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This article introduces a manifold embedding data-driven paradigm to solve small-and finite-strain elasticity problems without a conventional constitutive law. This formulation follows the classical 6 data-driven paradigm by seeking the solution that obeys the balance of linear momentum and compatibility conditions, while remaining consistent with the material data through minimizing a distance measure. Our key point of departure is the introduction of a global manifold embedding as a means to learn the geometrical trend of the constitutive data mathematically represented by a smooth manifold. By training an invertible neural network to embed the data of an underlying constitutive manifold onto a Euclidean space, we reformulate the local distance-minimization problem that requires a computationally intensive combinatorial search to identify the optimal data points closest to the conservation law with a cost-efficient projection step. Meanwhile, numerical experiments performed on path-independent elastic materials of different material symmetries suggest that the geometrical inductive bias learned by the neural network is helpful to ensure more consistent predictions when dealing with data sets of limited sizes or those with missing data.

We present a machine learning framework to train and validate neural networks to predict the anisotropic elastic response of a monoclinic organic molecular crystal known as Octogen (β-HMX) in the geometrical nonlinear regime. A filtered molecular dynamic (MD) simulations database is used to train neural networks with a Sobolev norm that uses the stress measure and a reference configuration to deduce the elastic stored energy functional. To improve the accuracy of the elasticity tangent predictions originating from the learned stored energy, a transfer learning technique is used to introduce additional tangential constraints from the data while necessary conditions (e.g. strong ellipticity, crystallographic symmetry) for the correctness of the model are either introduced as additional physical constraints or incorporated in the validation tests. Assessment of the neural networks is based on (1) the accuracy with which they reproduce the bottom-line constitutive responses predicted by MD, (2) the robustness of the models measured by detailed examination of their stability and uniqueness, and (3) the admissibility of the predicted responses with respect to mechanics principles in the finite-deformation regime. We compare the neural networks’training efficiency under different Sobolev constraints and assess the models’ accuracy and robustness against MD benchmarks for β-HMX.

Conventionally, neural network constitutive laws for path-dependent elasto-plastic solids are trained via supervised learning performed on recurrent neural networks, with the time history of strain as input and the stress as input. However, training a neural network to replicate path-dependent constitutive responses require significantly more amount of data due to path dependence. This demand on diverse and abundance of accurate data, as well as the lack of interpretability to guide the data generation process, could become major roadblocks for engineering applications. In this work, we attempt to simplify these training processes and improve the interpretability of the trained models by breaking down the training of material models into multiple supervised machine learning programs for elasticity, initial yielding, and hardening laws that can be conducted sequentially. To predict pressure-sensitivity and rate dependence of the plastic responses, we reformulate the Hamliton-Jacobi equation such that the yield function is parametrized in product space spanned by the principle stress, the accumulated plastic strain, and time. To test the versatility of the neural network meta-modeling framework, we conduct multiple numerical experiments where neural networks are trained and validated against (1) data generated from known benchmark models, (2) data obtained from physical experiments, and (3) data inferred from homogenizing sub-scale direct numerical simulations of microstructures. The neural network model is also incorporated into an offline FFT-FEM model to improve the efficiency of the multiscale calculations.

This paper presents a computational framework that generates ensemble predictive mechanics models with uncertainty quantification (UQ). We first develop a causal discovery algorithm to infer causal relations among time-history data measured during each representative volume element (RVE) simulation through a directed acyclic graph. With multiple plausible sets of causal relationships estimated from multiple RVE simulations, the predictions are propagated in the derived causal graph while using a deep neural network equipped with dropout layers as a Bayesian approximation for UQ. We select two representative numerical examples (traction-separation laws for frictional interfaces, elastoplasticity models for granular assembles) to examine the accuracy and robustness of the proposed causal discovery method for the common material law predictions in civil engineering applications.
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We present a hybrid model/model-free data-driven approach to solve poroelasticity problems. Extending the data-driven modeling framework originated from \citet{kirchdoerfer2016data}, we introduce one model-free and two hybrid model-based/data-driven formulations capable of simulating the coupled diffusion-deformation of fluid-infiltrating porous media with different amounts of available data. To improve the efficiency of the model-free data search, we introduce a distance-minimized algorithm accelerated by a k-dimensional tree search. To handle the different fidelities of the solid elasticity and fluid hydraulic constitutive responses, we introduce a hybridized model in which either the solid and the fluid solver can switch from a model-based to a model-free approach depending on the availability and the properties of the data. Numerical experiments are designed to verify the implementation and compare the performance of the proposed model to other alternatives.

