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

TL;DR

The paper tackles the data-generation bottleneck for nonlinear temporal PDEs used by neural operators by introducing HOPSS, a data-generation pipeline that acts on training-time time steps rather than thousands of iterations. HOPSS consists of three steps: base solution generation, homologous perturbation in the solution space, and RHS variation computation to produce new solution–RHS pairs, with the key update $u_{new} = u_i + \mu u_j + \xi$ and $f_{new} = f_i + \Delta f$. Theoretical analysis shows a cost reduction to approximately $O(N_b T n^2)$ for the data-generation phase, and experiments on Navier–Stokes, Burgers, and KdV demonstrate up to $10\times$ faster data generation with comparable predictive performance to conventional methods. These results indicate that HOPSS can enable scalable, data-rich training for neural operators in nonlinear temporal PDE settings, potentially broadening industrial applicability. Future work includes optimizing base-solution selection, active learning for coverage, and physics-informed metrics to quantify dataset quality beyond model performance.

Abstract

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 pairs\u2014the 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.

Figures (1)

• Figure 1: Overview of the HOPSS method. First, generating a series of solutions using the existing method and downsampling as base solutions. Then, randomly select two base solutions and apply the homologous perturbation to generate new solutions. Finally, calculating the RHS using the new solutions.