Abstract 

Solving differential problems using full order models (FOMs), such as the finite element method, usually results in prohibitive computational costs, particularly in real-time simulations and multi-query routines. Reduced order modeling aims to replace FOMs with reduced order models (ROMs) characterized by much lower complexity but still able to express the physical features of the system under investigation. Within this context, deep learning-based reduced order models (DL-ROMs) have emerged as a novel and comprehensive approach, offering efficient and accurate surrogates for solving parametrized time-dependent nonlinear PDEs. By leveraging both the mathematical properties and physical knowledge of the system, the accuracy and generalization capabilities of DL-based ROMs can be further enhanced. Building on this motivation, two main approaches to reduced order modeling of parametrized PDEs are introduced: latent dynamics models (LDMs) and pretrained physics-informed DL-ROMs (PTPI-DL-ROMs).

 

LDMs represent a novel mathematical framework in which the latent state is constrained to evolve according to an (unknown) ODE. A time-continuous setting is employed to derive error and stability estimates for the LDM approximation of the FOM solution. The impact of using an explicit Runge-Kutta scheme in a time-discrete setting is then analyzed, resulting in the ∆LDM formulation. Additionally, the learnable setting, ∆LDMθ, is explored, where deep neural networks approximate the discrete LDM components, ensuring a bounded approximation error with respect to the high-fidelity solution. Moreover, the framework demonstrates the capability to achieve a time-continuous approximation of the FOM solution in a multi-query context, thus being able to compute the LDM approximation at any given time instance while retaining a prescribed level of accuracy.

 

As the complexity of PDEs increases, however, the computational cost associated with generating synthetic data using high-fidelity solvers for training DL-based ROMs also intensifies. To address this  challenge, a significant extension of POD-DL ROMs is proposed, integrating the limited labeled data with the underlying physical laws to achieve reliable approximations in a small data context. The approach relies on a physics-informed loss formulation to compensate for data scarcity, providing the neural network with information about the underlying physics. By intertwining the contributions of data and physics, PTPI-DL-ROMs incorporate a novel training paradigm consisting of an efficient pre-training strategy that enables the optimizer to quickly approach the minimum in the loss landscape, followed by a fine-tuning phase that further enhances prediction accuracy.