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

arXiv: https://arxiv.org/abs/2504.13875 · HTML: https://mlsanity.com/paper/2504.13875/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 76 references that are in the map.
- First posted on arXiv 31 Mar 2025.

## Builds on (most cited first, 24 of 24)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics](https://mlsanity.com/paper/1912.11114/) (2019)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Collapse of Deep and Narrow Neural Nets](https://mlsanity.com/paper/1808.04947/) (2018)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Data-Driven Aerospace Engineering: Reframing the Industry with Machine Learning](https://mlsanity.com/paper/2008.10740/) (2020)
- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Finite Element Method-enhanced Neural Network for Forward and Inverse Problems](https://mlsanity.com/paper/2205.08321/) (2022)
- [Mesh sampling and weighting for the hyperreduction of nonlinear Petrov-Galerkin reduced-order models with local reduced-order bases](https://mlsanity.com/paper/2008.02891/) (2020)
- [PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs](https://mlsanity.com/paper/2405.08558/) (2024)
- [Hyper-reduction for Petrov-Galerkin reduced order models](https://mlsanity.com/paper/2309.16267/) (2023)
- [Parallel Reduced Order Modeling for Digital Twins using High-Performance Computing Workflows](https://mlsanity.com/paper/2409.09080/) (2024)
- [Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifolds](https://mlsanity.com/paper/2409.10126/) (2024)

## Built on by (most cited first, 2 of 2)

- [Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds](https://mlsanity.com/paper/2601.00834/) (2025)
- [Polyconvex neural-network surrogates with learned paired directional features for anisotropic hyperelastic homogenization](https://mlsanity.com/paper/2610.12801/) (2026)

## Abstract (arXiv)

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.