A Physics-Informed Neural Network Framework for Simulating Creep Buckling in Growing Viscoelastic Biological Tissues

TL;DR

This work introduces an energy-based, physics-informed neural network (PINN) framework (Deep Energy Method) to simulate nonlinear viscoelasticity, including creep, stress relaxation, creep buckling, and growth-driven morphogenesis in cylindrical geometries. By using a time-incremental approach that minimizes a total potential energy at each step and updates relaxation parameters, the method captures time-dependent instabilities without artificial perturbations and reproduces growth-induced folding patterns that resemble morphogenesis. Benchmark comparisons with FEM demonstrate accuracy for creep and relaxation, while buckling behavior emerges from optimizer dynamics rather than predefined imperfections, highlighting the potential of mesh-free, energy-minimizing neural approaches for complex, evolving biomechanics and soft-material problems. The framework offers a flexible tool for engineering and biological applications, with implications for design of soft tissues, biomaterials, and morphogenetic modeling, albeit with current limitations in full eigenvalue buckling analysis and computational efficiency.

Abstract

Modeling viscoelastic behavior is crucial in engineering and biomechanics, where materials undergo time-dependent deformations, including stress relaxation, creep buckling and biological tissue development. Traditional numerical methods, like the finite element method, often require explicit meshing, artificial perturbations or embedding customised programs to capture these phenomena, adding computational complexity. In this study, we develop an energy-based physics-informed neural network (PINN) framework using an incremental approach to model viscoelastic creep, stress relaxation, buckling, and growth-induced morphogenesis. Physics consistency is ensured by training neural networks to minimize the systems potential energy functional, implicitly satisfying equilibrium and constitutive laws. We demonstrate that this framework can naturally capture creep buckling without pre-imposed imperfections, leveraging inherent training dynamics to trigger instabilities. Furthermore, we extend our framework to biological tissue growth and morphogenesis, predicting both uniform expansion and differential growth-induced buckling in cylindrical structures. Results show that the energy-based PINN effectively predicts viscoelastic instabilities, post-buckling evolution and tissue morphological evolution, offering a promising alternative to traditional methods. This study demonstrates that PINN can be a flexible robust tool for modeling complex, time-dependent material behavior, opening possible applications in structural engineering, soft materials, and tissue development.

Figures (10)

• Figure 1: Viscoelastic materials and models. (a) Viscoelasticity is very common among different materials, e.g., hydrogels, bitumen Edgeworth.1984, and soft biological tissues. (b-c) The viscoelastic models used for mechanical modeling. (b) The general Maxwell model. (c) The standard three-parameter model.
• Figure 2: Diagram showing the decomposition of the deformation tensor when modeling tissue growth. The corresponding deformation gradient is decomposed into $\mathbf{F}_a$ and $\mathbf{F}_g$. A material point with the coordinate vector $\mathbf{X}$ in the initial configuration is mapped to $\mathbf{x}$ in the current configuration by $\chi(\mathbf{X}, t)$. The viscoelasticity leads to time-dependent deformation, which evolves with time.
• Figure 3: Diagram of the physics-informed neural networks.
• Figure 4: Viscoelastic creep and stress relaxation cases solved by PINN. (a) The schematic diagram of stress relaxation of a cantilever. The left boundary is fixed, and the right is fixed horizontally after stretching for a displacement $\Delta L$. (b) Comparison of the stress relaxation results at the time $t = 3 \tau$ from FEM and PINN. (c) The schematic diagram of creep of a cantilever. The left boundary is fixed, and the right boundary is subjected to a horizontal uniformly distributed tensile force. (d) Comparison of the creep results at the time $t = 3 \tau$ from FEM and PINN. (e) The training dynamics of the energy-based PINN. The potential energy loss decreases as training progress. Small oscillations may occur during the convergence process, which are brought in by the optimization algorithm itself.
• Figure 5: Creep buckling of a cantilever beam under axial compression. (a) Schematic diagram of a compressed cantilever beam. (b) Uniform discretization of the beam. (c) Displacement in the y-direction of the creep buckling cantilever beam at different times obtained from PINN under pressure $2\text{e}-2$ Pa, with Adam optimizer and learning rate $5\text{e}-3$. (d) Displacement in y-direction obtained from the Fourier feature-PINN (FF-PINN). The FF-PINN has three layers, and the feature number of each layer is 64 in this study. (e) Deformed cantilever beam obtained from FEM under pressure $2\text{e}-2$ Pa, with initial geometric imperfections. (f) Pointwise error of the y-displacement obtained from PINN and FEM at $t=2 \tau$. The error corresponds to the square error. (g) A comparison of the time-deflection curves obtained from PINN and FEM.