mlsanityData to 14 Oct 2026

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

arXiv 2504.13875 ↗First posted 31 Mar 2025cs.LGcited by 2

Not on the map. Why?

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.

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

24 of 76 references.

References · 24

  1. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  2. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  3. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  4. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  5. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  6. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  7. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 373
  8. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
  9. 2019Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systemscited by 205
  10. 2020A fast and accurate physics-informed neural network reduced order model with shallow masked autoencodercited by 80
  11. 2021Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equationscited by 68
  12. 2019Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamicscited by 45
12 more
  1. 2023A graph convolutional autoencoder approach to model order reduction for parametrized PDEscited by 44
  2. 2018Collapse of Deep and Narrow Neural Netscited by 37
  3. 2022Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reductioncited by 30
  4. 2020Data-Driven Aerospace Engineering: Reframing the Industry with Machine Learningcited by 27
  5. 2022Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifoldscited by 25
  6. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  7. 2022Finite Element Method-enhanced Neural Network for Forward and Inverse Problemscited by 13
  8. 2020Mesh sampling and weighting for the hyperreduction of nonlinear Petrov-Galerkin reduced-order models with local reduced-order basescited by 8
  9. 2024PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEscited by 4
  10. 2023Hyper-reduction for Petrov-Galerkin reduced order modelscited by 2
  11. 2024Parallel Reduced Order Modeling for Digital Twins using High-Performance Computing Workflowscited by 2
  12. 2024Data-free Non-intrusive Model Reduction for Nonlinear Finite Element Models via Spectral Submanifoldscited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 1.

2 papers.

  1. 2025Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifoldscited by 0
  2. 2026Polyconvex neural-network surrogates with learned paired directional features for anisotropic hyperelastic homogenizationcited by 0