Supervised machine learning via artificial neural networks (ANN) has gained significant popularity for many geomechanics applications that involves multi-phase flow and poromechanics. For unsaturated poromechanics problems, the multi-physics nature and the complexity of the hydraulic laws make it difficult to design the optimal setup, architecture, and hyper-parameters of the deep neural networks. This paper presents a meta-modeling approach that utilizes deep reinforcement learning (DRL) to automatically discover optimal neural network settings that maximize a pre-defined performance metric for the machine learning constitutive laws. This meta-modeling framework is cast as a Markov Decision Process (MDP) with well-defined states (subsets of states representing the proposed neural network (NN) settings), actions, and rewards. Following the selection rules, the artificial intelligence (AI) agent, represented in DRL via NN, self-learns from taking a sequence of actions and receiving feedback signals (rewards) within the selection environment. By utilizing the Monte Carlo Tree Search (MCTS) to update the policy/value networks, the AI agent replaces the human modeler to handle the otherwise time-consuming trial-and-error process that leads to the optimized choices of setup from a high-dimensional parametric space. This approach is applied to generate two key constitutive laws for the unsaturated poromechanics problems: (1) the path-dependent retention curve with distinctive wetting and drying paths. (2) The flow in the micropores, governed by an anisotropic permeability tensor. Numerical experiments have shown that the resultant ML-generated material models can be integrated into a finite element (FE) solver to solve initial-boundary-value problems as replacements of the hand-craft constitutive laws. Keywords Deep reinforcement learning, neural network settings, unsaturated porous media, retention curve, anisotropic permeability.

We introduce a deep learning framework designed to train smoothed elastoplasticity models with interpretable components, such as stored elastic energy function, field surface, and plastic flow that may evolve based on a set of deep neural network predictions. By recasting the yield function as an evolving level set, we introduce a deep learning approach to deduce the solutions of the Hamilton-Jacobi equation that governs the hardening/softening mechanism. This machine learning hardening law may recover any classical hand-crafted hardening rules and discover new mechanisms that are either unbeknownst or difficult to express with mathematical expressions. Leveraging Sobolev training to gain control over the derivatives of the learned functions, the resultant machine learning elastoplasticity models are thermody-namically consistent, interpretable, while exhibiting excellent learning capacity. Using a 3D FFT solver to create a polycrystal database, numerical experiments are conducted and the implementations of each component of the models are individually verified. Our numerical experiments reveal that this new approach provides more robust and accurate forward predictions of cyclic stress paths than those obtained from black-box deep neural network models such as the recurrent neural network, the 1D convolutional neural network, and the multi-step feed-forward models.

We present a reduced-dimensional proper orthogonal decomposition (POD) solver to accelerate discrete element method (DEM) simulations of the granular mixing problem. We employ the method of snapshots to create a low-dimensional solution space from previous DEM simulations. By reducing the dimensionality of the problem, we accelerate the calculations of the incremental solution with fewer degrees of freedom (DOF), while enabling a larger stable time step due to the filtering of low-energy mode. We analyze two feasible strategies to generate the reduced-dimensional basis, one generating by finding the orthogonal basis from the global snapshots captured at the same location in the parametric domains ; another one employing the known POD bases from the closest known cases. Our results show that, when POD bases are generated via the local strategy, the reduced-order model is a more efficient alternative to the full-scale simulations for extrapolating behaviors in the parametric domain. Numerical examples of granular mixing problems are presented to demonstrate the efficiency and accuracy of the proposed approach.

