PaperlayerView exact revision on arXiv

Source-linked AI summary

Learning physics-based reduced-order models for a single-injector combustion process

Renee Swischuk, Boris Kramer, Cheng Huang, Karen Willcox

arXiv:1908.03620v4physics.comp-phcs.LGcs.SYeess.SYmath.DSstat.ML

TL;DR

Effective and robust reduced models for complex combustion remain challenging, particularly when projection requires high-dimensional operators. This paper learns a quadratic ROM from transformed simulation data using operator inference, and reports five to six orders of magnitude in computational speedup over generating the training snapshots.

  • Problem

    Effective and robust ROMs for nonlinear, multi-scale combustion remain challenging, and projection-based methods can require access to high-dimensional operators.

    Source: §I · p. 2, C. Numerical model · p. 5

  • Method

    The method transforms simulation snapshots to expose quadratic structure and learns a low-dimensional ROM with operator inference, without requiring access to the high-fidelity model operators.

    Source: §I · p. 3, A. Projection preserves polynomial… · p. 6, §V · p. 19

  • Results

    Five to six orders of magnitude in computational speedup are reported for the ROMs compared with the approximately 200 hours needed to compute the first 10,000 training snapshots.

    Source: C. Learned reduced model performance · p. 16

  • Takeaways & Limitations

    The approach demonstrates non-intrusive learning of a structured polynomial ROM for a single-injector combustion model from high-fidelity simulation snapshots.

    Source: §I · p. 3, §V · p. 19

Abstract

from arXiv · show

This paper presents a physics-based data-driven method to learn predictive reduced-order models (ROMs) from high-fidelity simulations, and illustrates it in the challenging context of a single-injector combustion process. The method combines the perspectives of model reduction and machine learning. Model reduction brings in the physics of the problem, constraining the ROM predictions to lie on a subspace defined by the governing equations. This is achieved by defining the ROM in proper orthogonal decomposition (POD) coordinates, which embed the rich physics information contained in solution snapshots of a high-fidelity computational fluid dynamics (CFD) model. The machine learning perspective brings the flexibility to use transformed physical variables to define the POD basis. This is in contrast to traditional model reduction approaches that are constrained to use the physical variables of the high-fidelity code. Combining the two perspectives, the approach identifies a set of transformed physical variables that expose quadratic structure in the combustion governing equations and learns a quadratic ROM from transformed snapshot data. This learning does not require access to the high-fidelity model implementation. Numerical experiments show that the ROM accurately predicts temperature, pressure, velocity, species concentrations, and the limit-cycle amplitude, with speedups of more than five orders of magnitude over high-fidelity models. Our ROM simulation is shown to be predictive 200% past the training interval. Moreover, ROM-predicted pressure traces accurately match the phase of the pressure signal and yield good approximations of the limit-cycle amplitude.

Nomenclature

The paper addresses the challenge of building stable, efficient reduced models for complex combustion dynamics. It uses transformed variables and operator inference to learn structured polynomial ROMs from simulation snapshots without access to high-fidelity operators.

I. Introduction

  • Effective and robust ROMs for nonlinear, multi-scale combustion remain an open challenge, while existing projection-based methods can require access to high-dimensional operators and many modes.For the same single-injector simulation, prior work found that well over 100 modes were needed for stable ROMs and sufficient accuracy.Source: §I · p. 2
  • Non-intrusive learning fits a model from training data without explicit access to high-fidelity operators, but learned model parametrization affects predictive accuracy outside training data.Neural networks are one example, but their large data requirements can limit utility with expensive PDE simulations.Source: §I · p. 2
  • The approach uses variable transformations to expose polynomial structure, including auxiliary-variable lifting for systems that are not naturally polynomial.Some governing equations admit transformations such as the specific-volume representation of Euler equations [35].Source: §I · pp. 2–3
  • The paper learns structured polynomial ROMs from simulated snapshots by transforming variables to expose quadratic structure and applying operator inference in a low-dimensional subspace.The workflow obtains CFD snapshots, transforms their state variables, and learns the ROM with operator inference.Source: §I · p. 3

II. Combustion model

The combustion model is a simplified two-dimensional single-injector problem governed by conservation equations for flow, energy, and species. Its setup includes a one-step methane reaction and prescribed inlet and outlet conditions.

A. Computational domain

  • The computational domain is a simplified two-dimensional version of a single-injector combustor, with four locations used to monitor state variables.The domain is based on the GEMS combustion model implementation.Source: A. Computational domain · p. 3, §II · p. 3

B. Governing equations

  • The model governs mass, momentum, energy, and species mass fractions, with a one-step reaction among methane, oxygen, carbon dioxide, and water.The stoichiometric coefficients are −1, −2, 1, and 2, respectively; the reactant orders are 0.2 for methane and 1.3 for oxygen.Source: B. Governing equations · pp. 3–5
  • The setup imposes a non-reflecting downstream boundary while maintaining chamber pressure, no-slip top and bottom walls, and constant-mass-flow upstream inlets.The pressure boundary condition uses a 1.0 × 10^6 Pa reference pressure, amplitude 0.1, and frequency 5000 Hz.Source: B. Governing equations · p. 5

C. Numerical model

The numerical model discretizes the combustion equations into a high-dimensional nonlinear ODE system, making repeated solutions costly. The paper motivates a ROM approach that avoids the efficiency and robustness problems encountered with POD-DEIM in combustion applications.

