Analyses of internal structures and defects in materials using physics-informed neural networks

Effects of Surrogate Hybridization and Adaptive Sampling for Simulation-Based Optimization

Process simulators are essential for modeling of complex processes; however, optimization of expensive models remains challenging due to lack of equations, simulation cost, and lack of convergence guarantees. To tackle these challenges, surrogate modeling and surrogate-based optimization methods have been proposed. Most commonly, surrogates are treated as black-box models, while recently hybrid surrogates have gained popularity. In this work, we assess two main methodologies: (a) optimization of surrogates trained using a set of fixed a priori samples using deterministic solvers, and (b) adaptive sampling-based optimization, which leverages surrogate predictions to guide the search process. Across both methods, we systematically compare the effect of black-box versus hybrid surrogates, that utilize a "model-correction" architecture combining different fidelity data. Through mathematical benchmarks with up to ten dimensions, and two engineering case studies for process design of an extractive distillation simulation model and an adsorption simulation model, we present the effects of sampling quantity, dimensionality, formulation, and hybridization on solution convergence, reliability, and CPU efficiency. Our results show that hybrid modeling improves surrogate robustness and reduces solution variability with fewer samples, though it increases optimization costs. Additionally, adaptive sampling methods are more efficient and consistent than fixed-sampling surrogate strategies, even across different sampling and dimensionality scenarios.

Recent Advances in Food Drying Modeling: Empirical to Multiscale Physics-Informed Neural Networks

Food insecurity is a major global challenge. Food preservation, particularly through drying, presents a promising solution to enhance food security and minimize waste. Fruits and vegetables contain 80%-90% water, and much of this is removed during drying. However, structural changes across multiple length scales occur during drying, compromising stability and affecting quality. Understanding these changes is essential, and several modeling techniques exist to analyze them, including empirical modeling, physics-based computational methods, purely data-driven machine learning approaches, and physics-informed neural network (PINN) models. Although empirical methods are straightforward to implement, their limited generalizability and lack of physical insights have led to the development of physics-based computational methods. These methods can achieve high spatiotemporal resolution without requiring experimental investigations. However, their complexity and high computational costs have prompted the exploration of data-driven machine learning models for drying processes, which involve comparatively lower computational costs and are more straightforward to execute. Nonetheless, their poor predictive ability with sparse data has restricted their application, leading to a hybrid modeling approach: PINN, which merges physical insights with data-driven machine learning techniques. This method still holds significant potential for advancements in food drying modeling. Therefore, this study aims to conduct a comprehensive literature review of state-of-the-art conventional drying modeling techniques, such as empirical, physics-based computational, and pure data-driven machine learning techniques, and explores the potential of the PINN approach for overcoming the limitations associated with conventional drying modeling strategies.

Identifying Heterogeneous Micromechanical Properties of Biological Tissues via Physics-Informed Neural Networks

The heterogeneous micromechanical properties of biological tissues have profound implications across diverse medical and engineering domains. However, identifying full-field heterogeneous elastic properties of soft materials using traditional engineering approaches is fundamentally challenging due to difficulties in estimating local stress fields. Recently, there has been a growing interest in data-driven models for learning full-field mechanical responses, such as displacement and strain, from experimental or synthetic data. However, research studies on inferring full-field elastic properties of materials, a more challenging problem, are scarce, particularly for large deformation, hyperelastic materials. Here, a physics-informed machine learning approach is proposed to identify the elasticity map in nonlinear, large deformation hyperelastic materials. This study reports the prediction accuracies and computational efficiency of physics-informed neural networks (PINNs) in inferring the heterogeneous elasticity maps across materials with structural complexity that closely resemble real tissue microstructure, such as brain, tricuspid valve, and breast cancer tissues. Further, the improved architecture is applied to three hyperelastic constitutive models: Neo-Hookean, Mooney Rivlin, and Gent. The improved network architecture consistently produces accurate estimations of heterogeneous elasticity maps, even when there is up to 10% noise present in the training data.

