Equation Recast for Canonical Operator Learning Across Parametric PDEs

arXiv:2609.02982v1 Announce Type: new 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.

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.

Machine Learning (cs.LG); Computational Physics (physics.comp-ph); Plasma Physics (physics.plasm-ph)

arXiv:2609.02982 [cs.LG]

(or arXiv:2609.02982v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.02982 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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Equation Recast for Canonical Operator Learning Across Parametric PDEs

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