Abstract:Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim to provide universal surrogates, typical encoding-decoding approaches suffer from high pretraining costs and limited interpretability. In this paper, we propose Compositional Neural Operators (CompNO) for 2D systems, a framework that decomposes complex PDEs into a library of Foundation Blocks. Each block is a specialized Neural Operator pretrained on elementary physics. This modular library contains convection, diffusion, and nonlinear convection blocks as well as a Poisson Solver, enabling the framework to address the pressure-velocity coupling. These experts are assembled via an Adaptation Block featuring an Aggregator. This aggregator learns nonlinear interactions by minimizing data loss and physics-based residuals driven from governing equations. The proposed approach has been evaluated on the Convection-Diffusion equation, the Burgers' equation, and the Incompressible Navier-Stokes equation. Our results demonstrate that learning from elementary operators significantly improves adaptability, enhances model interpretability and facilitates the reuse of pretrained blocks when adapting to new physical systems.
| Comments: | Published as a conference paper at ICLR 2026 |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.11691 [cs.LG] |
| (or arXiv:2605.11691v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.11691 arXiv-issued DOI via DataCite (pending registration) |
From: Hamda Hmida [view email]
[v1]
Tue, 12 May 2026 07:48:03 UTC (148 KB)
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