HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs

TL;DR

HyCOP is a modular framework that learns interpretable PDE solution operators by composing simple modules, enabling hybrid surrogates with improved out-of-distribution performance and transferability.

Contribution

It introduces a query-conditioned, compositional approach to learning PDE operators with interpretability and modular transfer capabilities, supported by theoretical analysis.

Findings

HyCOP achieves order-of-magnitude OOD improvements over monolithic neural operators.

HyCOP produces interpretable programs across diverse PDE benchmarks.

The framework supports modular transfer through dictionary updates.

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.