The coupling of Proper Orthogonal Decomposition (POD) and deep learning-based ROMs (DL-ROMs) has proved to be a successful strategy to construct non-intrusive, highly accurate, surrogates for the real time solution of parametric nonlinear time-dependent PDEs. Inexpensive to evaluate, POD-DL-ROMs are also relatively fast to train, thanks to their limited complexity. However, POD-DL-ROMs account for the physical laws governing the problem at hand only through the training data, that are usually obtained through a full order model (FOM) relying on a high-fidelity discretization of the underlying equations. Moreover, the accuracy of POD-DL-ROMs strongly depends on the amount of available data. In this paper, we consider a major extension of POD-DL-ROMs by enforcing the fulfillment of the governing physical laws in the training process -- that is, by making them physics-informed -- to compensate for possible scarce and/or unavailable data and improve the overall reliability. To do that, we first complement POD-DL-ROMs with a trunk net architecture, endowing them with the ability to compute the problem's solution at every point in the spatial domain, and ultimately enabling a seamless computation of the physics-based loss by means of the strong continuous formulation. Then, we introduce an efficient training strategy that limits the notorious computational burden entailed by a physics-informed training phase. In particular, we take advantage of the few available data to develop a low-cost pre-training procedure; then, we fine-tune the architecture in order to further improve the prediction reliability. Accuracy and efficiency of the resulting pre-trained physics-informed DL-ROMs (PTPI-DL-ROMs) are then assessed on a set of test cases ranging from non-affinely parametrized advection-diffusion-reaction equations, to nonlinear problems like the Navier-Stokes equations for fluid flows.

翻译：将本征正交分解（POD）与基于深度学习的降阶模型（DL-ROMs）相结合，已被证明是构建非侵入式、高精度代理模型的成功策略，可用于实时求解参数化非线性含时偏微分方程。POD-DL-ROMs 不仅评估代价低廉，而且由于复杂度受限，其训练速度也相对较快。然而，POD-DL-ROMs 仅通过训练数据（通常依赖基于底层方程高保真离散的全阶模型获得）来捕捉问题的物理规律。此外，POD-DL-ROMs 的精度强烈依赖于可用数据量。本文对 POD-DL-ROMs 进行了重要扩展：通过在训练过程中强制满足物理定律（即赋予其物理信息性），以补偿可能的数据稀疏和/或缺失问题，并提升整体可靠性。为此，我们首先为 POD-DL-ROMs 补充主干网络架构，使其具备计算空间域内任意点问题解的能力，并最终通过强连续形式实现基于物理的损失函数的无缝计算。随后，我们引入了一种高效训练策略，以缓解物理信息训练阶段众所周知的巨大计算负担。具体而言，我们利用少量可用数据开发低成本预训练流程；然后通过微调架构进一步提升预测可靠性。最终形成的预训练物理信息深度学习降阶模型（PTPI-DL-ROMs）的精度与效率，通过一系列测试案例进行了评估：从非仿射参数化的对流-扩散-反应方程，到流体流动的纳维-斯托克斯方程等非线性问题。