Balance equations for physics-informed machine learning

Using traditional machine learning (ML) methods may produce results that are inconsistent with the laws of physics. In contrast, physics-based models of complex physical, biological, or engineering systems incorporate the laws of physics as constraints on ML methods by introducing loss terms, ensuring that the results are consistent with these laws. However, accurately deriving the nonlinear and high order differential equations to enforce various complex physical laws is non-trivial. There is a lack of comprehensive guidance on the formulation of residual loss terms. To address this challenge, this paper proposes a new framework based on the balance equations, which aims to advance the development of PIML across multiple domains by providing a systematic approach to constructing residual loss terms that maintain the physical integrity of PDE solutions. The proposed balance equation method offers a unified treatment of all the fundamental equations of classical physics used in models of mechanical, electrical, and chemical systems and guides the derivation of differential equations for embedding physical laws in ML models. We show that all of these equations can be derived from a single equation known as the generic balance equation, in conjunction with specific constitutive relations that bind the balance equation to a particular domain. We also provide a few simple worked examples how to use our balance equation method in practice for PIML. Our approach suggests that a single framework can be followed to incorporate physics into ML models. This level of generalization may provide the basis for more efficient methods of developing physics-based ML for complex systems.

Coagulo-Net: Enhancing the mathematical modeling of blood coagulation using physics-informed neural networks

Blood coagulation, which involves a group of complex biochemical reactions, is a crucial step in hemostasis to stop bleeding at the injury site of a blood vessel. Coagulation abnormalities, such as hypercoagulation and hypocoagulation, could either cause thrombosis or hemorrhage, resulting in severe clinical consequences. Mathematical models of blood coagulation have been widely used to improve the understanding of the pathophysiology of coagulation disorders, guide the design and testing of new anticoagulants or other therapeutic agents, and promote precision medicine. However, estimating the parameters in these coagulation models has been challenging as not all reaction rate constants and new parameters derived from model assumptions are measurable. Although various conventional methods have been employed for parameter estimation for coagulation models, the existing approaches have several shortcomings. Inspired by the physics-informed neural networks, we propose Coagulo-Net, which synergizes the strengths of deep neural networks with the mechanistic understanding of the blood coagulation processes to enhance the mathematical models of the blood coagulation cascade. We assess the performance of the Coagulo-Net using two existing coagulation models with different extents of complexity. Our simulation results illustrate that Coagulo-Net can efficiently infer the unknown model parameters and dynamics of species based on sparse measurement data and data contaminated with noise. In addition, we show that Coagulo-Net can process a mixture of synthetic and experimental data and refine the predictions of existing mathematical models of coagulation. These results demonstrate the promise of Coagulo-Net in enhancing current coagulation models and aiding the creation of novel models for physiological and pathological research. These results showcase the potential of Coagulo-Net to advance computational modeling in the study of blood coagulation, improving both research methodologies and the development of new therapies for treating patients with coagulation disorders.

Identifying heterogeneous micromechanical properties of biological tissues via physics-informed neural networks

The heterogeneous micromechanical properties of biological tissues have profound implications across diverse medical and engineering domains. However, identifying full-field heterogeneous elastic properties of soft materials using traditional engineering approaches is fundamentally challenging due to difficulties in estimating local stress fields. Recently, there has been a growing interest in using data-driven models to learn full-field mechanical responses such as displacement and strain from experimental or synthetic data. However, research studies on inferring full-field elastic properties of materials, a more challenging problem, are scarce, particularly for large deformation, hyperelastic materials. Here, we propose a physics-informed machine learning approach to identify the elasticity map in nonlinear, large deformation hyperelastic materials. We evaluate the prediction accuracies and computational efficiency of physics-informed neural networks (PINNs) by inferring the heterogeneous elasticity maps across three materials with structural complexity that closely resemble real tissue patterns, such as brain tissue and tricuspid valve tissue. We further applied our improved architecture to three additional examples of breast cancer tissue and extended our analysis to three hyperelastic constitutive models: Neo-Hookean, Mooney Rivlin, and Gent. Our selected network architecture consistently produced highly accurate estimations of heterogeneous elasticity maps, even when there was up to 10% noise present in the training data.

