Accelerating PDE Data Generation via Differential Operator Action in Solution Space

TL;DR

This paper tackles the data-generation bottleneck in data-driven PDE solvers by introducing DiffOAS, a two-stage method that replaces costly linear-system solves with operator actions on a basis of PDE solution functions. By generating physically meaningful basis functions and applying the discretized differential operator, DiffOAS achieves substantial data-generation speedups (up to about $300\times$ in large-scale tests) while preserving machine-precision data ($\sim 10^{-16}$). Theoretical analysis shows that the overall complexity scales as $O(n^2 N + m n^2 l + m^2 n l)$, yielding a near $O(n)$ improvement over GMRES-based generation, and experiments demonstrate that neural operators trained on DiffOAS data perform comparably to those trained on conventional data, even with only a fraction of the generation time. This approach significantly enhances the practicality and scalability of data-driven PDE solvers across applications such as Darcy flow, Helmholtz-type problems, and solute diffusion, by enabling rapid, high-fidelity data generation without loss of accuracy.

Abstract

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.

Figures (2)

• Figure 1: Above. Generation process of the PDE dataset. 1. Produce collection of random parameters derived from PDE. 2. Generate the relevant PDE using these parameters 3. Convert the PDE into linear equation systems using discretization methods. 4. Solve linear equations based on input parameters 5. Acquire solutions for the linear systems and translate them into solutions for the PDEs. 6. Compile the data into a dataset. Below. The generation cost of DiffOAS and GMRES varies with the dimension of the linear system. GMRES curves in the graph represent different truncation errors, where GMRES $10^{-5}$ indicates the algorithm's time cost with a truncation error of $10^{-5}$. In contrast, our DiffOAS maintains machine precision of $10^{-16}$. It can be observed that DiffOAS significantly speeds up the data generation process, achieving a speedup of up to $70,000$ times.
• Figure 2: Overview of the model architecture. The process of DiffOAS and existing method: Firstly, Transforming a PDE into a linear equation system. Secondly, (Above) Existing Method generates $f$, input its discretized form $\bm{b}$ into the linear equation system, and solve $\bm{x}$ accordingly. (Below) DiffOAS generates $u$ using basis functions, inputs its discretized form $\bm{x}$ into the linear equation system, and calculates $\bm{b}$ accordingly. Finally organizing the inputs and outputs of the linear equation system to create a complete PDE dataset.