Learning neural operators for solving partial differential equations (PDEs)
has attracted great attention due to its high inference efficiency. However,
training such operators requires generating a substantial amount of labeled
data, i.e., PDE problems together with their solutions. The data generation
process is exceptionally time-consuming, as it involves solving numerous
systems of linear equations to obtain numerical solutions to the PDEs. Many
existing methods solve these systems independently without considering their
inherent similarities, resulting in extremely redundant computations. To tackle
this problem, we propose a novel method, namely Sorting Krylov Recycling (SKR),
to boost the efficiency of solving these systems, thus significantly
accelerating data generation for neural operators training. To the best of our
knowledge, SKR is the first attempt to address the time-consuming nature of
data generation for learning neural operators. The working horse of SKR is
Krylov subspace recycling, a powerful technique for solving a series of
interrelated systems by leveraging their inherent similarities. Specifically,
SKR employs a sorting algorithm to arrange these systems in a sequence, where
adjacent systems exhibit high similarities. Then it equips a solver with Krylov
subspace recycling to solve the systems sequentially instead of independently,
thus effectively enhancing the solving efficiency. Both theoretical analysis
and extensive experiments demonstrate that SKR can significantly accelerate
neural operator data generation, achieving a remarkable speedup of up to 13.9
times.
Learning neural operators for solving partial differential equations (PDEs) is a highly efficient and effective approach. However, the process of training such operators involves generating a large amount of labeled data, which can be time-consuming. This is because it requires solving multiple systems of linear equations to obtain numerical solutions to the PDEs. Many existing methods solve these systems independently, leading to redundant computations and inefficiencies.
To address this issue, the authors propose a novel method called Sorting Krylov Recycling (SKR) that significantly improves the efficiency of solving these systems and accelerates data generation for neural operator training. SKR is the first attempt to address the time-consuming nature of data generation in learning neural operators. Its key technique is Krylov subspace recycling, which leverages the inherent similarities among interrelated systems to solve them more efficiently.
SKR employs a sorting algorithm to arrange the systems in a sequence, with adjacent systems exhibiting high similarities. By equipping the solver with Krylov subspace recycling, SKR solves the systems sequentially instead of independently, leading to a significant enhancement in solving efficiency. The authors provide both theoretical analysis and extensive experiments to demonstrate the effectiveness of SKR, achieving a remarkable speedup of up to 13.9 times in neural operator data generation.
This work highlights the multi-disciplinary nature of the concepts involved. It combines techniques from numerical algorithms, linear algebra, and machine learning to develop an innovative solution. The application of sorting algorithms and Krylov subspace recycling brings together ideas from computer science and mathematics to tackle the challenging problem of generating labeled data for neural operators training.
The significance of this research lies in its potential impact on accelerating the development and application of neural operators for solving PDEs. By reducing the time and resources required for data generation, researchers and engineers can more efficiently train neural networks to solve complex PDE problems. This, in turn, can lead to advancements in various fields, such as physics, engineering, and computational modeling.
Looking forward, this work opens doors for further exploration and improvements in data generation for learning neural operators. The SKR method sets a foundation for future research to develop more advanced techniques that exploit the inherent structure and similarities in PDE systems. Additionally, this work encourages collaborations between researchers in different disciplines to continue pushing the boundaries of efficient neural operator training.