Real-Time Reduced-Order Modeling of Stochastic Partial Differential Equations via Time-Dependent Subspaces

Abstract

We present a new methodology for the real-time reduced-order modeling of stochastic partial differential equations called the dynamically/bi-orthonormal (DBO) decomposition. In this method, the stochastic fields are approximated by a low-rank decomposition to spatial and stochastic subspaces. Each of these subspaces is represented by a set of orthonormal time-dependent modes. We derive exact evolution equations of these time-dependent modes and the evolution of the factorization of the reduced covariance matrix. We show that DBO is equivalent to the dynamically orthogonal (DO) and bi-orthogonal (BO) decompositions via linear and invertible transformation matrices that connect DBO to DO and BO. However, DBO shows several improvements compared to DO and BO: (i) DBO performs better than DO and BO for cases with ill-conditioned covariance matrix; (ii) In contrast to BO, the issue of eigenvalue crossing is not present in the DBO formulation; (iii) In contrast to DO, the stochastic modes are orthonormal, which leads to more accurate representation of the stochastic subspace. We study the convergence properties of the method and compare it to the DO and BO methods. For demonstration, we consider three cases: (i) stochastic linear advection equation, (ii) stochastic Burgers' equation, and (iii) stochastic incompressible flow over a bump in a channel. Overall we observe improvements in the numerical accuracy of DBO compared against DO and BO.

arXiv comment30 pages, 11 figures

References in corpus (3)

• A minimization principle for the description of time-dependent modes associated with transient instabilities
• Reduced-order Description of Transient Instabilities and Computation of Finite-Time Lyapunov Exponents
• A Scalable Observation-Driven Time-Dependent Basis for a Reduced Description of Transient Systems

Cited by in corpus (6)

• On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues
• On-the-fly Reduced Order Modeling of Passive and Reactive Species via Time-Dependent Manifolds
• Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases
• A Multifidelity deep operator network approach to closure for multiscale systems
• Low-rank representation of head impact kinematics: A data-driven emulator
• Skeletal Model Reduction with Forced Optimally Time Dependent Modes