# Gradient Preserving Operator Inference: Data-Driven Reduced-Order Models for Equations with Gradient Structure

CoRR（2024）

摘要

Hamiltonian Operator Inference has been introduced in [Sharma, H., Wang, Z.,
Kramer, B., Physica D: Nonlinear Phenomena, 431, p.133122, 2022] to learn
structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This
approach constructs a low-dimensional model using only data and knowledge of
the Hamiltonian function. Such ROMs can keep the intrinsic structure of the
system, allowing them to capture the physics described by the governing
equations. In this work, we extend this approach to more general systems that
are either conservative or dissipative in energy, and which possess a gradient
structure. We derive the optimization problems for inferring
structure-preserving ROMs that preserve the gradient structure. We further
derive an a priori error estimate for the reduced-order approximation. To
test the algorithms, we consider semi-discretized partial differential
equations with gradient structure, such as the parameterized wave and
Korteweg-de-Vries equations in the conservative case and the one- and
two-dimensional Allen-Cahn equations in the dissipative case. The numerical
results illustrate the accuracy, structure-preservation properties, and
predictive capabilities of the gradient-preserving Operator Inference ROMs.

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