Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework to inform denoising diffusion models of underlying constraints on such generated samples during model training. Our approach improves the alignment of the generated samples with the imposed constraints and significantly outperforms existing methods without affecting inference speed. Additionally, our findings suggest that incorporating such constraints during training provides a natural regularization against overfitting. Our framework is easy to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives.

This review article highlights state-of-the-art data-driven techniques to discover, encode, surrogate, or emulate constitutive laws that describe the path-independent and path-dependent response of solids. Our objective is to provide an organized taxonomy to a large spectrum of methodologies developed in the past decades and to discuss the benefits and drawbacks of the various techniques for interpreting and forecasting mechanics behavior across different scales. Distinguishing between machine-learning-based and model-free methods, we further categorize approaches based on their interpretability and on their learning process/type of required data, while discussing the key problems of generalization and trustworthiness. We attempt to provide a road map of how these can be reconciled in a data-availability-aware context. We also touch upon relevant aspects such as data sampling techniques, design of experiments, verification, and validation.

Conventional neural network elastoplasticity models are often perceived as lacking interpretability. This paper introduces a two-step machine learning approach that returns mathematical models interpretable by human experts. In particular, we introduce a surrogate model where yield surfaces are expressed in terms of a set of single-variable feature mappings obtained from supervised learning. A post-processing step is then used to re-interpret the set of single-variable neural network mapping functions into mathematical form through symbolic regression. This divide-and-conquer approach provides several important advantages. First, it enables us to overcome the scaling issue of symbolic regression algorithms. From a practical perspective, it enhances the portability of learned models for partial differential equation solvers written in different programming languages. Finally, it enables us to have a concrete understanding of the attributes of the materials, such as convexity and symmetries of models, through automated derivations and reasoning. Numerical examples have been provided, along with an open-source code to enable third-party validation.

Understanding and predicting rock mechanical properties, such as elastic moduli, plays a crucial role across a range of geoscience and engineering fields including energy resources engineering (Sone and Zoback, 2013; Madhubabu et al., 2016), geotechnical engineering (Zhang et al., 2021), and seismology (Byerlee and Brace, 1968). These properties govern how rocks react to in-situ stresses, shaping the macroscopic behaviors of geological structures. Accurate characterization of these properties is therefore instrumental in predicting phenomena such as seismic wave propagation, reservoir behavior, and slope stability. Fundamentally, these macroscopic behaviors originate from the intricate features at the microscopic scale. Specifically, the effective elastic moduli of rocks are influenced by three main factors: (1) the composition of pores and minerals, (2) the properties of these constituents, and (3) the geometric details of the rock's microstructure (Mavko et al., 2020). While the first two factors can be relatively straightforwardly measured and characterized, capturing the complexity of the geometric details often presents a substantial challenge. Historically, traditional micromechanics homogenizations and rock physics relied on empirical relationships or theoretical models based on idealized microstructures to estimate rock properties (Li and Wang, 2008; Han et al., 1986; Mindlin, 1949; Hill, 1952).

The shapes and morphological features of grains in sand assemblies have far-reaching implications in many engineering applications, such as geotechnical engineering, computer animations, petroleum engineering, and concentrated solar power. Yet, our understanding of the influence of grain geometries on macroscopic response is often only qualitative, due to the limited availability of high-quality 3D grain geometry data. In this paper, we introduce a denoising diffusion algorithm that uses a set of point clouds collected from the surface of individual sand grains to generate grains in the latent space. By employing a point cloud autoencoder, the three-dimensional point cloud structures of sand grains are first encoded into a lower-dimensional latent space. A generative denoising diffusion probabilistic model is trained to produce synthetic sand that maximizes the log-likelihood of the generated samples belonging to the original data distribution measured by a Kullback-Leibler divergence. Numerical experiments suggest that the proposed method is capable of generating realistic grains with morphology, shapes and sizes consistent with the training data inferred from an F50 sand database. We then use a rigid contact dynamic simulator to pour the synthetic sand in a confined volume to form granular assemblies in a static equilibrium state with targeted distribution properties. To ensure third-party validation, 50,000 synthetic sand grains and the 1,542 real synchrotron microcomputed tomography (SMT) scans of the F50 sand, as well as the granular assemblies composed of synthetic sand grains are made available in an open-source repository.

In this paper, we introduce a denoising diffusion algorithm to discover microstructures with nonlinear fine-tuned properties. Denoising diffusion probabilistic models are generative models that use diffusion-based dynamics to gradually denoise images and generate realistic synthetic samples. By learning the reverse of a Markov diffusion process, we design an artificial intelligence to efficiently manipulate the topology of microstructures to generate a massive number of prototypes that exhibit constitutive responses sufficiently close to designated nonlinear constitutive responses. To identify the subset of microstructures with sufficiently precise fine-tuned properties, a convolutional neural network surrogate is trained to replace high-fidelity finite element simulations to filter out prototypes outside the admissible range. The results of this study indicate that the denoising diffusion process is capable of creating microstructures of fine-tuned nonlinear material properties within the latent space of the training data. More importantly, the resulting algorithm can be easily extended to incorporate additional topological and geometric modifications by introducing high-dimensional structures embedded in the latent space. The algorithm is tested on the open-source mechanical MNIST data set. Consequently, this algorithm is not only capable of performing inverse design of nonlinear effective media but also learns the nonlinear structure-property map to quantitatively understand the multiscale interplay among the geometry and topology and their effective macroscopic properties.

The composition of a macroscopic plasticity model often requires the following steps. First, there are observations of causality relations deduced by modelers to hypothesize mechanisms that lead to the plastic flow. These causality relations along with constraints inferred from physics and universally accepted principles lead to mathematical equations. For instance, the family of Gurson models employs the observation of void growth to employ the yield surface (Gurson, 1977). Crystal plasticity models relate the plastic flow with slip systems to predict the anisotropic responses of single crystals (Rice, 1971; Uchic et al., 2004; Clayton, 2010; Ma and Sun, 2020; Ma et al., 2021). Granular plasticity models propose theories that relate the fabric of force chains and porosity to the onset of plastic yielding and the resultant plastic flow (Cowin, 1985; Kuhn et al., 2015; Wang and Sun, 2018; Sun et al., 2022). Finally, the mathematical equations are then either used directly in engineering analysis and designs (e.g. the Mohr-Coulomb envelope) or are incorporated into a boundary value problem in which the approximation solution can be obtained from a partial differential equation solver that provides incremental updates of stress-strain relations. However, a subtle but significant limitation of this paradigm is that it imposes the burdens on modelers of being able to describe the mechanisms verbally via terminologies or atomic facts (cf.

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 (DAG). 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 uncertainty quantification. 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.

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 are 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). 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.