🤖 AI Summary
Data-driven modeling of nonlinear time-varying partial differential equations (PDEs) suffers from prohibitive computational cost in generating training samples—conventional numerical solvers require thousands of time steps, vastly exceeding typical model training iterations.
Method: We propose Homologous Perturbation Sample Synthesis (HOPSS), the first approach to introduce homologous perturbation into PDE data generation. HOPSS leverages a small set of high-fidelity base solutions computed via an accurate solver, then efficiently synthesizes high-quality training samples through temporal downsampling, controlled stochastic noise injection, and inter-solution differencing to construct accurate right-hand-side (RHS) terms.
Results: On the Navier–Stokes equations, HOPSS generates 10,000 training samples using only 10% of the computational time required by conventional solvers, while preserving model accuracy. Its core contribution is a paradigm shift from expensive long-time numerical integration to high-fidelity, low-overhead data augmentation.
📝 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 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.