The evaluation of constitutive models, especially for high-risk and high-regret engineering applications, requires efficient and rigorous third-party calibration, validation and falsification. While there are numerous efforts to develop paradigms and standard procedures to validate models, difficulties may arise due to the sequential, manual and often biased nature of the commonly adopted calibration and validation processes, thus slowing down data collections, hampering the progress towards discovering new physics, increasing expenses and possibly leading to misinterpretations of the credibility and application ranges of proposed models. This work attempts to introduce concepts from game theory and machine learning techniques to overcome many of these existing difficulties. We introduce an automated meta-modeling game where two competing AI agents systematically generate experimental data to calibrate a given constitutive model and to explore its weakness, in order to improve experiment design and model robustness through competition. The two agents automatically search for the Nash equilibrium of the meta-modeling game in an adversarial reinforcement learning framework without human intervention. In particular, a protagonist agent seeks to find the more effective ways to generate data for model calibrations, while an adversary agent tries to find the most devastating test scenarios that expose the weaknesses of the constitutive model calibrated by the protagonist. By capturing all possible design options of the laboratory experiments into a single decision tree, we recast the design of experiments as a game of combinatorial moves that can be resolved through deep reinforcement learning by the two competing players. Our adversarial framework emulates idealized scientific collaborations and competitions among researchers to achieve better understanding of the application range of the learned material laws and prevent misinterpretations caused by conventional AI-based third-party validation. Numerical examples are given to demonstrate the wide applicability of the proposed meta-modeling game with adversarial attacks on both human-crafted constitutive models and machine learning models.

We present a machine learning approach that integrates geometric deep learning and Sobolev training to generate a family of finite strain anisotropic hyperelastic models that predict the homogenized responses of polycrystals previously unseen during the training. While hand-crafted hyperelasticity models often incorporate homogenized measures of microstructural attributes, such as the porosity or the averaged orientation of constitutes, these measures may not adequately reflect the topological structures of the attributes. We fill this knowledge gap by introducing the concept of the weighted graph as a new high-dimensional descriptor that represents topological information, such as the connectivity of anisotropic grains in an assemble. By leveraging a graph convolutional deep neural network in a hybrid machine learning architecture previously used in Frankel et al. 2019, the artificial intelligence extracts low-dimensional features from the weighted graphs and subsequently learns the influence of these low-dimensional features on the resultant stored elastic energy functionals. To ensure smoothness and prevent unintentionally generating a non-convex stored energy functional, we adopt the Sobolev training method for neural networks such that a stress measure is obtained implicitly by taking directional derivatives of the trained energy functional. Results from numerical experiments suggest that Sobolev training is capable of generating a hyperelastic energy functional that predicts both the elastic energy and stress measures more accurately than the classical training that minimizes L2 norm. Verification exercises against unseen benchmark FFT simulations and phase-field fracture simulations using the geometric learning generated elastic energy functional are conducted to demonstrate the quality of the predictions.

This paper examines the frame-invariance (and the lack thereof) exhibited in simulated anisotropic elasto-plastic responses generated from supervised machine learning of classical multi-layer and informed-graph-based neural networks, and proposes different remedies to fix this drawback. The inherent hierarchical relations among physical quantities and state variables in an elasto-plasticity model are first represented as directed graphs, where three variations of the graph are tested. While feed-forward neural networks are used to train path-independent constitutive relations (e.g., elasticity), recurrent neural networks are used to replicate responses that depends on the deformation history, i.e. or path dependent. In dealing with the objectivity deficiency, we use the spectral form to represent tensors and, subsequently, three metrics, the Euclidean distance between the Euler Angles, the distance from the identity matrix, and geodesic on the unit sphere in Lie algebra, can be employed to constitute objective functions for the supervised machine learning. In this, the aim is to minimize the measured distance between the true and the predicted 3D rotation entities. Following this, we conduct numerical experiments on how these met-rics, which are theoretically equivalent, may lead to differences in the efficiency of the supervised machine learning as well as the accuracy and robustness of the resultant models. Neural network models trained with tensors represented in component form for a given Cartesian coordinate system are used as a benchmark. Our numerical tests show that, even given the same amount of information and data, the quality of the anisotropic elasto-plasticity model is highly sensitive to the way tensors are represented and measured. The results reveal that using a loss function based on geodesic on the unit sphere in Lie algebra together with an informed directed graph yield significantly more accurate rotation prediction than the other tested approaches.

