Energy-Preserving Reduced Operator Inference for Efficient Design and Control

TL;DR

This work introduces Energy-Preserving Operator Inference (EP-OpInf), a non-intrusive method for learning reduced-order models of quadratic PDE systems whose quadratic nonlinearities conserve energy. By imposing a quadratic energy-preserving constraint on the reduced operator via constrained optimization, EP-OpInf retains desirable stability characteristics without requiring access to full model code. Numerical experiments on viscous Burgers' and Kuramoto–Sivashinsky equations show that EP-OpInf achieves accuracy similar to standard OpInf and intrusive projections while enforcing the energy-preserving structure. The approach enhances reliability for design and control tasks in many-query settings and lays groundwork for future stability guarantees and structure-preserving extensions.

Abstract

Many-query computations, in which a computational model for an engineering system must be evaluated many times, are crucial in design and control. For systems governed by partial differential equations (PDEs), typical high-fidelity numerical models are high-dimensional and too computationally expensive for the many-query setting. Thus, efficient surrogate models are required to enable low-cost computations in design and control. This work presents a physics-preserving reduced model learning approach that targets PDEs whose quadratic operators preserve energy, such as those arising in governing equations in many fluids problems. The approach is based on the Operator Inference method, which fits reduced model operators to state snapshot and time derivative data in a least-squares sense. However, Operator Inference does not generally learn a reduced quadratic operator with the energy-preserving property of the original PDE. Thus, we propose a new energy-preserving Operator Inference (EP-OpInf) approach, which imposes this structure on the learned reduced model via constrained optimization. Numerical results using the viscous Burgers' and Kuramoto-Sivashinksy equation (KSE) demonstrate that EP-OpInf learns efficient and accurate reduced models that retain this energy-preserving structure.

Figures (10)

• Figure 1: Visualization of Burgers' equation state evolution and flow field with $\mu = 0.1$ and reduced dimension of 10 with initial condition $x(\omega,0) = 0.8\sin(2\pi\omega -0.125)$.
• Figure 2: Relative energy lost from truncated modes for viscous Burgers' equation training data.
• Figure 3: Comparison of intrusive, standard OpInf, and EP-OpInf methods of relative state error for training data \ref{['eqn:rel-state-err']} (left) and test cases (right).
• Figure 4: Constraint violation of all three methods for Burgers' equation over reduced dimensions $r$.
• Figure 5: Relative energy lost from truncated modes for Kuramoto-Sivashinksy equation training data.