A discrete physics-informed training for projection-based reduced order models with neural networks

TL;DR

The paper tackles the computational bottleneck of high-fidelity simulations by fusing projection-based ROMs with physics-informed training. It introduces a discrete FEM residual loss, $\\mathcal{L}_{\\mathbf{R}}$, into the PROM-ANN framework, making the ROM learn not only from snapshot data but also from the physics residuals in a parameter-agnostic manner. Architectural refinements include scaling the reduced coefficients with SVD-derived matrices and reweighting the data-based loss to handle fast-decaying singular values, along with incorporating the residual loss into training. The approach is validated on a nonlinear hyperelastic cantilever under multi-axial loading, showing consistent improvements in snapshot reconstruction and modest gains in ROM accuracy, albeit with increased offline training time; the results motivate further exploration of residual-driven ROMs and scalable, residual-aware architectures for nonlinear problems.

Abstract

This paper presents a physics-informed training framework for projection-based Reduced Order Models (ROMs). We extend the 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. Key contributions include: (1) a parameter-agnostic, discrete residual loss applicable to non-linear 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 non-linear 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 non-linear 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.

Figures (11)

• Figure 1: Mesh of the cantilever with line loads
• Figure 2: Examples of deformation in the cantilever for six different random combinations of line loads
• Figure 3: Singular values' energies for the first 200 modes of $\mathbf{S}_u$.
• Figure 4: Boxplots showing the spread of the values for each mode of $\mathbf{q}$, for all the samples in our training dataset. (a) shows the case where $\mathbf{q} = \boldsymbol{\boldsymbol{\Phi}}^T \mathbf{u^*}$ and (b) the case where $\mathbf{q} = \boldsymbol{\Xi}^{-1}\boldsymbol{\boldsymbol{\Phi}}^T \mathbf{u^*}$. The ranges are much more uniform in the latter case, which should be beneficial when training the neural network.
• Figure 5: Boxplots showing the spread of the mean value per epoch of the applied gradients during training, four different training strategies. (a) Shows the results when no rescaling of the loss is applied. (b) Shows the results when the rescaling factors $e_{\text{POD},d}$ and $e_{\text{POD},\mathcal{R}}$ are applied. The latter case is invariant to the choice of latent space size, contrary to the first one.

Theorems & Definitions (1)

• proof