Physics-Informed Neural Networks (PINNs) have gained considerable interest in
diverse engineering domains thanks to their capacity to integrate physical laws
into deep learning models. Recently, geometry-aware PINN-based approaches that
employ the strong form of underlying physical system equations have been
developed with the aim of integrating geometric information into PINNs. Despite
ongoing research, the assessment of PINNs in problems with various geometries
remains an active area of investigation. In this work, we introduce a novel
physics-informed framework named the Geometry-Aware Deep Energy Method (GADEM)
for solving structural mechanics problems on different geometries. As the weak
form of the physical system equation (or the energy-based approach) has
demonstrated clear advantages compared to the strong form for solving solid
mechanics problems, GADEM employs the weak form and aims to infer the solution
on multiple shapes of geometries. Integrating a geometry-aware framework into
an energy-based method results in an effective physics-informed deep learning
model in terms of accuracy and computational cost. Different ways to represent
the geometric information and to encode the geometric latent vectors are
investigated in this work. We introduce a loss function of GADEM which is
minimized based on the potential energy of all considered geometries. An
adaptive learning method is also employed for the sampling of collocation
points to enhance the performance of GADEM. We present some applications of
GADEM to solve solid mechanics problems, including a loading simulation of a
toy tire involving contact mechanics and large deformation hyperelasticity. The
numerical results of this work demonstrate the remarkable capability of GADEM
to infer the solution on various and new shapes of geometries using only one
trained model.

GADEM 是一种物理信息深度学习框架，适用于解决不同几何形状的结构力学问题，通过弱形式和几何感知框架将几何信息整合到深度学习模型中，提高了准确性和计算成本。

深度能量方法的几何感知框架：一种应用于超弹性材料的结构力学

Geometry-aware framework for deep energy method: an application to  structural mechanics with hyperelastic materials

Recent work in data-driven modeling has demonstrated that a weak formulation
of model equations enhances the noise robustness of a wide range of
computational methods. In this paper, we demonstrate the power of the weak form
to enhance the LaSDI (Latent Space Dynamics Identification) algorithm, a
recently developed data-driven reduced order modeling technique.
We introduce a weak form-based version WLaSDI (Weak-form Latent Space
Dynamics Identification). WLaSDI first compresses data, then projects onto the
test functions and learns the local latent space models. Notably, WLaSDI
demonstrates significantly enhanced robustness to noise. With WLaSDI, the local
latent space is obtained using weak-form equation learning techniques. Compared
to the standard sparse identification of nonlinear dynamics (SINDy) used in
LaSDI, the variance reduction of the weak form guarantees a robust and precise
latent space recovery, hence allowing for a fast, robust, and accurate
simulation. We demonstrate the efficacy of WLaSDI vs. LaSDI on several common
benchmark examples including viscid and inviscid Burgers', radial advection,
and heat conduction. For instance, in the case of 1D inviscid Burgers'
simulations with the addition of up to 100% Gaussian white noise, the relative
error remains consistently below 6% for WLaSDI, while it can exceed 10,000% for
LaSDI. Similarly, for radial advection simulations, the relative errors stay
below 15% for WLaSDI, in stark contrast to the potential errors of up to
10,000% with LaSDI. Moreover, speedups of several orders of magnitude can be
obtained with WLaSDI. For example applying WLaSDI to 1D Burgers' yields a 140X
speedup compared to the corresponding full order model.
Python code to reproduce the results in this work is available at
(this https URL) and
(this https URL).

基于弱形式方程的数据驱动建模技术在 LaSDI 算法中显示出显著的噪声鲁棒性提高，引入的 WLaSDI 相较于 LaSDI 具有更强的鲁棒性和精度，速度也得到了数个数量级的提升。