Intrusive ROMs via Model-form Preserving Least-Squares Projections with Variable Transformation (MP-LSVT)

  • Test established ROM methods for a wide range of rocket combustor models

  • Rigorous analysis of single-injector models

  • Explore sources and solutions of ROM instability and loss of accuracy

  • Ongoing testing and analysis for 2D multi-injector and 3D single-injector models


  1. Huang, C., Wentland, C.R., Duraisamy, K., Merkle, C. Model Reduction for Multi-Scale Transport Problems using Model-form Preserving Least-Squares Projections with Variable Transformation, Journal of Computational Physics. 2021.

  2. Wentland, C.R., Huang, C., and Duraisamy, K. Investigation of sampling strategies for reduced-order models of rocket combustors, AIAA Scitech Forum, 2021.

  3. Huang, C., Duraisamy, K., and Merkle, C., Investigations and Improvement of Robustness of Reduced-Order Models of Reacting Flow, AIAA Journal, 2019.

Modeling with Mesh-aware Neural Networks

  • Conditional parameterization (CP) builds numerical discretization information and hierarchical relations between physical quantities into the network

  • Drop-in CP modification improves existing methods on physical modeling tasks

  • Conditionally Parameterized Graph Neural Network (GP-GNet) models chemical source terms, irregular mesh discretizations, and different boundary conditions.

  • State-of-the-art performance on problems of different complexities, including flow on unseen geometries and complex reacting flow


  1. Xu, J., Pradhan, A., & Duraisamy, K. (2021). Conditionally Parameterized, Discretization-Aware Neural Networks for Mesh-Based Modeling of Physical Systems, arXiv preprint arXiv:2109.09510.

High-fidelity Predictions Multi-fidelity Predictions

Multi-Fidelity Framework for Full-Scale Engine

  • To address problems in which full order solutions are not available for system of interest

  • Train ROMs for computationally expensive regions of flow field (e.g. injectors)

  • Couple ROM regions to benign, coarsely-resolved regions computed with standard high-fidelity solvers (e.g. downstream)

  • Generate modular bases for repeated features, such as injectors


  1. Huang, C., Anderson, W.E., Merkle, C.L., and Sankaran, V., Multi-Fidelity Framework for Modeling Combustion Instability, AIAA Journal, 2019.

  2. Xu, J., Huang, C., and Duraisamy, K., Reduced-Order Modeling Framework for Combustor Instabilities Using Truncated Domain Training. AIAA Journal, 2019.

Adjoint Petrov-Galerkin ROMs

  • A mathematically-rigorous framework for formally closing ROMs of dynamical systems, using a Markovian finite memory assumption within the Mori-Zwanzig formalism.

  • Demonstrated improvements in accuracy and robustness over Galerkin ROMs, and improvements in accuracy and efficiency over the least-squares Petrov-Galerkin method in most cases.


  1. Parish, E., Wentland, C., and Duraisamy, K., The Adjoint Petrov-Galerkin Method for Non-Linear Model Reduction, CMAME, Vol. 365, 2020.

  2. Wentland, C.R., Huang, C., and Duraisamy, K., Closure of Reacting Flow Reduced-Order Models via the Adjoint Petrov-Galerkin Method, 2019 AIAA Aviation Forum, 2019.

Improvements in Sampling and Subspace Extraction

  • Problems with strong convection, shocks, and sharp gradients exhibit complex behavior for which linear model reduction methods fail.

  • We construct nonlinear approximations of solution manifolds via, e.g., subspace adaptation during time stepping, transformations, transport maps, and deep neural networks.

  • Nonlinear approximation techniques provide accurate predictions of combustion dynamics with orders of magnitude reductions in the number of degrees of freedom compared to linear reduced models and high-fidelity models.


  1. Peherstorfer, B., Model reduction for transport-dominated problems via online adaptive bases and adaptive sampling, SIAM Journal on Scientific Computing, Vol. 45, No. 5, pp. A2803-2836, 2020.

  2. Peherstorfer, B., Drmac, Z., and Gugercin, S., Stability of discrete empirical interpolation and gappy proper orthogonal decomposition with randomized and deterministic sampling points, SIAM Journal on Scientific Computing, Vol. 42, No. 5, pp.A2837-A2864, 2020.

Successfully tested for future-state prediction of 2D single-injector combustion dynamics

Lifting Transformations and Learning Nonlinear Reduced Models

  • Lifting variable transformations of the nonlinear system expose polynomial structure, obviating the need for sparse sampling in model reduction.

  • Data-driven operator inference learns reduced models in lifted variable representation.

  • Scalable Python implementation as well as a MATLAB implementation are available on GitHub.

  • More detail at the Willcox Research Group website.


  1. Swischuk, R., Kramer, B., Huang, C., Willcox, K., Learning physics-based reduced-order models for a single-injector combustion process, AIAA Journal, 2020.

  2. Qian, E, Kramer, B., Marques, A., and Willcox, K., Transform & Learn: A data-driven approach to nonlinear model reduction. 2019 AIAA Aviation Forum, 2019.

  3. Kramer, B., and Willcox, K., Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA Journal, 2019.

  4. Swischuk, R., Mainini, L., Peherstorfer, B., and Willcox, K. Projection-based model reduction: Formulations for physics-based machine learning, Computers and Fluids, Vol. 179, pp. 704-717, 2019.

High-Fidelity Simulations

  • Developing test cases for multi-injector combustor models

  • Ensuring that combustor dynamics are indicative of a nominal rocket combustor configuration while remaining amenable to ROM training

  • Providing diagnostics of flow field, reaction, and acoustic dynamics

Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics

  • A multi-level deep learning based framework for the direct prediction of spatio-temporal dynamics for new global paramters and/or future states

  • Evaluated on a range of problems involving discontinuities, wave propagation, strong transients, and coherent structures

  • Significantly improved predictive capability over POD-based methods

  • Dealing with arbitrary number of dimensions/variables without changing network architecture


  1. Xu, J. and Duraisamy, K., Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics, CMAME, Vol. 372, 2020.

Standardized Hierarchy of Test Problems

  • Developing a set of standardized chemically-reacting flow cases for the broader ROM community to test novel methods

  • Formalizing 1D advecting contact surface and 1D advecting viscous flame cases

  • Establishing test metrics for accurately comparing benchmarks

CLoVER (Contour Location Via Entropy Reduction)

  • Algorithm that combines data from multiple information sources to locate contours of expensive functions at low cost

  • Contour entropy: measure of uncertainty about the location of the zero contour of function approximated by statistical surrogate model

  • Decision mechanism: maximizes average reduction of contour entropy via one-step lookahead approach


  1. Marques, A., Lam, R. and Willcox, K., Contour location via entropy reduction leveraging multiple information sources, Advances in Neural Information Processing Systems 31 (NeurIPS), 2018.