mlsanityData to 7 Oct 2026

Parameterized Physics-informed Neural Networks for Parameterized PDEs

arXiv 2408.09446 ↗First posted 18 Aug 2024cs.LG cs.NA math.NAcited by 35

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Abstract (arXiv)

Complex physical systems are often described by partial differential equations (PDEs) that depend on parameters such as the Reynolds number in fluid mechanics. In applications such as design optimization or uncertainty quantification, solutions of those PDEs need to be evaluated at numerous points in the parameter space. While physics-informed neural networks (PINNs) have emerged as a new strong competitor as a surrogate, their usage in this scenario remains underexplored due to the inherent need for repetitive and time-consuming training. In this paper, we address this problem by proposing a novel extension, parameterized physics-informed neural networks (P2INNs). P2INNs enable modeling the solutions of parameterized PDEs via explicitly encoding a latent representation of PDE parameters. With the extensive empirical evaluation, we demonstrate that P2INNs outperform the baselines both in accuracy and parameter efficiency on benchmark 1D and 2D parameterized PDEs and are also effective in overcoming the known "failure modes".

Key facts. Cited by 35 papers in the map (data through 7 Oct 2026). Builds on 24 of its 40 references that are in the map. First posted on arXiv 18 Aug 2024.

Builds on

24 of 40 references.

References · 24

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2017Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semanticscited by 1,345
  3. 2018On the Spectral Bias of Neural Networkscited by 1,321
  4. 2021E(n) Equivariant Graph Neural Networkscited by 951
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  6. 2017An Overview of Multi-Task Learning in Deep Neural Networkscited by 791
  7. 2019Hamiltonian Neural Networkscited by 702
  8. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  9. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  10. 2020Lagrangian Neural Networkscited by 388
  11. 2020Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanismcited by 231
  12. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
12 more
  1. 2019Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learningcited by 205
  2. 2022From data to functa: Your data point is a function and you can treat it like onecited by 183
  3. 2019Hamiltonian Generative Networkscited by 136
  4. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  5. 2023Separable Physics-Informed Neural Networkscited by 64
  6. 2021Machine learning structure preserving brackets for forecasting irreversible processescited by 51
  7. 2020DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolationcited by 45
  8. 2023Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networkscited by 42
  9. 2021A novel meta-learning initialization method for physics-informed neural networkscited by 35
  10. 2024PINNACLE: PINN Adaptive ColLocation and Experimental points selectioncited by 31
  11. 2023An operator preconditioning perspective on training in physics-informed machine learningcited by 27
  12. 2022SVD-PINNs: Transfer Learning of Physics-Informed Neural Networks via Singular Value Decompositioncited by 13

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 5; 2025: 14; 2026: 16.

35 papers.

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  1. 2025FP64 is All You Need: Rethinking Failure Modes in Physics-Informed Neural Networkscited by 11
  2. 2025Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunitiescited by 10
  3. 2025Sub-Sequential Physics-Informed Learning with State Space Modelcited by 10
  4. 2026Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operatorscited by 6
  5. 2025Adaptive Physics-informed Neural Networks: A Surveycited by 5
  6. 2024Physics-informed reduced order model with conditional neural fieldscited by 4
  7. 2025Governing Equation Discovery from Data Based on Differential Invariantscited by 4
  8. 2026OpInf-LLM: Parametric PDE Solving with LLMs via Operator Inferencecited by 4
  9. 2026DiffusionRollout: Uncertainty-Aware Rollout Planning in Long-Horizon PDE Solvingcited by 4
  10. 2024Solving Differential Equations with Constrained Learningcited by 3
  11. 2024Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?cited by 3
  12. 2025Enhanced Vascular Flow Simulations in Aortic Aneurysm via Physics-Informed Neural Networks and Deep Operator Networkscited by 3
23 more
  1. 2024Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimizationcited by 1
  2. 2025Physics-Informed Deep B-Spline Networkscited by 1
  3. 2025Unveiling the Potential of Superexpressive Networks in Implicit Neural Representationscited by 1
  4. 2025Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learningcited by 1
  5. 2026A deep learning framework for jointly solving transient Fokker-Planck equations with arbitrary parameters and initial distributionscited by 1
  6. 2026Compositional Meta-Learning for Mitigating Task Heterogeneity in Physics-Informed Neural Networkscited by 1
  7. 2026Adaptive Hard-Soft Physics-Informed Neural Networks for Robust Boundary-Constrained PDE Solvingcited by 1
  8. 2026Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEscited by 1
  9. 2024MaD-Scientist: AI-based Scientist solving Convection-Diffusion-Reaction Equations Using Massive PINN-Based Prior Datacited by 0
  10. 2025Engineering application of physics-informed neural networks for Saint-Venant torsioncited by 0
  11. 2025Beyond Static Models: Hypernetworks for Adaptive and Generalizable Forecasting in Complex Parametric Dynamical Systemscited by 0
  12. 2025IP-Basis PINNs: Efficient Multi-Query Inverse Parameter Estimationcited by 0
  13. 2025PIANO: Physics-informed Dual Neural Operator for Precipitation Nowcastingcited by 0
  14. 2025PGOT: A Physics-Geometry Operator Transformer for Complex PDEscited by 0
  15. 2026Parameterized Temporal Extrapolation of PDEs via Disentangled Latent Dynamics Manifold Fusioncited by 0
  16. 2026Sparsely-Supervised Data Assimilation via Physics-Informed Schr"odinger Bridgecited by 0
  17. 2026Material-agnostic temperature field prediction for metal additive manufacturing via a parametric PINN frameworkcited by 0
  18. 2026Conflict-Aware Harmonized Rotational Gradient for Multiscale Kinetic Regimescited by 0
  19. 2026Differentiable Chemistry in PINNs for Solving Parameterized and Stiff Reaction Systemscited by 0
  20. 2026Architecture Shapes Transfer Specificity in Implicit Neural Representationscited by 0
  21. 2026Physics-Informed Foresight Pruning for Sparse PINN Solvers of Nonlinear PDEscited by 0
  22. 2026Physics and Data Driven Transformer-Mamba Framework for Flow Fieldcited by 0
  23. 2026Beyond Simulation: Retain-and-Repair Neural Operators for Real-World Adaptationcited by 0