Identifying constitutive parameters for complex hyperelastic materials using physics-informed neural networks

Identifying constitutive parameters in engineering and biological materials, particularly those with intricate geometries and mechanical behaviors, remains a longstanding challenge. The recent advent of physics-informed neural networks (PINNs) offers promising solutions, but current frameworks are often limited to basic constitutive laws and encounter practical constraints when combined with experimental data. In this paper, we introduce a robust PINN-based framework designed to identify material parameters for soft materials, specifically those exhibiting complex constitutive behaviors, under large deformation in plane stress conditions. Distinctively, our model emphasizes training PINNs with multi-modal synthetic experimental datasets consisting of full-field deformation and loading history, ensuring algorithm robustness even with noisy data. Our results reveal that the PINNs framework can accurately identify constitutive parameters of the incompressible Arruda-Boyce model for samples with intricate geometries, maintaining an error below 5%, even with an experimental noise level of 5%. We believe our framework provides a robust modulus identification approach for complex solids, especially for those with geometrical and constitutive complexity.

Spiking neural networks for nonlinear regression

Spiking neural networks (SNN), also often referred to as the third generation of neural networks, carry the potential for a massive reduction in memory and energy consumption over traditional, second-generation neural networks. Inspired by the undisputed efficiency of the human brain, they introduce temporal and neuronal sparsity, which can be exploited by next-generation neuromorphic hardware. Energy efficiency plays a crucial role in many engineering applications, for instance, in structural health monitoring. Machine learning in engineering contexts, especially in data-driven mechanics, focuses on regression. While regression with SNN has already been discussed in a variety of publications, in this contribution, we provide a novel formulation for its accuracy and energy efficiency. In particular, a network topology for decoding binary spike trains to real numbers is introduced, using the membrane potential of spiking neurons. Several different spiking neural architectures, ranging from simple spiking feed-forward to complex spiking long short-term memory neural networks, are derived. Since the proposed architectures do not contain any dense layers, they exploit the full potential of SNN in terms of energy efficiency. At the same time, the accuracy of the proposed SNN architectures is demonstrated by numerical examples, namely different material models. Linear and nonlinear, as well as history-dependent material models, are examined. While this contribution focuses on mechanical examples, the interested reader may regress any custom function by adapting the published source code.

Myofascial trigger point (MTrP) size and elasticity properties can be used to differentiate characteristics of MTrPs in lower back skeletal muscle

Myofascial trigger points (MTrPs) are localized contraction knots that develop after muscle overuse or an acute trauma. Significant work has been done to understand, diagnose, and treat MTrPs in order to improve patients suffering from their effects. However, effective non-invasive diagnostic tools are still a missing gap in both understanding and treating MTrPs. Effective treatments for patients suffering from MTrP mediated pain require a means to measure MTrP properties quantitatively and diagnostically both prior to and during intervention. Further, quantitative measurements of MTrPs are often limited by the availability of equipment and training. Here we develop ultrasound (US) based diagnostic metrics that can be used to distinguish the biophysical properties of MTrPs, and show how those metrics can be used by clinicians during patient diagnosis and treatment. We highlight the advantages and limitations of previous US-based approaches that utilize elasticity theory. To overcome these previous limitations, we use a hierarchical approach to distinguish MTrP properties by patients' reported pain and clinician measured palpation. We show how US-based measurements can characterize MTrPs with this approach. We demonstrate that MTrPs tend to be smaller, stiffer, and deeper in the muscle tissue for patients with pain compared to patients without pain. We provide evidence that more than one MTrP within a single US-image field increases the stiffness of neighboring MTrPs. Finally, we highlight a combination of metrics (depth, thickness, and stiffness) that can be used by clinicians to evaluate individual MTrPs in combination with standard clinical assessments.

