Operator Inference of Non-Markovian Terms for Learning Reduced Models from Partially Observed State Trajectories

Uy, Wayne Isaac Tan; Peherstorfer, Benjamin

doi:10.1007/s10915-021-01580-2

Cited by 11 publications

(7 citation statements)

References 56 publications

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“…The governing PDE is spatially discretized using n = 500 grid points leading to a discretized state y ∈ R 1000 . The FOM is numerically integrated for total time T = 10 using the implicit midpoint rule (40), a symplectic method, with ∆t = 0.01. To propagate the system forward in time, we need to solve a system of 2n = 1000 linear equations at every time step.…”

Section: Motivationmentioning

confidence: 99%

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Hamiltonian Operator Inference: Physics-preserving Learning of Reduced-order Models for Canonical Hamiltonian Systems

Sharma¹,

Wang²,

Kramer³

2021

Preprint

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This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Hamiltonian systems. Traditional intrusive projection-based model reduction approaches utilize symplectic Galerkin projection to construct Hamiltonian reduced models by projecting Hamilton's equations of the full model onto a symplectic subspace. This symplectic projection requires complete knowledge about the full model operators and full access to manipulate the computer code. In contrast, the proposed Hamiltonian operator inference approach embeds the physics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying symplectic structure. Our method exploits knowledge of the Hamiltonian functional to define and parametrize a Hamiltonian ROM form which can then be learned from data projected via symplectic projectors. The proposed method is 'gray-box' in that it utilizes knowledge of the Hamiltonian structure at the partial differential equation level, as well as knowledge of spatially local components in the system. However, it does not require access to computer code, only data to learn the models. Our numerical results demonstrate Hamiltonian operator inference on a linear wave equation, the cubic nonlinear Schrödinger equation, and a nonpolynomial sine-Gordon equation. Accurate long-time predictions far outside the training time interval for nonlinear examples illustrate the generalizability of our learned models.

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Section: Motivationmentioning

confidence: 99%

“…The approach has also been extended to a grey-box setting in [38] where analytical expressions for the nonpolynomial nonlinear terms are known and the remaining operators are learned via operator inference. Convergence and accuracy certificates were developed in [39,40].…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian Operator Inference: Physics-preserving Learning of Reduced-order Models for Canonical Hamiltonian Systems

Sharma¹,

Wang²,

Kramer³

2021

Preprint

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“…Building on the philosophy of inducing time-dependent shifts into the ROM framework, we extend the non-intrusive projection-based operator inference ROM framework [43] towards advectiondominated systems by transforming the dynamics to a moving coordinate frame. Standard operator inference has successfully been applied to diverse applications such as combustion [57,34], chemical reactors [10], ocean flows [61], Hamiltonian systems [55] and general reaction systems in the presence of incomplete data [59]. Since operator inference learns the operators that would be obtained through intrusive Galerkin projection (which can be done exactly with additional data pre-processing, see [42]) it inherits problems that intrusive Galerkin ROMs face in the presence of strong advection.…”

Section: Introductionmentioning

confidence: 99%

Predicting Solar Wind Streams from the Inner-Heliosphere to Earth via Shifted Operator Inference

Issan¹,

Kramer²

2022

Preprint

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Solar wind conditions are predominantly predicted via three-dimensional numerical magnetohydrodynamic (MHD) models. Despite their ability to produce highly accurate predictions, MHD models require computationally intensive high-dimensional simulations. This renders them inadequate for making time-sensitive predictions and for large-ensemble analysis required in uncertainty quantification. This paper presents a new data-driven reduced-order model (ROM) capability for forecasting heliospheric solar wind speeds. Traditional model reduction methods based on Galerkin projection have difficulties with advection-dominated systems-such as solar winds-since they require a large number of basis functions and can become unstable. A core contribution of this work addresses this challenge by extending the non-intrusive operator inference ROM framework to exploit the translational symmetries present in the solar wind caused by the Sun's rotation. The numerical results show that our method can adequately emulate the MHD simulations and outperforms a reduced-physics surrogate model, the Heliospheric Upwind Extrapolation model.

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“…In particular, estimation of dynamic structures under partial observability is an important problem due to its generality and to potential applications (cf. Menda et al [2020], Tsiamis and Pappas [2019], Tsiamis et al [2020], Lale et al [2020], Lee [2022], Adams et al [2021], Bhouri and Perdikaris [2021], Ouala et al [2020], Uy and Peherstorfer [2021], Subramanian et al [2022], Bennett and Kallus [2021], Lee et al [2020]). Inheriting accumulated results of the controls literature, most of the work on provably correct methods for partially observed dynamical systems consider additive Gaussian noise and controllable/observable linear system; some work further adds restrictions on spectral radius of the linear system to be strictly less than one.…”

Section: Introductionmentioning

confidence: 99%

Dynamic Structure Estimation from Bandit Feedback

Ohnishi¹,

Ishikawa²,

Kuroki³

et al. 2022

Preprint

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This work present novel method for structure estimation of an underlying dynamical system. We tackle problems of estimating dynamic structure from bandit feedback contaminated by sub-Gaussian noise. In particular, we focus on periodically behaved discrete dynamical system in the Euclidean space, and carefully identify certain obtainable subset of full information of the periodic structure. We then derive a sample complexity bound for periodic structure estimation. Technically, asymptotic results for exponential sums are adopted to effectively average out the noise effects while preventing the information to be estimated from vanishing. For linear systems, the use of the Weyl sum further allows us to extract eigenstructures. Our theoretical claims are experimentally validated on simulations of toy examples, including Cellular Automata.

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Operator Inference of Non-Markovian Terms for Learning Reduced Models from Partially Observed State Trajectories

Cited by 11 publications

References 56 publications

Hamiltonian Operator Inference: Physics-preserving Learning of Reduced-order Models for Canonical Hamiltonian Systems

Hamiltonian Operator Inference: Physics-preserving Learning of Reduced-order Models for Canonical Hamiltonian Systems

Predicting Solar Wind Streams from the Inner-Heliosphere to Earth via Shifted Operator Inference

Dynamic Structure Estimation from Bandit Feedback

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