HYDRA: Symbolic feature engineering of overparameterized Eulerian hyperelasticity models for fast inference time

We introduce HYDRA, a learning algorithm that generates symbolic hyperelasticity models designed for running in 3D Eulerian hydrocodes that require fast and robust inference time. Classical deep learning methods require a large number of neurons to express a learned hyperelasticity model adequately. Large neural network models may lead to slower inference time when compared to handcrafted models expressed in symbolic forms. This expressivity-speed trade-off is not desirable for high-fidelity hydrocodes that require one inference per material point per time step. Pruning techniques may speed up inference by removing/deactivating less important neurons, but often at a non-negligible expense of expressivity and accuracy. In this work, we introduce a post-hoc procedure to convert a neural network model into a symbolic one to reduce inference time. Rather than directly confronting NP-hard symbolic regression in the ambient strain space, HYDRA leverages a data-driven projection to map strain onto a hyperplane and a neural additive model to parameterize the hyperplane via univariate bases. This setting enables us to convert the univariate bases into symbolic forms via genetic programming with explicit control of the expressivity-speed trade-off. Additionally, the availability of analytical models provides the benefits of ensuring the enforcement of physical constraints (e.g., material frame indifference, material symmetry, growth condition) and enabling symbolic differentiation that may further reduce the memory requirement of high-performance solvers. Benchmark numerical examples of material point simulations for shock loading in $beta$-octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine ($beta$-HMX) are performed to assess the practicality of using the discovered machine learning models for high-fidelity simulations.