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.

翻译：我们提出了HyCOP，一个通过以查询条件化的方式组合简单模块（对流、扩散、学习型闭合、边界处理）来学习参数化偏微分方程解算子的模块化框架。HyCOP并非学习一个整体映射，而是学习一个关于短程序（即应用哪个模块及其持续时间）的策略，该策略以区域特征和状态统计量为条件。模块可以是数值子求解器或学习型组件，从而能够在任意查询时刻无需自回归滚动即可评估混合代理模型。在多个偏微分方程基准测试中，HyCOP生成可解释的程序，相比整体神经算子实现了数量级的分布外性能提升，并通过字典更新（例如边界交换、残差增广）支持模块化迁移。我们的理论刻画了表达能力，并给出了一个误差分解，该分解将组合误差与模块误差分离，同时作为过程级诊断工具。