MD-PNOP: Equation-Recast Neural Operators for Minimal-Data Extrapolation and PDE Solver Acceleration

Abstract

The computational overhead of traditional numerical solvers for partial differential equations (PDEs) remains a critical bottleneck for large-scale parametric studies and design optimization. We introduce a Minimal-Data Parametric Neural Operator Preconditioning (MD-PNOP) framework, which establishes a new strategy for accelerating parametric PDE solvers while strictly preserving physical constraints. To address the extrapolation limitation of neural operators, parameter-induced operator difference is recast as additional source terms and incorporated into an iterative solution scheme using a pretrained neural operator. This equation-recast formulation enables systematic parameter extrapolation from a single training configuration to a broad range of unseen parameter settings without retraining. The neural operator predictions are then embedded into iterative PDE solvers as improved initial guesses, thereby reducing convergence iterations without sacrificing accuracy. Unlike purely data-driven approaches, MD-PNOP guarantees that the governing equations remain fully enforced, eliminating concerns regarding loss of physics or interpretability. The framework is architecture-agnostic and is demonstrated using both DeepONet and FNO for Boltzmann transport equation solvers in neutron transport applications. Numerical results demonstrate that neural operators trained on a single set of constant parameters successfully accelerate solutions with heterogeneous, sinusoidal, and discontinuous parameter distributions. Moreover, MD-PNOP consistently achieves approximately 50% reduction in computational time while maintaining full-order fidelity for fixed-source, single-group eigenvalue, and multigroup coupled eigenvalue problems.