AI 中文总结
本研究提出FS-ENGD-PINN，通过固定应力分裂解耦耦合系统、能量自然梯度下降优化及三场混合格式，解决PINN应用于Biot模型时的性能问题，其精度与鲁棒性优于标准PINN，为多孔弹性问题提供可靠求解器。
AI 中文摘要
物理信息神经网络（PINNs）作为一种无网格偏微分方程求解框架，近来受到广泛关注。然而将其应用于强耦合多物理场系统（如Biot固结模型）时，由于优化地形严重病态，性能会下降。本研究提出一种鲁棒的PINN求解器，命名为FS-ENGD-PINN，它协同集成了基于物理的解耦与几何感知优化。具体而言，采用固定应力（FS）分裂方案将耦合的多孔弹性系统分解为收缩力学与流动子问题，从而显著提升训练稳定性与收敛性；为进一步加速优化，采用能量自然梯度下降（ENGD），该方法在函数空间中近似牛顿方向，有效缓解刚度诱导的缓慢收敛；此外，为解决近不可压缩区域出现的体积闭锁问题，在PINN框架中引入带有额外总压力变量的三场混合格式。大量数值实验表明，所提出的FS-ENGD-PINN在精度和鲁棒性上始终优于标准PINN格式，为宽范围材料参数下的多孔弹性问题提供了统一且可靠的基于学习的求解器。
英文摘要
Physics-Informed Neural Networks (PINNs) have recently gained considerable attention as a mesh-free framework for solving partial differential equations. Nevertheless, their performance deteriorates when applied to strongly coupled multiphysics systems, such as Biot's consolidation model, due to severely ill-conditioned optimization landscapes. In this work, we propose a robust PINN-based solver, termed FS-ENGD-PINN, which synergistically integrates physics-based decoupling with geometry-aware optimization. Specifically, the Fixed-Stress (FS) splitting scheme is employed to decompose the coupled poroelastic system into contractive mechanics and flow subproblems, thereby significantly improving training stability and convergence. To further accelerate optimization, we adopt Energy Natural Gradient Descent (ENGD), which approximates the Newton direction in function space effectively mitigates stiffness-induced slow convergence. Moreover, to address volumetric locking arising in the nearly incompressible regime, we incorporate a three-field mixed formulation with an additional total pressure variable into the PINN framework. Extensive numerical experiments demonstrate that the proposed FS-ENGD-PINN consistently outperforms standard PINN formulations in terms of accuracy and robustness, providing a unified and reliable learning-based solver for poroelasticity across a wide range of material parameters.
Comments11 figures. Accepted by CiCP
