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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2603.27929 |
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| _version_ | 1866908920216289280 |
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| author | Zeraatkar, Ehsan Podorozhny, Rodion Tešić, Jelena |
| author_facet | Zeraatkar, Ehsan Podorozhny, Rodion Tešić, Jelena |
| contents | Reconstructing continuous physical fields from sparse, irregular observations is a central challenge in scientific machine learning, particularly for systems governed by partial differential equations (PDEs). Existing physics-informed methods typically enforce governing equations as soft penalty terms during optimization, often leading to gradient imbalance, instability, and degraded physical consistency under limited data. We introduce the Physics-Guided Transformer (PGT), a neural architecture that embeds physical structure directly into the self-attention mechanism. Specifically, PGT incorporates a heat-kernel-derived additive bias into attention logits, encoding diffusion dynamics and temporal causality within the representation. Query coordinates attend to these physics-conditioned context tokens, and the resulting features are decoded using a FiLM-modulated sinusoidal implicit network that adaptively controls spectral response. We evaluate PGT on the one-dimensional heat equation and two-dimensional incompressible Navier-Stokes systems. In sparse 1D reconstruction with 100 observations, PGT achieves a relative L2 error of 5.9e-3, significantly outperforming both PINNs and sinusoidal representations. In the 2D cylinder wake problem, PGT uniquely achieves both low PDE residual (8.3e-4) and competitive relative error (0.034), outperforming methods that optimize only one objective. These results demonstrate that embedding physics within attention improves stability, generalization, and physical fidelity under data-scarce conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27929 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Physics-Guided Transformer (PGT): Physics-Aware Attention Mechanism for PINNs Zeraatkar, Ehsan Podorozhny, Rodion Tešić, Jelena Machine Learning Artificial Intelligence Reconstructing continuous physical fields from sparse, irregular observations is a central challenge in scientific machine learning, particularly for systems governed by partial differential equations (PDEs). Existing physics-informed methods typically enforce governing equations as soft penalty terms during optimization, often leading to gradient imbalance, instability, and degraded physical consistency under limited data. We introduce the Physics-Guided Transformer (PGT), a neural architecture that embeds physical structure directly into the self-attention mechanism. Specifically, PGT incorporates a heat-kernel-derived additive bias into attention logits, encoding diffusion dynamics and temporal causality within the representation. Query coordinates attend to these physics-conditioned context tokens, and the resulting features are decoded using a FiLM-modulated sinusoidal implicit network that adaptively controls spectral response. We evaluate PGT on the one-dimensional heat equation and two-dimensional incompressible Navier-Stokes systems. In sparse 1D reconstruction with 100 observations, PGT achieves a relative L2 error of 5.9e-3, significantly outperforming both PINNs and sinusoidal representations. In the 2D cylinder wake problem, PGT uniquely achieves both low PDE residual (8.3e-4) and competitive relative error (0.034), outperforming methods that optimize only one objective. These results demonstrate that embedding physics within attention improves stability, generalization, and physical fidelity under data-scarce conditions. |
| title | Physics-Guided Transformer (PGT): Physics-Aware Attention Mechanism for PINNs |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2603.27929 |