2026 Theses Doctoral

Graph-Theoretic and Machine Learning Frameworks for Material Modeling: From Discrete Decisions to Continuous Expressions

Phan, Nhon N.

The advancement of computational mechanics is increasingly dependent on our ability to bridge the gap between discrete experimental data and continuous constitutive laws while maintaining both mathematical rigor and physical interpretability. Although deep learning has offered unprecedented flexibility in approximating complex material responses, the transition from black-box models to trustworthy, efficient, and interpretable modeling tools remains a significant challenge. This dissertation presents a comprehensive modeling paradigm that automates the discovery, calibration, and characterization of material models across scales, focusing on a workflow that progresses from optimal experimental design decisions to the generation of expressive symbolic constitutive equations.

First, we introduce a deep reinforcement learning framework integrated with an enhanced Kalman filter to automate the design of experiments for the calibration of history-dependent models. By recasting the experimental protocol as a Markov decision process, we utilize a reinforcement learning agent to maximize information gain (measured via the Kullback–Leibler divergence), thereby transforming a sequence of discrete experimental decisions into an optimal path for minimizing parametric uncertainty in complex models such as elastoplasticity.

Second, we apply graph representation learning to contact networks of a dense sand assembly imaged by synchrotron micro-computed tomography throughout drained triaxial compression. Order embeddings and Monte Carlo tree search are combined to count and mine network motifs with up to 15 nodes—a scale previously unexplored in granular matter. This analysis reveals how chains, cycles, and densely connected subgraphs evolve across distinct strain regimes and how grain morphology modulates these topological signatures. The identified descriptors provide a microstructural basis for future constitutive models.

Third, we present a neural network guide for calibrating high-fidelity thermoelasticity models for the monoclinic energetic crystal 𝛽-1,3,5,7-tetranitro-1,3,5,7-tetrazocane. A physics-constrained neural network trained on molecular dynamics data is used to extract a complete set of isothermal third- and fourth-order elastic and higher-order thermal stress coefficients via Taylor expansion, yielding analytical constitutive approximations that respect the monoclinic material symmetry and provide significantly faster inference for computationally expensive continuum simulations.

To further resolve the expressivity-speed trade-off in high-performance solvers, we introduce HYDRA, an algorithm that generates symbolic hyperelasticity models. By leveraging a learnable projection to map strain onto a hyperplane and utilizing neural additive models to parameterize univariate bases, HYDRA distills overparameterized neural networks into compact symbolic expressions suitable for deployment in three-dimensional hydrocodes.

Finally, we provide a theoretical foundation for HYDRA by proving that projected neural additive models are universal approximators. By establishing their polynomial reproducing property through mathematical induction and invoking the Stone–Weierstrass theorem, the proof demonstrates that a linear combination of single-variable functions of projected inputs can approximate any continuous multi-variable function, bridging the gap between neural network expressivity and symbolic simplicity.

Collectively, these contributions form a modeling paradigm spanning data-efficient experimental design and state variable quantification to expressive, deployment-ready constitutive models, advancing the state of the art in computational mechanics at the intersection of graph theory, machine learning, and materials science.