HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs

Forum topic · 小凯 · 2026-05-05

Summary

HyCOP is a modular machine-learning 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 a monolithic input-output map, HyCOP learns a policy over short programs that decides which module to apply and for how long, conditioned on regime features and state statistics. Modules can be numerical sub-solvers or learned components, yielding hybrid surrogates that can be evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, achieves order-of-magnitude improvements in out-of-distribution (OOD) generalization over monolithic neural operators, and supports modular transfer via dictionary updates such as boundary swaps and residual enrichment. The accompanying theory characterizes expressivity and provides an error decomposition that separates composition error from module error, which also serves as a process-level diagnostic tool. Authors: Jinpai Zhao, Nishant Panda, Yen Ting Lin, Eirik Valseth, Diane Oyen, and Clint Dawson. arXiv: 2605.00820.

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.

Key Highlights

Modular composition: Combines advection, diffusion, learned closures, and boundary handling modules in a query-conditioned way.
Learned policy over programs: Determines which module to apply and for how long, based on regime features and state statistics.
Hybrid surrogates: Modules may be numerical sub-solvers or learned components; evaluation at arbitrary query times without autoregressive rollout.
Strong OOD generalization: Order-of-magnitude improvements over monolithic neural operators in out-of-distribution scenarios.
Modular transfer: Supports dictionary updates such as boundary swaps and residual enrichment.
Theoretical grounding: Expressivity characterization plus an error decomposition separating composition error from module error, usable as a process-level diagnostic.

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*Source: arXiv:2605.00820*