Equation Recast for Canonical Operator Learning Across Parametric PDEs

• URL: http://arxiv.org/abs/2609.02982v1
• Date: Wed, 02 Sep 2026 13:15:50 GMT
• Title: Equation Recast for Canonical Operator Learning Across Parametric PDEs
• Abstract summary: We introduce equation recast, which reformulates parametric operator learning as the learning of a single canonical operator.<n>In high-fidelity tokamak simulations for nuclear fusion, the framework unifies electron-temperature data across four device geometries.
• Score: 59.44573033543284
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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