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.

翻译：在工程学和生物力学中，粘弹性行为的建模至关重要，因为材料会经历随时间变化的变形，包括应力松弛、蠕变屈曲和生物组织发育。传统数值方法（如有限元法）通常需要显式网格划分、人工扰动或嵌入定制程序来捕捉这些现象，从而增加了计算复杂性。本研究开发了一种基于能量的物理信息神经网络（PINN）框架，采用增量方法模拟粘弹性蠕变、应力松弛、屈曲以及生长诱导的形态发生。通过训练神经网络最小化系统势能泛函，隐式满足平衡和本构定律，从而确保物理一致性。我们证明该框架能够自然捕捉蠕变屈曲而无需预先引入缺陷，利用其固有的训练动力学触发失稳。此外，我们将该框架扩展至生物组织生长与形态发生，预测了圆柱结构中的均匀膨胀和差异生长诱导的屈曲。结果表明，基于能量的PINN能有效预测粘弹性失稳、后屈曲演化及组织形态演变，为传统方法提供了一种有前景的替代方案。本研究证明PINN可成为模拟复杂时变材料行为的灵活稳健工具，为结构工程、软材料和组织发育等领域开辟了潜在应用前景。