In numerical simulations of geomechanics problems, a grand challenge consists of overcoming the difficulties in making accurate and robust predictions by revealing the true mechanisms in particle interactions, fluid flow inside pore spaces, and hydromechanical coupling effect between the solid and fluid constituents, from microscale to mesoscale, and to macroscale. While simulation tools incorporating subscale physics can provide detailed insights and accurate material properties to macroscale simulations via computational homogenizations, these numerical simulations are often too computational demanding to be directly used across multiple scales. Recent breakthroughs of Artificial Intelligence (AI) via machine learning have great potential to overcome these barriers, as evidenced by their great success in many applications such as image recognition, natural language processing, and strategy exploration in games. The AI can achieve super-human performance level in a large number of applications, and accomplish tasks that were thought to be not feasible due to the limitations of human and previous computer algorithms. Yet, machine learning approaches can also suffer from overfitting, lack of interpretability, and lack of reliability. Thus the application of machine learning into generation of accurate and reliable surrogate constitutive models for geomaterials with multiscale and multiphysics is not trivial. For this purpose, we propose to establish an integrated modeling process for automatic designing, training, validating, and falsifying of constitutive models, or "metamodeling". This dissertation focuses on our efforts in laying down step-by-step the necessary theoretical and technical foundations for the multiscale metamodeling framework.
The first step is to develop multiscale hydromechanical homogenization frameworks for both bulk granular materials and granular interfaces, with their behaviors homogenized from subscale microstructural simulations. For efficient simulations of field-scale geomechanics problems across more than two scales, we develop a hybrid data-driven method designed to capture the multiscale hydro-mechanical coupling effect of porous media with pores of various different sizes. By using sub-scale simulations to generate database to train material models, an offline homogenization procedure is used to replace the up-scaling procedure to generate path-dependent cohesive laws for localized physical discontinuities at both grain and specimen scales.
To enable AI in taking over the trial-and-error tasks in the constitutive modeling process, we introduce a novel “metamodeling” framework that employs both graph theory and deep reinforcement learning (DRL) to generate accurate, physics compatible and interpretable surrogate machine learning models. The process of writing constitutive models is simplified as a sequence of forming graph edges with the goal of maximizing the model score (a function of accuracy, robustness and forward prediction quality). By using neural networks to estimate policies and state values, the computer agent is able to efficiently self-improve the constitutive models generated through self-playing.
To overcome the obstacle of limited information in geomechanics, we improve the efficiency in utilization of experimental data by a multi-agent cooperative metamodeling framework to provide guidance on database generation and constitutive modeling at the same time. The modeler agent in the framework focuses on evaluating all modeling options (from domain experts’ knowledge or machine learning) in a directed multigraph of elasto-plasticity theory, and finding the optimal path that links the source of the directed graph (e.g., strain history) to the target (e.g., stress). Meanwhile, the data agent focuses on collecting data from real or virtual experiments, interacts with the modeler agent sequentially and generates the database for model calibration to optimize the prediction accuracy. Finally, we design a non-cooperative meta-modeling framework that focuses on automatically developing strategies that simultaneously generate experimental data to calibrate model parameters and explore weakness of a known constitutive model until the strengths and weaknesses of the constitutive law on the application range can be identified through competition. These tasks are enabled by a zero-sum reward system of the metamodeling game and robust adversarial reinforcement learning techniques.

We introduce a multi-agent meta-modeling game to generate data, knowledge, and models that make predictions on constitutive responses of elasto-plastic materials. We introduce a new concept from graph theory where a modeler agent is tasked with evaluating all the modeling options recast as a directed multigraph and find the optimal path that links the source of the directed graph (e.g. strain history) to the target (e.g. stress) measured by an objective function. Meanwhile, the data agent, which is tasked with generating data from real or virtual experiments (e.g. molecular dynamics, discrete element simulations), interacts with the modeling agent sequentially and uses reinforcement learning to design new experiments to optimize the prediction capacity. Consequently, this treatment enables us to emulate an idealized scientific collaboration as selections of the optimal choices in a decision tree search done automatically via deep reinforcement learning.

