Geometry-Informed Data-Driven Geomechanics with Physics Constraints

Columbia University

This talk gives an overview of geometry concepts introduced to empower machine learning (ML) algorithms in robustly predicting responses of natural materials. These algorithms fulfill physics principles, such as conservation laws, symmetry, and thermodynamics, and are effective for both big and limited data.

For fluid-infiltrating solids, experimental data for fluid flow and solid mechanics are of different qualities andquantities. To handle this multi-fidelity, a hybridized model-free/model-based paradigm is formulated forporomechanics applications where a model-free approach can avoid ad-hoc assumptions while model-based predictions improve the robustness. To enable predictions without constitutive laws, balance principles are considered as constraints in an inverse problem that searches for optimized data points. The concept of hyperplane is utilized to partition and store high dimensional data in a tree structure, which improves the scalability of the model-free solver for big data. This reduces the computation time from linear to logarithmic with respect to the number of data points. This model-free paradigm is then reformulated by imposing more geometrical properties consistent with the underlying geometry of data. These properties are learned via a manifold embedding scheme, which helps us handle scenarios where data are scarce, unevenly distributed, and noisy.

The second part of the talk focuses on geometric concepts and the enforcement of physics principles that make ML material models trustworthy. First, an interpretable ML-based algorithm is introduced that leverages the scalability of neural networks and the interpretability of symbolic regression. This algorithm uses a divide-and-conquer approach to discover fully interpretable, human-readable, and physics-constrained models. This framework is applied to discover plastic yield surfaces for geomaterials and hyperelastic models for soft materials. Then, reasoning mechanisms are incorporated by introducing a causal discovery algorithm. This algorithm identifies a directed acyclic graph that represents the causal relationships among material and topological descriptors, such as porosity, Fabric tensor, and network coordination number. This computational framework produces an ensemble of predictive models to incorporate uncertainty quantification. The application of this method is demonstrated by modeling the behavior of granular material assemblies.

Bahador Bahmani has recently obtained his PhD. in Civil Engineering and Engineering Mechanics from Columbia University. Bahador’s research lies at the intersection of computational geomechanics and machine learning, focusing on developing scalable and robust algorithms for forward and inverse problems in data-driven solid mechanics under multiphysics interactions such as heat transfer and fluid flow. He also has industrial research experience in computational mechanics, computational geometry, machine learning, and software development.