Physics-informed shape optimization using coordinate projection

The rapid growth of artificial intelligence is revolutionizing classical engineering society, offering novel approaches to material and structural design and analysis. Among various scientific machine learning techniques, physics-informed neural network (PINN) has been one of the most researched subjects, for its ability to incorporate physics prior knowledge into model training. However, the intrinsic continuity requirement of PINN demands the adoption of domain decomposition when multiple materials with distinct properties exist. This greatly complicates the gradient computation of design features, restricting the application of PINN to structural shape optimization. To address this, we present a novel framework that employs neural network coordinate projection for shape optimization within PINN. This technique allows for direct mapping from a standard shape to its optimal counterpart, optimizing the design objective without the need for traditional transition functions or the definition of intermediate material properties. Our method demonstrates a high degree of adaptability, allowing the incorporation of diverse constraints and objectives directly as training penalties. The proposed approach is tested on magnetostatic problems for iron core shape optimization, a scenario typically plagued by the high permeability contrast between materials. Validation with finite-element analysis confirms the accuracy and efficiency of our approach. The results highlight the framework's capability as a viable tool for shape optimization in complex material design tasks.

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials.

Fill in the Blank: Transferrable Deep Learning Approaches to Recover Missing Physical Field Information

Solving materials engineering tasks is often hindered by limited information, such as in inverse problems with only boundary data information or design tasks with a simple objective but a vast search space. To address these challenges, multiple deep learning (DL) architectures are leveraged to predict missing mechanical information given limited known data in part of the domain, and further characterize the composite geometries from the recovered mechanical fields for 2D and 3D complex microstructures. In 2D, a conditional generative adversarial network (GAN) is utilized to complete partially masked field maps and predict the composite geometry with convolutional models with great accuracy and generality by making precise predictions on field data with mixed stress/strain components, hierarchical geometries, distinct materials properties and various types of microstructures including ill-posed inverse problems. In 3D, a Transformer-based architecture is implemented to predict complete 3D mechanical fields from input field snapshots. The model manifests excellent performance regardless of microstructural complexity and recovers the entire bulk field even from a single surface field image, allowing internal structural characterization from only boundary measurements. The whole frameworks provide efficient ways for analysis and design with incomplete information and allow the direct inverse translation from properties back to materials structures.

Deep learning-accelerated computational framework based on Physics Informed Neural Network for the solution of linear elasticity

The paper presents an efficient and robust data-driven deep learning (DL) computational framework developed for linear continuum elasticity problems. The methodology is based on the fundamentals of the Physics Informed Neural Networks (PINNs). For an accurate representation of the field variables, a multi-objective loss function is proposed. It consists of terms corresponding to the residual of the governing partial differential equations (PDE), constitutive relations derived from the governing physics, various boundary conditions, and data-driven physical knowledge fitting terms across randomly selected collocation points in the problem domain. To this end, multiple densely connected independent artificial neural networks (ANNs), each approximating a field variable, are trained to obtain accurate solutions. Several benchmark problems including the Airy solution to elasticity and the Kirchhoff-Love plate problem are solved. Performance in terms of accuracy and robustness illustrates the superiority of the current framework showing excellent agreement with analytical solutions. The present work combines the benefits of the classical methods depending on the physical information available in analytical relations with the superior capabilities of the DL techniques in the data-driven construction of lightweight, yet accurate and robust neural networks. The models developed herein can significantly boost computational speed using minimal network parameters with easy adaptability in different computational platforms.

