hermann von, 1821-1894
        Type: general
      – SubjectFull: Orthogonal functions
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      – TitleFull: Physics Informed Learning of Orthogonal Features With Applications in Solving Partial Differential Equations.
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              M: 09
              Text: Sep2026
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              Y: 2026
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              Value: 42
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