Capturing path- and rate-dependent behaviors of solids, such as creeping, plastic deformation, damage, and fracture, often requires interpreting and quantifying relationships among the histories of variables, such as dislocation density and porosity. This relational information is then represented by mathematical models composed of differentiable functions. In this talk, we explore (1) how these relationships can be represented by directed graphs and deduced through deep reinforcement learning1, (2) how they can be trained on a library of existing material models represented as multidimensional meshes2, and (3) how they can be re-interpreted as classical material models through symbolic pruning3. To understand the mechanisms of constitutive behaviors, we represent physical quantities as vertices and determine their relations using Monte Carlo tree search. To facilitate learning from limited data4, we leverage existing models to construct priors that enhance the likelihood of plausible predictions. By representing material models as meshes, we train a latent diffusion model that uses previous material models and experimental data to guide the reverse generation of new models. For applications where fast inference and interpretability are essential, we introduce a projected neural additive method for learning symbolic constitutive models as expression trees. This approach expresses the material response as a linear combination of univariate functions of projected variables. This technique enables us to search for hyperelasticity in high-dimensional spaces without sacrificing neural network expressivity. We show that the proposed model can reproduce polynomials of arbitrary order and dimension and thus achieve universal approximation by the Stone-Weierstrass theorem. Through a series of 1D post-hoc symbolic regressions, we obtain symbolic material models that significantly reduce the inference time for hydrocodes. The pros and cons of these techniques for various practical applications will be discussed. Together, these examples highlight how efficient representation of materials is critical for balancing the need for accuracy, interpretability, and computational efficiency in the learning of constitutive models.

Reference
1. Wang, K. & Sun, W. Meta-modeling game for deriving theory-consistent, microstructure-based traction–separation laws via deep reinforcement learning. Comput. Methods Appl. Mech. Eng. 346, 216–241 (2019).
2. Xiao, M., Poliner, J. & Sun, W. Geometric learning for computational mechanics Part IV: Efficient mesh-based plasticity from a domain-specific foundation model. Comput. Methods Appl. Mech. Eng. 446, 118310 (2025).
3. Phan, N. N., Sun, W. & Clayton, J. D. HYDRA: Symbolic feature engineering of overparameterized Eulerian hyperelasticity models for fast inference time. Comput. Methods Appl. Mech. Eng. 437, 117792 (2025).
4. Gavris, G. B. & Sun, W. Discovering neural elastoplasticity from kinematic observations. Proc. Natl. Acad. Sci. 122, e2508732122 (2025).