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.
Quadratic manifold-based model order reduction offers a viable pathway to circumvent the limitations of linear subspaces for linear control systems characterized by slow Kolmogorov $n$-width decay. However, a system-theoretic framework for constructing such quadratic approximations remains absent from the literature. T...
Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approac...
J. A. Hernández, S. A. de Parga, R. Rossi· 1 citation
We use a large database of direct numerical simulations to investigate the transition of the Rayleigh--Taylor instability to turbulence and its evolution toward a late-time self-similar regime. In addition to tracking the growth of the mixing layer through the mean heavy-fluid concentration profile, we analyze one-dime...
Téo Granger, B. Nadiga, B. Gréa et al.· 0 citations
A coupled perturbation analysis for the entire PPMD procedure identifies how the principal errors interact and which components limit the accuracy of the nonlinear reduced order model.
A compact nonlinear-manifold representation, based on Gaussian splatting, for solving partial differential equations with discontinuities and thin shear layers, enabling sharp features to be captured with far fewer parameters than conventional numerical or neural-network-based solvers.
Xu Han, Kuang-Xu Chen, Rui Wang et al.· Nature Communications· 0 citations
This work develops efficient computational expressions for invariant manifolds and their reduced dynamics that depend smoothly on system parameters, without shifting coordinates or introducing parameters as auxiliary variables and demonstrates the effectiveness of the method on a range of high-fidelity numerical models...
