Towards a Foundation Model for Partial Differential Equations Across Physics Domains
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915640293457920 |
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| author | Soares, Eduardo Brazil, Emilio Vital Shirasuna, Victor de Carvalho, Breno W. S. R. Malossi, Cristiano |
| author_facet | Soares, Eduardo Brazil, Emilio Vital Shirasuna, Victor de Carvalho, Breno W. S. R. Malossi, Cristiano |
| contents | We present PDE-FM, a modular foundation model for physics-informed machine learning that unifies spatial, spectral, and temporal reasoning across heterogeneous partial differential equation (PDE) systems. PDE-FM combines spatial-spectral tokenization, physics-aware conditioning, and a Mamba-based state-space backbone with an operator-theoretic decoder, enabling scalable and data-efficient modeling of complex physical dynamics. In contrast to task-specific neural operators, PDE-FM is pretrained once on diverse PDE datasets and can be transferred to new physical regimes without architectural or data-specific modifications. Evaluated on twelve 2D and 3D datasets from The Well benchmark - spanning hydrodynamic, radiative, elastic, and astrophysical phenomena - PDE-FM achieves state-of-the-art accuracy in six domains, reducing mean VRMSE by 46% relative to prior operator-learning baselines. The model demonstrates robust cross-physics generalization, excelling in turbulent and radiative systems while maintaining strong performance in linear and steady-state regimes. These results suggest that large-scale pretraining across diverse physical processes can yield transferable representations of dynamics, marking a step toward unified, foundation-level surrogates for multi-physics simulation and scientific discovery. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_21861 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Towards a Foundation Model for Partial Differential Equations Across Physics Domains Soares, Eduardo Brazil, Emilio Vital Shirasuna, Victor de Carvalho, Breno W. S. R. Malossi, Cristiano Machine Learning Artificial Intelligence We present PDE-FM, a modular foundation model for physics-informed machine learning that unifies spatial, spectral, and temporal reasoning across heterogeneous partial differential equation (PDE) systems. PDE-FM combines spatial-spectral tokenization, physics-aware conditioning, and a Mamba-based state-space backbone with an operator-theoretic decoder, enabling scalable and data-efficient modeling of complex physical dynamics. In contrast to task-specific neural operators, PDE-FM is pretrained once on diverse PDE datasets and can be transferred to new physical regimes without architectural or data-specific modifications. Evaluated on twelve 2D and 3D datasets from The Well benchmark - spanning hydrodynamic, radiative, elastic, and astrophysical phenomena - PDE-FM achieves state-of-the-art accuracy in six domains, reducing mean VRMSE by 46% relative to prior operator-learning baselines. The model demonstrates robust cross-physics generalization, excelling in turbulent and radiative systems while maintaining strong performance in linear and steady-state regimes. These results suggest that large-scale pretraining across diverse physical processes can yield transferable representations of dynamics, marking a step toward unified, foundation-level surrogates for multi-physics simulation and scientific discovery. |
| title | Towards a Foundation Model for Partial Differential Equations Across Physics Domains |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2511.21861 |