用稀疏方法高效学习微分方程解算子，显著减少求解次数。
Learning solution operators of PDEs with sparse approximation methods
- 结合乘积基与正交匹配追踪，构建稀疏高维近似框架
- 所需方程求解次数比前人方法减少超一半，精度仍保持良好
- 结果可解释性强，能揭示关键变量与参数关系
我们研究了利用稀疏高维技术逼近偏微分方程（PDE）解算子的方法。基于维度增量框架，将乘积基展开与稀疏恢复方法（特别是正交匹配追踪，OMP）结合，显著降低了相对于先前基于积分法的样本需求量。我们在多个实例上进行了数值评估，对比了该方法与基于积分的稀疏逼近及傅里叶神经算子在准确率、运行时间和样本量方面的表现。实验表明，本方法相比前人工作大幅减少了所需的PDE求解次数，同时保持了竞争力的精度，尤其当解在所选基下具有稀疏表示时效果更优。此外，恢复出的稀疏索引集提供了对相关变量和参数交互的可解释洞察。
原文摘要 · Abstract (English)
We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques. Building on a dimension-incremental framework, we combine product basis expansions with sparse recovery methods, specifically orthogonal matching pursuit (OMP), to substantially reduce the required sample size compared with a previously considered cubature-based approach. We evaluate the resulting method numerically on several examples, comparing it against both cubature-based sparse approximation and Fourier neural operators in terms of accuracy, runtime, and sample size. The experiments show that our approach considerably reduces the number of required PDE solves relative to its predecessor while maintaining competitive accuracy, particularly when the solution admits a sparse representation in the chosen basis. Furthermore, the recovered sparse index sets yield interpretable insights into the relevant variables and parameter interactions.
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