HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs
2026-05-01 • Computational Engineering, Finance, and Science
Computational Engineering, Finance, and ScienceMachine Learning
AI summaryⓘ
The authors developed HyCOP, a system that solves complex equations describing physical processes by combining small, simple pieces like advection and diffusion modules. Instead of learning one big solution all at once, their method learns how to choose and sequence these pieces based on the current problem conditions. This approach allows HyCOP to solve problems more accurately, even outside its training range, and to adapt by swapping or updating modules. Their theory also helps understand errors made by the system at different stages.
parametric PDEsolution operatoradvectiondiffusionneural operatormodular frameworkquery conditioningout-of-distributionhybrid surrogateerror decomposition
Authors
Jinpai Zhao, Nishant Panda, Yen Ting Lin, Eirik Valseth, Diane Oyen, Clint Dawson
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.