C. Numerical model

  • Solving the high-dimensional nonlinear ODEs is expensive, while their nonlinear multi-scale dynamics make ROM construction challenging.GEMS uses finite volumes and primitive variables as solution variables; the discretized state and boundary input define the nonlinear ODE system.Source: C. Numerical model · p. 5
  • POD-DEIM has encountered combustion-application problems, including the need for many interpolation points that erodes computational efficiency and challenges with robustness and stability.The paper presents non-intrusive ROM learning as an alternative that exposes system structure through variable transformations and removes the need for DEIM.Source: C. Numerical model · p. 6

III. Non-intrusive learning of a combustion reduced model

The method represents the governing dynamics in polynomial form and uses projection to retain that structure in reduced coordinates. Variable choices and auxiliary variables can expose polynomial terms for operator inference.

A. Projection preserves polynomial structure in the governing equations

  • The governing system can be written with linear, quadratic, cubic, input, constant, and higher-order terms, making its polynomial structure explicit.The higher-order terms include quartic and higher powers, as well as nonlinear terms that cannot be expressed polynomially.Source: A. Projection preserves polynomial… · p. 6
  • The polynomial form depends on the state variables chosen, so terms that are linear in one representation may be quadratic in another.For example, the mass-flux term involving density and x-velocity is quadratic in those state variables.Source: A. Projection preserves polynomial… · p. 6
  • Galerkin projection of the state onto a low-dimensional basis preserves the polynomial form, yielding reduced operators for the same terms.The reduced state is represented in a basis V with dimension r much smaller than the full state dimension.Source: A. Projection preserves polynomial… · p. 6
  • Variable transformations and auxiliary-variable lifting can convert non-polynomial terms into polynomial form, while higher-order terms can increase operator size substantially.The number of operator elements scales with r^4 for a cubic operator and r^5 for a quartic operator; block sparsity can limit computational cost.Source: A. Projection preserves polynomial… · p. 6

B. Operator inference for learning reduced models

Operator inference learns reduced operators from high-fidelity snapshots after projecting the data onto a POD subspace. The least-squares fit uses projected state and derivative data without requiring access to the original high-dimensional operators.

  • The workflow collects state and input snapshots, defines a POD subspace, projects the snapshots, and obtains projected state-derivative data.Derivatives may be approximated from projected snapshots or obtained by collecting and projecting high-fidelity model derivatives.Source: B. Operator inference for learning… · p. 7
  • Operator inference fits reduced operators by minimizing the residual between projected dynamics and data, without explicit access to the original high-dimensional operators.Source: B. Operator inference for learning… · p. 7
  • For K > r + r^2 + m + 1, the least-squares problem has a unique solution and separates into r independent problems, supporting efficient, scalable inference.Source: B. Operator inference for learning… · p. 7
  • L2 regularization penalizes off-diagonal entries of the linear operator and all entries of the other operators to reduce overfitting and promote ROM stability.The regularization parameter is problem-specific; too much regularization can fit the data poorly.Source: B. Operator inference for learning… · p. 7

C. A structure-exploiting ROM learning formulation for GEMS

The GEMS formulation uses transformed physical variables to expose quadratic structure in many governing-equation terms, making non-intrusive quadratic ROM learning possible. Some terms remain non-polynomial, and fully lifting the equations would add many auxiliary variables and algebraic constraints.

  • Non-intrusive operator inference allows transformed physical variables to define the ROM, avoiding changes to the high-fidelity simulator that an intrusive approach would require.This flexibility is useful for unusual variable choices that can make rewriting the simulator time-consuming and mathematically difficult.Source: C. A structure-exploiting ROM learning… · p. 8
  • The GEMS equations admit a transformation in which the specific-volume and velocity dynamics are quadratic in the learning variables.The cited derivation identifies ξ, vx, and vy as variables for the quadratic equations.Source: C. A structure-exploiting ROM learning… · pp. 8–9
  • A complete polynomial lifting is possible but would require many auxiliary variables and algebraic equations; lifting Arrhenius terms also adds algebraic constraints that complicate model reduction.The paper therefore uses a transformation that makes many, rather than all, governing-equation terms polynomial.Source: C. A structure-exploiting ROM learning… · p. 8
  • Pressure and species-concentration equations still contain non-polynomial terms in the chosen learning variables.Source: C. A structure-exploiting ROM learning… · p. 8

IV. Numerical Results

The numerical-results section introduces the GEMS combustion dataset and its preparation for ROM learning. It describes a large spatial discretization, a 1 ms training simulation, variable scaling, and derivative estimation.

