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. This paper presents a system-agnostic, optimization-free framework for the direct construction of quadratic projection matrices. We prove that the synthesized reduced-order model matches the nonlinear moments of the full-order system and preserves its exact center manifold mapping, thereby ensuring asymptotic tracking of steady-state outputs under specific input classes. Numerical results on transport-dominated benchmark problems, namely, the one-dimensional damped wave and advection equations, show that the proposed framework achieves high-fidelity trajectory reconstruction within a significantly reduced-dimensional state space, yielding substantial online computational savings.
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 approaches employ weights that remain fixed over the solution manifold. We contend that this restriction leaves hyperreduction potential untapped: allowing the weights to vary continuously and nonlinearly with the latent coordinates can further decrease the number of sampled spatial entities. To exploit this possibility, we propose the Manifold-Adaptive-Weight Empirical Cubature Method (MAW-ECM). Starting from a feasible fixed-weight ECM rule, a greedy pruning strategy removes sampled entities through convex quadratic weight-redistribution problems enforcing local conditions and positivity. The method is assessed on two nonlinear benchmarks: homogenization of a metamaterial unit cell exhibiting negative incremental stiffness, and a history-dependent continuum-damage problem. In both cases, the nonlinear manifold is constructed from an initial linear compression followed by an input-informed identification of the latent coordinates as general linear combinations of the retained modal amplitudes, incorporating graph information when relevant to seek the intrinsic dimensionality of the solution manifold. We show that combining the nonlinear-manifold representation with MAW-ECM reduces the number of sampled integration points by more than two orders of magnitude relative to the corresponding standard linear reduced model. Relative to the fixed-weight manifold models alone, the adaptive weights eliminate approximately 80% of the remaining points in the homogenization benchmark and more than 97% in the damage benchmark, while essentially preserving their accuracy.
J. A. Hernández, S. A. de Parga, R. Rossi· 0 citations
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-dimensional profiles of turbulent kinetic energy and dissipation, two key quantities in classical turbulent-mixing models. We consider two reduced-order modeling strategies that differ in where nonlinearity is introduced: either in the construction of the latent space or in the description of its temporal evolution. The first method uses a linear encoder--decoder obtained using Proper Orthogonal Decomposition (POD), with nonlinear reduced dynamics learned by a physics-informed neural network (PINN). The second uses a nonlinear encoder--decoder learned by an autoencoder, while constraining the latent dynamics to remain linear and satisfy physical constraints. Both approaches achieve satisfactory performance in reconstructing, interpolating, and extrapolating the dynamics of the Rayleigh--Taylor instability.
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.
Conventional linear discretizations, including high-order schemes, often require prohibitively many degrees of freedom to resolve sharp, localized features, and the practical advantages of nonlinear, physics-informed representations over fixed linear spaces remain unclear. Here we introduce a compact nonlinear-manifold representation, based on Gaussian splatting, for solving partial differential equations with discontinuities and thin shear layers. Mobile, anisotropic Gaussian kernels are evolved or optimized to satisfy the governing equations: time-dependent problems through a projection-based scheme on the manifold, and steady forward and inverse problems through direct residual minimization. The representation offers closed-form spatial derivatives and a highly compact description of sharp features. Across smooth, shock-dominated and shear-layer benchmarks, it attains accuracy comparable to classical solvers for smooth flows while remaining markedly more compact in strongly nonlinear regimes, and it displays an approximate error invariance as the Reynolds number increases—departing from the error accumulation inherent to linear representations. Simulating shocks and thin shear layers typically requires extremely fine computational grids. Here, the authors adapt mobile, stretchable Gaussian kernels originally developed in computer graphics to directly satisfy the governing equations, 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
Invariant manifolds provide a solid foundation for rigorous model reduction and analysis of complex dynamical behavior in high-dimensional nonlinear systems. Their existence and smooth parameter dependence are well established theoretically; yet the practical computation of parameter-dependent invariant manifolds remains challenging, particularly for systems with nontrivial equilibria, large state dimension, or limited access to governing equations. In this work, we develop 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. This approach enables formal Taylor and Taylor–Fourier expansions of parameter-dependent equilibria, invariant manifolds, and reduced-order models directly from the governing equations. We show how our expressions can be implemented nonintrusively for systems with affine parameter dependence and up to cubic nonlinearities. We demonstrate the effectiveness of the method on a range of high-fidelity numerical models, where the resulting parameter-dependent reduced-order models enable the analysis of local bifurcations, including buckling, flutter, period-doubling, and internal and parametric resonances, at computational costs far below those of the full system. The framework provides a practical bridge between rigorous invariant manifold theory and large-scale engineering applications, with particular relevance to design and optimization.
