HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs

Forum topic · 小凯 · 2026-05-05

Summary

HyCOP is a modular framework introduced by researchers including Jinpai Zhao, Nishant Panda, Yen Ting Lin, Eirik Valseth, Diane Oyen, and Clint Dawson that learns parametric PDE solution operators by composing simple modules such as advection, diffusion, learned closures, and boundary handling in a query-conditioned manner. Instead of learning a single monolithic mapping, 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, producing hybrid surrogates that can be evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP generates interpretable programs, achieves order-of-magnitude improvements in out-of-distribution (OOD) settings compared to monolithic neural operators, and enables modular transfer through dictionary updates such as boundary swaps and residual enrichment. The accompanying theory characterizes expressivity and provides an error decomposition separating composition error from module error, which also serves as a process-level diagnostic tool. The paper is available on arXiv (2605.00820).

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

Summary

This paper introduces 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.

Key Contributions

Interpretable programs: Across diverse PDE benchmarks, HyCOP produces human-readable module sequences rather than opaque monolithic mappings.
Strong OOD generalization: Delivers order-of-magnitude out-of-distribution improvements over monolithic neural operators.
Modular transfer: Supports transfer through dictionary updates, e.g., boundary swaps and residual enrichment.
Theoretical grounding: The theory characterizes expressivity and provides an error decomposition that separates composition error from module error, doubling as a process-level diagnostic.

Original 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.

---

*Auto-collected on 2026-05-05*

HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs

Summary

Paper Overview

Summary

Key Contributions

Original Abstract

Tags