# Accelerating PDE Data Generation via Differential Operator Action in Solution Space

arXiv: https://arxiv.org/abs/2402.05957 · HTML: https://mlsanity.com/paper/2402.05957/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 4 Feb 2024.

## Builds on (most cited first, 11 of 11)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservation](https://mlsanity.com/paper/2212.14365/) (2022)
- [A Deep Conjugate Direction Method for Iteratively Solving Linear Systems](https://mlsanity.com/paper/2205.10763/) (2022)
- [Accelerating Data Generation for Neural Operators via Krylov Subspace Recycling](https://mlsanity.com/paper/2401.09516/) (2024)

## Built on by (most cited first, 15 of 15)

- [Apollo-MILP: An Alternating Prediction-Correction Neural Solving Framework for Mixed-Integer Linear Programming](https://mlsanity.com/paper/2503.01129/) (2025)
- [Mixture-of-Experts Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2510.25803/) (2025)
- [Distinguished Quantized Guidance for Diffusion-based Sequence Recommendation](https://mlsanity.com/paper/2501.17670/) (2025)
- [SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic Preconditioning](https://mlsanity.com/paper/2510.24170/) (2025)
- [CoCo-MILP: Inter-Variable Contrastive and Intra-Constraint Competitive MILP Solution Prediction](https://mlsanity.com/paper/2511.09209/) (2025)
- [On Efficiency-Effectiveness Trade-off of Diffusion-based Recommenders](https://mlsanity.com/paper/2510.17245/) (2025)
- [STNet: Spectral Transformation Network for Solving Operator Eigenvalue Problem](https://mlsanity.com/paper/2510.23986/) (2025)
- [RoME: Domain-Robust Mixture-of-Experts for MILP Solution Prediction across Domains](https://mlsanity.com/paper/2511.02331/) (2025)
- [Accelerating Data Generation for Nonlinear temporal PDEs via homologous perturbation in solution space](https://mlsanity.com/paper/2510.21592/) (2025)
- [Accelerating Eigenvalue Dataset Generation via Chebyshev Subspace Filter](https://mlsanity.com/paper/2510.23215/) (2025)
- [Learning Neural Operators from Partial Observations via Latent Autoregressive Modeling](https://mlsanity.com/paper/2601.15547/) (2026)
- [Accelerating IC Thermal Simulation Data Generation via Block Krylov and Operator Action](https://mlsanity.com/paper/2510.23221/) (2025)
- [Learning-Guided Integration Contours Construction for Fast Large-Scale Generalized Eigensolvers](https://mlsanity.com/paper/2511.01927/) (2025)
- [HGATSolver: A Heterogeneous Graph Attention Solver for Fluid-Structure Interaction](https://mlsanity.com/paper/2601.09251/) (2026)
- [MGDiff: Multi-Interest Sequence Recommendation with Masking GNN-Guided Diffusion](https://mlsanity.com/paper/2609.01619/) (2026)

## Abstract (arXiv)

Recent advancements in data-driven approaches, such as Neural Operator (NO), have demonstrated their effectiveness in reducing the solving time of Partial Differential Equations (PDEs). However, one major challenge faced by these approaches is the requirement for a large amount of high-precision training data, which needs significant computational costs during the generation process. To address this challenge, we propose a novel PDE dataset generation algorithm, namely Differential Operator Action in Solution space (DiffOAS), which speeds up the data generation process and enhances the precision of the generated data simultaneously. Specifically, DiffOAS obtains a few basic PDE solutions and then combines them to get solutions. It applies differential operators on these solutions, a process we call 'operator action', to efficiently generate precise PDE data points. Theoretical analysis shows that the time complexity of DiffOAS method is one order lower than the existing generation method. Experimental results show that DiffOAS accelerates the generation of large-scale datasets with 10,000 instances by 300 times. Even with just 5% of the generation time, NO trained on the data generated by DiffOAS exhibits comparable performance to that using the existing generation method, which highlights the efficiency of DiffOAS.