We will extend and apply symmetry-equivariant neural networks (ENNs) to develop accurate, efficient, and physically grounded surrogate models across atomistic and continuum scales:

Extend ENN framework. We will enhance existing ENN libraries (developed by PI Smidt) by: (1) Integrating angular monomial bases and proper orthogonal descriptors from PI Nguyen’s ML-POD package, improving representational richness beyond spherical harmonics; and (2) Improving performance on irregular and regular meshes for upstream constitutive models that can predict atomistic response for constitutive modeling.

Environment-adaptive modeling. We will implement the environment decomposition strategies developed within CESMIX using local ENNs blended via partition-of-unity functions to enable efficient and transferable models across diverse physicochemical conditions.

Machine-learned interatomic potentials (MLIPs). We will develop charge-aware ENN-based MLIPs that conserve total charge, requiring DFT datasets with computed partial charges. To achieve this we plan to use active learning and subset selection tools (e.g., from CESMIX) to minimize DFT evaluations and optimize the training process; build

interfaces for ENN MLIPs to be integrated with LAMMPS; and use Julia packages InteratomicPotentials.jl, PotentialLearning.jl, and Cairn.jl or similar tools.

Continuum constitutive modeling. We will: (1) use ENNs to learn tensorial constitutive models from MD simulation data and experimental macroscopic property measurements; (2) develop models for C and SiC based on deforma- tion tensors and thermo-chemo-mechanical responses; (3) integrate ENN-based constitutive models into continuum solvers such as ΣMIT; and (4) validate against experimental data. This multi-scale framework will enable symmetry- consistent, data-efficient surrogate modeling for complex materials under extreme conditions.

Manifold learning surrogates. Online computation of detailed chemistry in continuum thermo-chemo-mechanics simulations is computationally infeasible, but manifold learning techniques—particularly diffusion maps—offer effi- cient, physically interpretable surrogates by projecting solutions onto a low-dimensional manifold that respects phys- ical constraints. We have demonstrated an initial success when applied to model SiC oxidation transitions, capturing key regimes. The approach will be extended to create surrogate models for C pyrolysis, SiC oxidation, and nitri- dation, enabling tractable online chemistry integration in continuum simulations, validated against spectroscopic gas measurements.

Reduced order modeling. Reduced order models (ROMs), based on approximation spaces defined from data, can be highly effective in accelerating forward models due to their accuracy, minimal training data requirements, and ability to be equipped with a posteriori error estimates. We will develop both projection-based intrusive and interpolation-based non-intrusive ROMs for simulating hypersonic flow and solid mechanics interactions. We will develop models for the separate fluid and structure simulations, while also coupling these models to accurately represent the interactions between the flow and material. We will implement multiphysics, multiscale reduced basis methods to improve the stability and enforcement of boundary and interface conditions. Additionally, we will employ dimension reduction techniques for nonlinear transport phenomena via optimal transport and hyper-reduction, while providing inexpensive and rigorous error estimates using the adjoint method. ROMs will support our uncertainty quantification (UQ) efforts.