We propose a data-driven latent space dynamics identification method (tLaSDI)
that embeds the first and second principles of thermodynamics. The latent
variables are learned through an autoencoder as a nonlinear dimension reduction
model. The dynamics of the latent variables are constructed by a neural
network-based model that preserves certain structures to respect the
thermodynamic laws through the GENERIC formalism. An abstract error estimate of
the approximation is established, which provides a new loss formulation
involving the Jacobian computation of autoencoder. Both the autoencoder and the
latent dynamics are trained to minimize the new loss. Numerical examples are
presented to demonstrate the performance of tLaSDI, which exhibits robust
generalization ability, even in extrapolation. In addition, an intriguing
correlation is empirically observed between the entropy production rates in the
latent space and the behaviors of the full-state solution.

基于数据驱动的潜空间动力学识别方法通过嵌入热力学的第一和第二原理，利用自编码器作为非线性降维模型学习潜变量，并通过基于神经网络模型构建潜变量的动力学，遵循广义热动力学原理，从而产生了一种新的损失公式。潜编码器和潜动力学均通过最小化新损失进行训练，其泛化能力强，即使在外推情况下也表现出鲁棒性。此外，实证观察到潜空间中熵产生速率与完全状态解的行为之间存在着有趣的相关性。