Operator inference for non-intrusive model reduction with quadratic manifolds

Abstract

This paper proposes a novel approach for learning a data-driven quadratic manifold from highdimensional data, then employing the quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping comprises two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projectionbased model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.

Error estimate of the Non-Intrusive Reduced Basis (NIRB) two-grid method with parabolic equations

“…We remind that (11) is well posed (see [10] for the existence and the uniqueness of solutions to problem (11)) and we refer to the notations of [10].…”

“…The semi-discrete form of the variational problem (11) writes for the fine mesh (similarly for the coarse mesh):…”

“…The full discrete form of the variational problem (11) for the fine mesh with implicit Euler scheme writes:…”

“…Let Ω be a bounded domain in R d , with d ≤ 3 and a smooth enough boundary ∂Ω, and consider a parametric problem P on Ω. Non-Intrusive Reduced Basis (NIRB) methods are an alternative to classical Reduced Basis Methods (RBMs) for approximating the solutions of such problems where the parameter is denoted as µ, in a given set G [8,9] (see also different NIRB methods [5,1,11] from the two-grid method). From an engineering point of view, they may be more practical to implement than intrusive RBMs, as they only require the execution of the High-Fidelity (HF) code as a "black-box" solver.…”

Physics-Informed Deep Neural Operator Networks

“…For arbitrary inputs, which are out of the distribution (OOD), further training is required but this may be relatively light if the input space is sampled sufficiently. We note here that a neural operator is very different from a reduced order model (ROM) that is restricted to a very small subset of conditions and lacks generalization properties due to the under-parameterization in such methods [10,11]. Another important property of DeepONet is that it is based on a universal approximation theorem for operators [12,9], and more recent theoretical work [13] has shown that DeepONet can break the curse of dimensionality in the input space, unlike approaches based on ROM for parameterized PDEs [14].…”