Residual-Driven Lifting Identification for Nonlinear-Manifold Reduced-Order Models of Parametrized Linear PDEs

Abstract

We introduce a residual-driven procedure for training nonlinear-manifold reduced-order models for parametrized linear partial differential equations that, given prescribed latent and lifting spaces, identifies the nonlinear lifting without high-fidelity solution snapshots. The approximation is represented by a low-dimensional latent coordinate together with a nonlinear lifting into a richer reduced space. Rather than fitting the lifting to snapshot data, we determine it by minimizing a computable residual-based upper bound for the state error. For affinely parametrized operators, the resulting training objective admits an efficient offline--online decomposition, and the lifting update reduces to a sequence of low-dimensional weighted least-squares problems. Numerical evaluations on an advection--diffusion problem and a plane-strain elasticity benchmark show that the proposed approach substantially improves accuracy over linear subspaces. The resulting nonlinear models achieve accuracy comparable to snapshot-driven nonlinear-manifold training while avoiding high-fidelity snapshots in the lifting-identification stage.

Disclosure

“on-affine, nonlinear, time-dependent, or noncoercive problems would broaden the applicability of the method, but may require empirical interpolation, hyper- reduction, or alternative stability estimates. Acknowledgements The authors used ChatGPT and Claude for language editing and stylistic sug- gestions. The authors reviewed and are responsible for all scientific content, derivations, code, results, and conclusions. References [1] Peter Benner, Mario Ohlberger, Albert Cohen, a”

PDF page 30

Classification

Rewriting existing author-written text

Multiplier

4

Verified

2026-08-23T19:12:41.754933Z

Structural counts