Discrete Physics-Informed Training for Projection-Based Reduced-Order Models with Neural Networks

This paper presents a physics-informed training framework for projection-based Reduced-Order Models (ROMs). We extend the original PROM-ANN architecture by complementing snapshot-based training with a FEM-based, discrete physics-informed residual loss, bridging the gap between traditional projection-based ROMs and physics-informed neural networks (PINNs). Unlike conventional PINNs that rely on analytical PDEs, our approach leverages FEM residuals to guide the learning of the ROM approximation manifold. Our key contributions include the following: (1) a parameter-agnostic, discrete residual loss applicable to nonlinear problems, (2) an architectural modification to PROM-ANN improving accuracy for fast-decaying singular values, and (3) an empirical study on the proposed physics-informed training process for ROMs. The method is demonstrated on a nonlinear hyperelasticity problem, simulating a rubber cantilever under multi-axial loads. The main accomplishment in regards to the proposed residual-based loss is its applicability on nonlinear problems by interfacing with FEM software while maintaining reasonable training times. The modified PROM-ANN outperforms POD by orders of magnitude in snapshot reconstruction accuracy, while the original formulation is not able to learn a proper mapping for this use case. Finally, the application of physics-informed training in ANN-PROM modestly narrows the gap between data reconstruction and ROM accuracy; however, it highlights the untapped potential of the proposed residual-driven optimization for future ROM development. This work underscores the critical role of FEM residuals in ROM construction and calls for further exploration on architectures beyond PROM-ANN.