Equation Recast for Canonical Operator Learning Across Parametric PDEs
Abstract
Learning solution operators across broad parameter ranges can require substantial coverage of both input functions and physical parameters, particularly for purely data-driven parametric models. In addition, the resulting models may fail silently outside the training distribution. We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator. Parameter-induced operator variations are derived analytically from the governing equation and absorbed into effective sources, enabling zero-shot prediction across new parameter regimes. Across multi-parameter, nonlinear, and singular PDE settings, equation recast supports extrapolation, integrates sparse heterogeneous datasets in a shared canonical representation, and uses loss of convergence as an internal warning signal for failure of the recast iteration. In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries through canonical-domain mapping within one jointly trained operator. Equation recast provides a route toward reusable neural PDE solvers combining equation-guided transfer, data efficiency, and monitorable inference.
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BibTeX
@misc{cheng2026equation,
title = {{Equation Recast for Canonical Operator Learning Across Parametric PDEs}},
author = {Qiyun Cheng and Valentin Duruisseaux and Cesar F. Clauser and Md Hossain Sahadath and Huihua Yang and Shaowu Pan and Nathaniel Ferraro and Anima Anandkumar and Wei Ji and Cristina Rea},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02982},
eprint = {2609.02982},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02982},
}
RIS
TY - UNPB
TI - Equation Recast for Canonical Operator Learning Across Parametric PDEs
AU - Qiyun Cheng
AU - Valentin Duruisseaux
AU - Cesar F. Clauser
AU - Md Hossain Sahadath
AU - Huihua Yang
AU - Shaowu Pan
AU - Nathaniel Ferraro
AU - Anima Anandkumar
AU - Wei Ji
AU - Cristina Rea
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02982
UR - https://arxiv.org/abs/2609.02982
AB - Learning solution operators across broad parameter ranges can require substantial coverage of both input functions and physical parameters, particularly for purely data-driven parametric models. In addition, the resulting models may fail silently outside the training distribution. We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator. Parameter-induced operator variations are derived analytically from the governing equation and absorbed into effective sources, enabling zero-shot prediction across new parameter regimes. Across multi-parameter, nonlinear, and singular PDE settings, equation recast supports extrapolation, integrates sparse heterogeneous datasets in a shared canonical representation, and uses loss of convergence as an internal warning signal for failure of the recast iteration. In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries through canonical-domain mapping within one jointly trained operator. Equation recast provides a route toward reusable neural PDE solvers combining equation-guided transfer, data efficiency, and monitorable inference.
ER -