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

arXiv: https://arxiv.org/abs/2509.01416 · HTML: https://mlsanity.com/paper/2509.01416/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 49 references that are in the map.
- First posted on arXiv 1 Sep 2025.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [Plasma Surrogate Modelling using Fourier Neural Operators](https://mlsanity.com/paper/2311.05967/) (2023)
- [Graph Neural Preconditioners for Iterative Solutions of Sparse Linear Systems](https://mlsanity.com/paper/2406.00809/) (2024)
- [Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagation](https://mlsanity.com/paper/2304.10242/) (2023)
- [Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers](https://mlsanity.com/paper/2306.17486/) (2023)
- [MetaNO: How to Transfer Your Knowledge on Learning Hidden Physics](https://mlsanity.com/paper/2301.12095/) (2023)
- [Neural Preconditioning Operator for Efficient PDE Solves](https://mlsanity.com/paper/2502.01337/) (2025)

## Built on by (most cited first, 2 of 2)

- [Surrogate Modeling for Neutron Transport: A Neural Operator Approach](https://mlsanity.com/paper/2602.15890/) (2026)
- [Learning features from Newton's algorithm: a way to accelerate nonlinear parametrized PDE solvers](https://mlsanity.com/paper/2607.28036/) (2026)

## Abstract (arXiv)

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.