提出新架构与理论,实现多算子高效学习。
A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory
- 设计MNO和MONet架构,统一建模参数化算子族
- 证明三类算子的通用逼近性,给出网络规模增长规律
- 适合需学习多个函数映射的科学计算场景
机器学习中多数问题关注有限维空间间的映射学习,但科学应用需要对函数空间间的映射(即算子)进行近似。本文研究算子集合的学习问题,区分两种情形:(i) 多算子学习,即单个网络表示由参数函数定义的算子连续体;(ii) 学习若干独立的单一算子,各算子分别学习。针对第一类,提出MNO和MONet两种新架构,并在连续、可积或Lipschitz算子三种设定下建立通用逼近定理。对于第二类,进一步推导显式尺度律,量化达到目标逼近精度所需网络规模的增长关系。针对多个独立算子,提出跨子网络架构复杂度平衡框架,揭示逼近阶数对计算效率的影响。在参数化偏微分方程基准测试上的实验验证了所提架构的强大表达力与高效性。整体工作为多算子神经算子学习提供了统一的理论与实践基础。
原文摘要 · Abstract (English)
While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning collections of operators and provide both theoretical and empirical advances. We distinguish between two regimes: (i) multiple operator learning, where a single network represents a continuum of operators parameterized by a parametric function, and (ii) learning several distinct single operators, where each operator is learned independently. For the multiple operator case, we introduce two new architectures, $\mathrm{MNO}$ and $\mathrm{MONet}$, and establish universal approximation results in three settings: continuous, integrable, or Lipschitz operators. For the latter, we further derive explicit scaling laws that quantify how the network size must grow to achieve a target approximation accuracy. For learning several single operators, we develop a framework for balancing architectural complexity across subnetworks and show how approximation order determines computational efficiency. Empirical experiments on parametric PDE benchmarks confirm the strong expressive power and efficiency of the proposed architectures. Overall, this work establishes a unified theoretical and practical foundation for scalable neural operator learning across multiple operators.
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