提出高效学习椭圆伪微分算子的方法并证明收敛速度。
Convergence Rates for Learning Pseudo-Differential Operators
- 基于小波-伽辽金框架，将算子学习建模为多尺度稀疏回归。
- 设计稀疏估计器，实现计算与数据双重高效，收敛率可证。
- 所学算子可直接用于稳定高效的数值求解，统计误差即数值误差。
本文建立了学习椭圆伪微分算子的收敛速率，这类算子在偏微分方程和数学物理中具有基础地位。在小波-伽辽金框架下，我们将该类算子的学习问题建模为具有多尺度稀疏性的无限维结构化回归问题。基于此结构，提出一种稀疏、数据与计算高效的估计器，其利用针对学习任务定制的新矩阵压缩方案，并采用嵌套支撑策略平衡近似误差与估计误差。除了获得估计器的收敛速率外，还证明所学算子可诱导出高效且稳定的伽辽金求解器，其数值误差与统计精度一致。研究结果推动了算子学习、数据驱动求解器与小波方法在科学计算中的融合。
原文摘要 · Abstract (English)
This paper establishes convergence rates for learning elliptic pseudo-differential operators, a fundamental operator class in partial differential equations and mathematical physics. In a wavelet-Galerkin framework, we formulate learning over this class as a structured infinite-dimensional regression problem with multiscale sparsity. Building on this structure, we propose a sparse, data- and computation-efficient estimator, which leverages a novel matrix compression scheme tailored to the learning task and a nested-support strategy to balance approximation and estimation errors. In addition to obtaining convergence rates for the estimator, we show that the learned operator induces an efficient and stable Galerkin solver whose numerical error matches its statistical accuracy. Our results therefore contribute to bringing together operator learning, data-driven solvers, and wavelet methods in scientific computing.
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