Accelerating Data Generation for Nonlinear temporal PDEs via homologous perturbation in solution space
This addresses a bottleneck in data generation for PDE-solving deep learning methods, offering a more efficient approach for researchers and practitioners in computational science and engineering, though it is incremental as it optimizes an existing process rather than introducing a new paradigm.
The paper tackles the high computational cost of generating training data for neural operators solving nonlinear temporal PDEs by proposing HOPSS, a data generation algorithm that reduces time steps needed, achieving a 90% reduction in generation time for 10,000 samples on the Navier-Stokes equation while maintaining comparable training performance.
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.