HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEs
Jinpai Zhao, Nishant Panda, Yen Ting Lin, Eirik Valseth, Diane Oyen + 1 more
TLDR
HyCOP is a modular framework that learns parametric PDE solution operators by composing simple modules, leading to interpretable and robust solutions.
Key contributions
- Learns PDE solution operators by composing simple, query-conditioned modules.
- Employs a policy to select and apply modules, combining numerical solvers and learned components.
- Achieves order-of-magnitude out-of-distribution improvements over monolithic neural operators.
- Supports modular transfer and provides an error decomposition for process-level diagnostics.
Why it matters
This paper introduces a novel approach to solving PDEs that combines interpretability with high performance. By learning to compose simple modules, HyCOP significantly improves out-of-distribution generalization compared to traditional neural operators. Its modularity also enables flexible transfer and diagnostics.
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.
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