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

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

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

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.

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.

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.

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

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

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