HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs

2026-05-01

AI Summary

This work addresses the poor interpretability, weak extrapolation capability, and limited transferability of conventional neural operators for solving partial differential equations (PDEs) by proposing an interpretable hybrid PDE solver framework. The approach combines reusable, physically motivated modules—such as advection, diffusion, learned closures, and boundary handling—through procedural composition, with a policy network dynamically scheduling their execution order and duration based on query conditions. By integrating numerical sub-solvers with learnable components, the method enables non-autoregressive, high-accuracy solutions at arbitrary time points. Evaluated across multiple PDE benchmarks, it significantly outperforms existing monolithic neural operators in out-of-domain generalization and supports modular transfer, dictionary updates, and error decomposition for diagnostic purposes, thereby achieving, for the first time, a PDE solver that simultaneously delivers high performance and strong interpretability.

Abstract

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.

Problem

• interpretable learning
• PDE solution operators
• out-of-distribution generalization
• modular transfer
• neural operators

Innovation

• Hybrid composition
• Interpretable PDE learning
• Modular neural operators
• Query-conditioned policy
• Out-of-distribution generalization