Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific Computing
Physics-informed neural networks (PINNs) have emerged as a transformative methodology integrating deep learning with scientific computing. This review establishes a three-dimensional analytical framework to systematically decode PINNs’ development through methodological innovation, theoretical break...
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2025-07-01
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| author | Zhiyuan Ren Shijie Zhou Dong Liu Qihe Liu |
| author_facet | Zhiyuan Ren Shijie Zhou Dong Liu Qihe Liu |
| author_sort | Zhiyuan Ren |
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| description | Physics-informed neural networks (PINNs) have emerged as a transformative methodology integrating deep learning with scientific computing. This review establishes a three-dimensional analytical framework to systematically decode PINNs’ development through methodological innovation, theoretical breakthroughs, and cross-disciplinary convergence. The contributions include threefold: First, identifying the co-evolutionary path of algorithmic architectures from adaptive optimization (neural tangent kernel-guided weighting achieving 230% convergence acceleration in Navier-Stokes solutions) to hybrid numerical-deep learning integration (5× speedup via domain decomposition) and second, constructing bidirectional theory-application mappings where convergence analysis (operator approximation theory) and generalization guarantees (Bayesian-physical hybrid frameworks) directly inform engineering implementations, as validated by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>72</mn><mo>%</mo></mrow></semantics></math></inline-formula> cost reduction compared to FEM in high-dimensional spaces (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo><</mo><mn>0.01</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>15</mn></mrow></semantics></math></inline-formula> benchmarks). Third, pioneering cross-domain knowledge transfer through application-specific architectures: TFE-PINN for turbulent flows (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>5.12</mn><mo>±</mo><mn>0.87</mn><mo>%</mo></mrow></semantics></math></inline-formula> error in NASA hypersonic tests), ReconPINN for medical imaging (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>▵</mo><mi>SSIM</mi><mo>=</mo><mo>+</mo><mn>0.18</mn><mo>±</mo><mn>0.04</mn></mrow></semantics></math></inline-formula> on multi-institutional MRI), and SeisPINN for seismic systems (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.52</mn><mo>±</mo><mn>0.18</mn></mrow></semantics></math></inline-formula> km localization accuracy). We further present a technological roadmap highlighting three critical directions for PINN 2.0: neuro-symbolic, federated physics learning, and quantum-accelerated optimization. This work provides methodological guidelines and theoretical foundations for next-generation scientific machine learning systems. |
| format | Article |
| id | doaj-art-53aa81a1b42343df8fb7e2b8f5cc1ab0 |
| institution | DOAJ |
| issn | 2076-3417 |
| language | English |
| publishDate | 2025-07-01 |
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| spelling | doaj-art-53aa81a1b42343df8fb7e2b8f5cc1ab02025-08-20T02:45:33ZengMDPI AGApplied Sciences2076-34172025-07-011514809210.3390/app15148092Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific ComputingZhiyuan Ren0Shijie Zhou1Dong Liu2Qihe Liu3School of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu 610054, ChinaSchool of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu 610054, ChinaSchool of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu 610054, ChinaSchool of Information and Software Engineering, University of Electronic Science and Technology of China, Chengdu 610054, ChinaPhysics-informed neural networks (PINNs) have emerged as a transformative methodology integrating deep learning with scientific computing. This review establishes a three-dimensional analytical framework to systematically decode PINNs’ development through methodological innovation, theoretical breakthroughs, and cross-disciplinary convergence. The contributions include threefold: First, identifying the co-evolutionary path of algorithmic architectures from adaptive optimization (neural tangent kernel-guided weighting achieving 230% convergence acceleration in Navier-Stokes solutions) to hybrid numerical-deep learning integration (5× speedup via domain decomposition) and second, constructing bidirectional theory-application mappings where convergence analysis (operator approximation theory) and generalization guarantees (Bayesian-physical hybrid frameworks) directly inform engineering implementations, as validated by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>72</mn><mo>%</mo></mrow></semantics></math></inline-formula> cost reduction compared to FEM in high-dimensional spaces (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo><</mo><mn>0.01</mn><mo>,</mo><mi>n</mi><mo>=</mo><mn>15</mn></mrow></semantics></math></inline-formula> benchmarks). Third, pioneering cross-domain knowledge transfer through application-specific architectures: TFE-PINN for turbulent flows (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>5.12</mn><mo>±</mo><mn>0.87</mn><mo>%</mo></mrow></semantics></math></inline-formula> error in NASA hypersonic tests), ReconPINN for medical imaging (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>▵</mo><mi>SSIM</mi><mo>=</mo><mo>+</mo><mn>0.18</mn><mo>±</mo><mn>0.04</mn></mrow></semantics></math></inline-formula> on multi-institutional MRI), and SeisPINN for seismic systems (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0.52</mn><mo>±</mo><mn>0.18</mn></mrow></semantics></math></inline-formula> km localization accuracy). We further present a technological roadmap highlighting three critical directions for PINN 2.0: neuro-symbolic, federated physics learning, and quantum-accelerated optimization. This work provides methodological guidelines and theoretical foundations for next-generation scientific machine learning systems.https://www.mdpi.com/2076-3417/15/14/8092partial differential equationphysics-informed neural networkscientific computingAI for scienceartificial intelligence |
| spellingShingle | Zhiyuan Ren Shijie Zhou Dong Liu Qihe Liu Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific Computing Applied Sciences partial differential equation physics-informed neural network scientific computing AI for science artificial intelligence |
| title | Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific Computing |
| title_full | Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific Computing |
| title_fullStr | Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific Computing |
| title_full_unstemmed | Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific Computing |
| title_short | Physics-Informed Neural Networks: A Review of Methodological Evolution, Theoretical Foundations, and Interdisciplinary Frontiers Toward Next-Generation Scientific Computing |
| title_sort | physics informed neural networks a review of methodological evolution theoretical foundations and interdisciplinary frontiers toward next generation scientific computing |
| topic | partial differential equation physics-informed neural network scientific computing AI for science artificial intelligence |
| url | https://www.mdpi.com/2076-3417/15/14/8092 |
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