HyCOP用模块化组合学习可解释的偏微分方程求解器
HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs
- 通过条件化模块选择与时长，构建可解释的PDE求解程序
- 在多种测试中实现比传统神经算子高一个数量级的外推性能
- 支持模块替换与迁移，适合需要可解释性的科学计算场景
我们提出HyCOP，一种模块化框架，通过条件化地组合简单模块（如输运、扩散、学习型闭包、边界处理）来学习参数化偏微分方程（PDE）的解算子。不同于学习单一映射，HyCOP学习基于状态特征和统计量的模块应用策略——决定使用哪个模块及其持续时间。模块可为数值子求解器或学习组件，使混合代理能在任意查询时间评估，无需自回归推演。在多个典型PDE基准上，HyCOP生成可解释的程序，相比单体神经算子实现数量级的分布外泛化提升，并支持通过字典更新实现模块化迁移（如边界更换、残差增强）。理论分析揭示了其表达能力，并提供误差分解，分离出组合误差与模块误差，同时作为过程级诊断工具。
原文摘要 · Abstract (English)
We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.
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