Towards a Foundation Model for Partial Differential Equations Across Physics Domains

• URL: http://arxiv.org/abs/2511.21861v1
• Date: Wed, 26 Nov 2025 19:36:15 GMT
• Title: Towards a Foundation Model for Partial Differential Equations Across Physics Domains
• Authors: Eduardo Soares, Emilio Vital Brazil, Victor Shirasuna, Breno W. S. R. de Carvalho, Cristiano Malossi,
• Abstract summary: We present PDE-FM, a modular foundation model for physics-informed machine learning.<n>It unifies spatial, spectral, and temporal reasoning across heterogeneous partial differential equation (PDE) systems.<n>PDE-FM is pretrained once on diverse PDE datasets and can be transferred to new physical regimes without architectural or data-specific modifications.
• Score: 1.7115425267046014
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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