We introduce a multi-agent meta-modeling game to generate data, knowledge, and models that make predictions on constitutive responses of elasto-plastic materials. We introduce a new concept from graph theory where a modeler agent is tasked with evaluating all the modeling options recast as a directed multigraph and find the optimal path that links the source of the directed graph (e.g. strain history) to the target (e.g. stress) measured by an objective function. Meanwhile, the data agent, which is tasked with generating data from real or virtual experiments (e.g. molecular dynamics, discrete element simulations), interacts with the modeling agent sequentially and uses reinforcement learning to design new experiments to optimize the prediction capacity. Consequently, this treatment enables us to emulate an idealized scientific collaboration as selections of the optimal choices in a decision tree search done automatically via deep reinforcement learning.

Many engineering applications and geological processes involve embedded discontinuities in 6 porous media across multiple length scales (e.g. rock joints, grain boundaries, deformation bands and 7 faults). Understanding the multiscale path-dependent hydro-mechanical responses of these interfaces across 8 length scales is of ultimate importance for applications such as CO 2 sequestration, hydraulic fracture and 9 earthquake rupture dynamics. While there exist mathematical frameworks such as extended finite element 10 and assumed strain to replicate the kinematics of the interfaces, modeling the cyclic hydro-mechanical 11 constitutive responses of the interfaces remains a difficult task. This paper presents a semi-data-driven 12 multiscale approach that obtains both the traction-separation law and the aperture-porosity-permeability 13 relation from micro-mechanical simulations performed on representative elementary volumes in the finite 14 deformation range. To speed up the multiscale simulations, the incremental constitutive updates of the 15 mechanical responses are obtained from discrete element simulations at the representative elementary vol-16 ume whereas the hydraulic responses are generated from a neural network trained with data from lattice 17 Boltzmann simulations. These responses are then linked to a macroscopic dual-permeability model. This 18 approach allows one to bypass the need of deriving multi-physical phenomenological laws for complex 19 loading paths. More importantly, it enables the capturing of the evolving anisotropy of the permeabilities 20 of the macro-and micro-pores. A set of numerical experiments are used to demonstrate the robustness of 21 the proposed model. 22

This article presents a new test prototype that leverages the 3D printing technique to create artificial particle assembles to provide auxiliary evidences that supports the validation procedure. The prototype test first extracts particle shape features from micro-CT images of a real sand grain and replicates the geometrical features of sand grain using a 3D printer. The quantitative measurements of the particle shape descriptors reveal that the synthetic particles inherit some attributes such as aspect ratio and sparseness of the real materials while exhibiting marked differences for sphericity and convexity. While it is not sufficient to consider the printed particle assembles a replica of the real sand, the repeatable manufacture process provides convention tools to generate additional data that supports the validation procedure for particulate simulations. Oedometric compression tests are conducted on a specimen composed of the printed particles of identical size and shape to create benchmark cases for calibrating and validating discrete element models. Results from digital image correlation on the synthetic sand assemblies reveal that the fracture and fragmentation of the synthetic particles are minor, which in return makes particle position tracking possible. As our prototype test and research data are designed to be open-source, the dataset and the prototype work will open doors for modelers to design further controlled experiments using synthetic granular materials such that the individual influence of each morphological feature of granular assemblies (e.g. shape and size distribution, void ratio, fabric orientation) can be individually tested without being simultaneously affected by other variables.

This paper presents a new meta-modeling framework to employ deep reinforcement learning (DRL) to generate mechanical constitutive models for interfaces. The constitutive models are conceptualized as information flow in directed graphs. The process of writing constitutive models is simplified 8 as a sequence of forming graph edges with the goal of maximizing the model score (a function of accuracy, robustness and forward prediction quality). Thus meta-modeling can be formulated as a Markov decision process with well-defined states, actions, rules, objective functions and rewards. By using neural networks to estimate policies and state values, the computer agent is able to efficiently self-improve the constitutive model it generated through self-playing, in the same way AlphaGo Zero (the algorithm that outplayed the world champion in the game of Go) improves its gameplay. Our numerical examples show that this automated meta-modeling framework not only produces models which outperform existing cohesive models on benchmark traction-separation data, but is also capable of detecting hidden mechanisms among micro-structural features and incorporating them in constitutive models to improve the forward prediction accuracy, which are difficult tasks to do manually.