Intrusive ROMs via Model-form Preserving Least-Squares Projections with Variable Transformation (MP-LSVT)
The approach allows for the selection of an arbitrary, but complete, set of solution variables while preserving the structure of the governing equations.
Least-squares-based minimization is leveraged to guarantee symmetrization and discrete consistency with the full-order model (FOM).
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
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.
Wentland, C.R., Huang, C., and Duraisamy, K. Investigation of sampling strategies for reduced-order models of rocket combustors, AIAA Scitech Forum, 2021.
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
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
Huang, C., Anderson, W.E., Merkle, C.L., and Sankaran, V., Multi-Fidelity Framework for Modeling Combustion Instability, AIAA Journal, 2019.
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.
Parish, E., Wentland, C., and Duraisamy, K., “The Adjoint Petrov-Galerkin Method for Non-Linear Model Reduction, CMAME, Vol. 365, 2020.
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.
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.
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.
Swischuk, R., Kramer, B., Huang, C., Willcox, K., Learning physics-based reduced-order models for a single-injector combustion process, AIAA Journal, 2020.
Qian, E, Kramer, B., Marques, A., and Willcox, K., Transform & Learn: A data-driven approach to nonlinear model reduction. 2019 AIAA Aviation Forum, 2019.
Kramer, B., and Willcox, K., Nonlinear model order reduction via lifting transformations and proper orthogonal decomposition, AIAA Journal, 2019.
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.
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
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
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.