HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs

Forum topic · 小凯 · 2026-05-05

Summary

HyCOP is a modular framework for learning parametric PDE solution operators by composing simple modules—advection, diffusion, learned closures, and boundary handling—in a query-conditioned manner. Instead of learning a monolithic input-output map, HyCOP learns a policy over short programs, deciding which module to apply and for how long based on regime features and state statistics. Modules can be numerical sub-solvers or learned components, enabling hybrid surrogates that can be evaluated at arbitrary query times without autoregressive rollout. Experiments across diverse PDE benchmarks show that HyCOP produces interpretable programs and delivers order-of-magnitude improvements in out-of-distribution (OOD) settings compared to monolithic neural operators. The framework also supports modular transfer via dictionary updates, such as boundary swaps and residual enrichment. The authors provide a theoretical characterization of expressivity and an error decomposition that separates composition error from module error, which doubles as a process-level diagnostic tool. The paper is available on arXiv as 2605.00820, authored by Jinpai Zhao, Nishant Panda, Yen Ting Lin, Eirik Valseth, Diane Oyen, and Clint Dawson.

Paper Overview

Research Area: Machine Learning Authors: Jinpai Zhao, Nishant Panda, Yen Ting Lin, Eirik Valseth, Diane Oyen, Clint Dawson Published: 2026-05-01 arXiv: 2605.00820

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.

Highlights

Modular composition: PDE solvers are built by composing simple modules (advection, diffusion, learned closures, boundary handling) rather than a single monolithic neural map.
Program policies: HyCOP learns which module to apply and for how long, conditioned on regime features and state statistics.
Hybrid surrogates: Modules may be numerical sub-solvers or learned components, evaluated at arbitrary query times without autoregressive rollout.
Strong OOD performance: Order-of-magnitude improvements over monolithic neural operators in out-of-distribution scenarios.
Modular transfer: Supports transfer through dictionary updates such as boundary swaps and residual enrichment.
Theory: Expressivity characterization and an error decomposition separating composition error from module error, usable as a process-level diagnostic.

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