A physics-informed neural network based on mixed data sampling for solving modified diffusion equations

We developed a physics-informed neural network based on a mixture of Cartesian grid sampling and Latin hypercube sampling to solve forward and backward modified diffusion equations. We optimized the parameters in the neural networks and the mixed data sampling by considering the squeeze boundary condition and the mixture coefficient, respectively. Then, we used a given modified diffusion equation as an example to demonstrate the efficiency of the neural network solver for forward and backward problems. The neural network results were compared with the numerical solutions, and good agreement with high accuracy was observed. This neural network solver can be generalized to other partial differential equations.

Detecting Microstructural Criticality/Degeneracy through Hybrid Learning Strategies Trained by Molecular Dynamics Simulations

Efficient microstructure design can strongly accelerate the development of materials. However, the complexity of the microstructure-behavior relation renders the criticalities and degeneracies within the microstructure space highly possible. Criticality means that a slight microstructural change can lead to a dramatic transition in material behavior, while degeneracy means that very different microstructures may lead to similar behaviors. To investigate these microstructural characteristics of the fiber/matrix interface within composite materials, we have proposed a hybrid deep-learning-based framework by integrating the supervised feed-forward neural network and the unsupervised autoencoder, which are trained by the molecular dynamics (MD) simulation results. The well-trained model continuously maps the elemental density images within the interfacial area into a low-dimensional latent space. Assisted by the extracted latent features, we can easily detect the criticalities and degeneracies within the original microstructure space of the composite's interface. The predicted microstructural criticalities and degeneracies are validated by investigating their atomistic origins through MD simulations. The proposed framework can be employed for the interfacial microstructure design of composite materials by identifying certain interfacial microstructures that might lead to undesirable behaviors.

Interfacing finite elements with deep neural operators for fast multiscale modeling of mechanics problems

Multiscale modeling is an effective approach for investigating multiphysics systems with largely disparate size features, where models with different resolutions or heterogeneous descriptions are coupled together for predicting the system's response. The solver with lower fidelity (coarse) is responsible for simulating domains with homogeneous features, whereas the expensive high-fidelity (fine) model describes microscopic features with refined discretization, often making the overall cost prohibitively high, especially for time-dependent problems. In this work, we explore the idea of multiscale modeling with machine learning and employ DeepONet, a neural operator, as an efficient surrogate of the expensive solver. DeepONet is trained offline using data acquired from the fine solver for learning the underlying and possibly unknown fine-scale dynamics. It is then coupled with standard PDE solvers for predicting the multiscale systems with new boundary/initial conditions in the coupling stage. The proposed framework significantly reduces the computational cost of multiscale simulations since the DeepONet inference cost is negligible, facilitating readily the incorporation of a plurality of interface conditions and coupling schemes. We present various benchmarks to assess the accuracy and efficiency, including static and time-dependent problems. We also demonstrate the feasibility of coupling of a continuum model (finite element methods, FEM) with a neural operator, serving as a surrogate of a particle system (Smoothed Particle Hydrodynamics, SPH), for predicting mechanical responses of anisotropic and hyperelastic materials. What makes this approach unique is that a well-trained over-parametrized DeepONet can generalize well and make predictions at a negligible cost.

G2Φnet: Relating genotype and biomechanical phenotype of tissues with deep learning

Many genetic mutations adversely affect the structure and function of load-bearing soft tissues, with clinical sequelae often responsible for disability or death. Parallel advances in genetics and histomechanical characterization provide significant insight into these conditions, but there remains a pressing need to integrate such information. We present a novel genotype-to-biomechanical phenotype neural network (G2Φnet) for characterizing and classifying biomechanical properties of soft tissues, which serve as important functional readouts of tissue health or disease. We illustrate the utility of our approach by inferring the nonlinear, genotype-dependent constitutive behavior of the aorta for four mouse models involving defects or deficiencies in extracellular constituents. We show that G2Φnet can infer the biomechanical response while simultaneously ascribing the associated genotype by utilizing limited, noisy, and unstructured experimental data. More broadly, G2Φnet provides a powerful method and a paradigm shift for correlating genotype and biomechanical phenotype quantitatively, promising a better understanding of their interplay in biological tissues.