A. GEMS Dataset

  • The GEMS domain uses 38,523 spatial points, and the CFD state has dimension 308,184; the setup uses methane fuel and an oxidizer stream of 42% O2 and 58% H2O.The fuel and oxidizer mass flow rates are 5.0 kg s and 0.37 kg s, respectively.Source: A. GEMS Dataset · p. 9
  • The training simulation spans 1 ms at a 1 × 10^-7 s time step, yielding 10,000 snapshots that are transformed into the learning variables.Generating the high-fidelity training data took approximately 200 CPU hours.Source: A. GEMS Dataset · p. 9
  • Variables are scaled to [−1, 1] before POD because the data span large scales, including pressure of order 10^6 and species concentrations as low as 10^-12.Source: A. GEMS Dataset · p. 9

IV. Numerical Results

  • Projected-state derivatives use a fourth-order accurate five-point approximation, with first-order forward and backward Euler estimates at the time-series endpoints.Source: A. GEMS Dataset · p. 10

B. Learning a quadratic reduced-order model

The learning implementation combines POD basis construction, a reduced least-squares formulation, and regularization. It removes duplicated quadratic features and uses training-data richness and an L-curve to guide numerical choices.

B. Learning a quadratic reduced-order model

1. Singular value decomposition implementation and POD basis selection

  • A randomized SVD computes the leading 500 singular vectors for large snapshot matrices, and the POD dimension is selected using cumulative energy and relative projection error.Source: §1 · p. 10

2. Removing redundant terms in least squares problem

  • Repeated terms in the quadratic Kronecker product are removed before fitting, replacing the redundant operator representation with a smaller equivalent least-squares formulation.Source: §2 · pp. 10–11

3. Regularization

  • L2 regularization trades data fit against smaller operators to curb unstable overfitting, while the L-curve helps select its parameter.In this application, 10,000 training snapshots were required for a sufficiently rich POD basis and accurate ROMs; the condition number remained manageable.Source: §3 · p. 11

4. Error measures

The section defines scale-aware errors for reconstructed ROM fields and evaluates predictions across the training boundary and state variables.

4. Error measures

  • Pressure and temperature use relative error, while species concentrations and velocities use normalized absolute error to handle tiny values and zero-crossing velocities.The errors are computed after reconstructing ROM solutions to the full CFD dimension and scaling variables back to their original ranges.Source: §4 · pp. 11–12

C. Learned reduced model performance

V. Conclusion

The conclusion presents a non-intrusive operator-inference approach that uses variable transformations to expose quadratic structure and learn ROMs for complex combustion dynamics. The learned ROM predicts quantities of interest and conserves species accurately, although some ROMs remain unstable.

V. Conclusion

  • The learned quadratic ROM predicts relevant quantities of interest and conserves species accurately in the combustion application.The quadratic form is an approximation for this application.Source: §V · p. 19
  • Variable transformations expose quadratic structure in the combustion PDEs, and projection preserves that structure to justify learning a quadratic ROM with operator inference.Source: §V · p. 19
  • The learning is non-intrusive: it uses high-fidelity state solutions without requiring access to discretized governing-equation operators or modifications to the CFD code.Transformations can be applied as post-processing to simulation data.Source: §V · p. 19
  • Some learned ROMs remain unstable despite regularization, motivating future work on alternative approaches and theory guaranteeing stability.Source: §V · p. 19

Appendix A: Lifting chemical source terms

The appendix lifts chemical source-term dynamics by introducing auxiliary variables, expressing the source terms and lifted-variable dynamics in polynomial form.

Appendix A: Lifting chemical source terms

  • Auxiliary variables are introduced to lift the chemical source-term system into polynomial form.Source: Appendix A · p. 19
  • The chemical source-term dynamics are cubic in w1, w3, and w5.Source: Appendix A · p. 19
  • The appendix derives auxiliary-variable dynamics, including the example for w1 and a system of corresponding dynamics.Source: Appendix A · p. 19
  • The lifted-variable dynamics are quintic in w1 through w6; adding w7 = w1w3w5 yields a further system of equations.Source: Appendix A · p. 20

Appendix B: Equations for pressure and chemical species

The appendix gives governing-equation relationships for species concentrations and pressure in the chosen learning variables. It notes that exact polynomial form can require many auxiliary variables, so the ROM instead uses a quadratic approximation.

Appendix B: Equations for pressure and chemical species

  • The governing-equation setup identifies conservative variables g1 through g8, including density, velocity components, energy, and species mass fractions.Source: Appendix B · p. 20
  • Species molar concentrations are related to mass fractions by cl = ρYl, and their chemical source terms are cubic in lifted auxiliary states.Source: Appendix B · p. 20
  • Including density ρ as a lifted variable alongside specific volume ξ would make the noted divergence terms quadratic in the lifted state.Source: Appendix B · p. 20
  • The appendix starts the pressure derivation from the energy equation and states that it directly yields an evolution equation for pressure.Source: Appendix B · p. 20
  • The pressure equation remains nonlinear in the learning variables because enthalpy and its time derivatives depend nonlinearly on temperature; the ROM uses a quadratic approximation instead of adding many auxiliary variables.Source: Appendix B · p. 21
Loading 1908.03620v4…