Physics-informed machine learning for nonlinear deformations in soft solids

Mild traumatic brain injury (mTBI), often referred to as concussion, is a significant public health concern, particularly in contact sports, vehicular collisions and similar scenarios. The key mechanisms behind mTBI involve the transmission of shear shock waves through the brain tissue following an impact. These shock waves induce rapid, nonlinear deformations within the soft, heterogeneous structure of the brain, disrupting the integrity of the brain. The complexity of these injuries necessitates highly accurate biomechanical models that can capture the intricate response of brain tissue under loading. Traditionally, hyperelastic models such as the Neo-Hookean (NH) and Mooney-Rivlin (MR) have been employed to simulate brain tissue mechanics. However, these models fall short of accurately describing the highly nonlinear and rate-dependent behaviour observed in mTBI scenarios. A fourth-order Landau (LA) hyperelastic model offers a more suitable alternative, as it can capture the material stiffening effects and large strain behaviours intrinsic to brain tissue, making it particularly well-suited for modelling shear shock wave propagation. Finite Element Methods (FEM) have been the standard for solving the partial differential equations (PDEs) that govern such biomechanical systems. These solvers provide highly accurate predictions of tissue deformation and stress distributions. However, their computational intensity, especially when incorporating spatial resolution and nonlinear material behaviour, makes them impractical for real-time applications such as injury prediction in sports helmets or real-time diagnostics in clinical settings. To overcome this limitation, Physics-Informed Neural Networks (PINNs) have emerged as a compelling alternative. PINNs offer the potential to simulate real-time brain deformation under impact conditions with far lower computational costs than traditional FEM based solvers. PINNs are a class of deep learning models that embed the governing PDEs of physical systems directly into the training process of a neural network. Instead of relying on labelled data, PINNs minimise the residuals of the PDEs and enforce initial and boundary conditions during training. This allows the neural network to learn solutions that are consistent with physical laws, even with sparse or noisy data. PINNs are being used to solve partial differential equations in various physical systems. PINNs have garnered significant interest due to their applications in solving partial differential equations that govern various physical phenomena. In 2019, Raissi et al. proposed the concept of PINN, which has since led to hundreds of publications with applications in various disciplines. However, baseline PINN face several challenges, leading to inaccurate solutions in many scenarios. This work proposes a mesh-free Causal Marching Physics-Informed Neural Networks (CMPINN) model for various hyperelastic models to capture their nonlinear mechanical response of higher-order hyperelastic materials. It marks the first attempt to develop physics-informed neural networks to capture complex fourth-order deformation behaviours in soft biological tissues such as the brain. The developed CMPINN framework introduces a ”multinet” architecture, which is designed to handle multimaterial domains effectively. This structure enables the simultaneous modelling of different materials within a single computational domain, which is crucial for realistic simulations involving heterogeneous tissues. Enforcing material incompressibility can be numerically challenging. Another key advancement of CMPINN is the tailored enforcement of incompressibility to address floating-point errors, ensuring more stable and accurate training. Additionally, CMPINN introduces an automated model selection strategy across temporal or load-stepping iterations. The CMPINN framework autonomously selects the most accurate model for each test case by continuously monitoring training loss during these steps. To further refine performance, the study introduces a hyperparameter tuning strategy designed to efficiently identify the optimal set of training parameters. The proposed CMPINN framework is developed for a cube undergoing homogeneous isotropic, incompressible and canonical deformations: uniaxial tension/compression, simple shear, biaxial tension/compression, and pure shear. Three other tests for scenarios involving spatially varying material properties and inhomogeneous deformations are performed and benchmarked with numerical solutions. The approach systematically identifies optimal values for network depth, learning rate, loss weighting, and optimiser training epochs. The developed CMPINN model is mesh-free and accurately captures the mechanical deformation without labelled data. Once trained, the model can rapidly respond to any spatial coordinate within the physical domain. We develop the CMPINN framework to describe the propagation of nonlinear shear waves in soft solids, inspired by Tripathi et al.. The training of the CMPINN was performed without any labelled data to incorporate the causality of the wave propagation. Different tests involving linear and nonlinear shear propagation were performed, and the results were benchmarked against the numerical solution. In summary, this work advances the state of the art in PINN-based modelling by creating a stable and adaptive framework capable of handling the fourth-order hyperelastic deformation in soft solids. It opens new avenues for efficient and accurate simulation in biomechanics, particularly for applications such as real-time brain injury prediction.

Funder

College of Science and Engineering, University of Galway

Publisher

University of Galway

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CC BY-NC-ND

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University of Galway Theses (PhD Theses)