# Accelerating Data Generation for Nonlinear temporal PDEs via homologous perturbation in solution space

arXiv: https://arxiv.org/abs/2510.21592 · HTML: https://mlsanity.com/paper/2510.21592/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 30 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 13 of 13)

- [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)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Learning Neural PDE Solvers with Convergence Guarantees](https://mlsanity.com/paper/1906.01200/) (2019)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)
- [Accelerating PDE Data Generation via Differential Operator Action in Solution Space](https://mlsanity.com/paper/2402.05957/) (2024)
- [Strategies for Pretraining Neural Operators](https://mlsanity.com/paper/2406.08473/) (2024)
- [Accelerating Data Generation for Neural Operators via Krylov Subspace Recycling](https://mlsanity.com/paper/2401.09516/) (2024)
- [Pretraining a Neural Operator in Lower Dimensions](https://mlsanity.com/paper/2407.17616/) (2024)

## Built on by (most cited first, 1 of 1)

- [Accelerating Eigenvalue Dataset Generation via Chebyshev Subspace Filter](https://mlsanity.com/paper/2510.23215/) (2025)

## Abstract (arXiv)

Data-driven deep learning methods like neural operators have advanced in solving nonlinear temporal partial differential equations (PDEs). However, these methods require large quantities of solution pairsu2014the solution functions and right-hand sides (RHS) of the equations. These pairs are typically generated via traditional numerical methods, which need thousands of time steps iterations far more than the dozens required for training, creating heavy computational and temporal overheads. To address these challenges, we propose a novel data generation algorithm, called HOmologous Perturbation in Solution Space (HOPSS), which directly generates training datasets with fewer time steps rather than following the traditional approach of generating large time steps datasets. This algorithm simultaneously accelerates dataset generation and preserves the approximate precision required for model training. Specifically, we first obtain a set of base solution functions from a reliable solver, usually with thousands of time steps, and then align them in time steps with training datasets by downsampling. Subsequently, we propose a "homologous perturbation" approach: by combining two solution functions (one as the primary function, the other as a homologous perturbation term scaled by a small scalar) with random noise, we efficiently generate comparable-precision PDE data points. Finally, using these data points, we compute the variation in the original equation's RHS to form new solution pairs. Theoretical and experimental results show HOPSS lowers time complexity. For example, on the Navier-Stokes equation, it generates 10,000 samples in approximately 10% of traditional methods' time, with comparable model training performance.