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Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)

arXiv 2106.13361 ↗First posted 25 Jun 2021physics.comp-ph cs.LGcited by 25

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Abstract (arXiv)

Multifidelity simulation methodologies are often used in an attempt to judiciously combine low-fidelity and high-fidelity simulation results in an accuracy-increasing, cost-saving way. Candidates for this approach are simulation methodologies for which there are fidelity differences connected with significant computational cost differences. Physics-informed Neural Networks (PINNs) are candidates for these types of approaches due to the significant difference in training times required when different fidelities (expressed in terms of architecture width and depth as well as optimization criteria) are employed. In this paper, we propose a particular multifidelity approach applied to PINNs that exploits low-rank structure. We demonstrate that width, depth, and optimization criteria can be used as parameters related to model fidelity, and show numerical justification of cost differences in training due to fidelity parameter choices. We test our multifidelity scheme on various canonical forward PDE models that have been presented in the emerging PINNs literature.

Key facts. Cited by 25 papers in the map (data through 7 Oct 2026). Builds on 16 of its 46 references that are in the map. First posted on arXiv 25 Jun 2021.

Builds on

16 of 46 references.

References · 16

  1. 2014On the Number of Linear Regions of Deep Neural Networkscited by 378
  2. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  3. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  4. 2016On the Expressive Power of Deep Neural Networkscited by 302
  5. 2016Benefits of depth in neural networkscited by 248
  6. 2019Complexity of Linear Regions in Deep Networkscited by 166
  7. 2017Bounding and Counting Linear Regions of Deep Neural Networkscited by 161
  8. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  9. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  10. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  11. 2018Survey of multifidelity methods in uncertainty propagation, inference, and optimizationcited by 141
  12. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
4 more
  1. 2013On the number of response regions of deep feed forward networks with piece-wise linear activationscited by 94
  2. 2015Representation Benefits of Deep Feedforward Networkscited by 84
  3. 2020nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applicationscited by 36
  4. 2013A Radial Basis Function (RBF)-Finite Difference Method for the Simulation of Reaction-Diffusion Equations on Stationary Platelets within the Augmented Forcing Methodcited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 5; 2023: 4; 2024: 5; 2025: 5; 2026: 5.

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  1. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  2. 2022Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problemscited by 57
  3. 2024Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problemscited by 35
  4. 2021A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEscited by 31
  5. 2023Stacked networks improve physics-informed training: applications to neural networks and deep operator networkscited by 21
  6. 2022AutoIP: A United Framework to Integrate Physics into Gaussian Processescited by 20
  7. 2022State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learningcited by 12
  8. 2023Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlookcited by 11
  9. 2024Multifidelity Kolmogorov-Arnold Networkscited by 9
  10. 2022Combining physics-based and data-driven techniques for reliable hybrid analysis and modeling using the corrective source term approachcited by 6
  11. 2022Cost-effective Framework for Gradual Domain Adaptation with Multifidelitycited by 3
  12. 2025Active Learning with Bayesian Multi-Fidelity Laplace Neural Operators for Oscillatory Parametric PDEscited by 2
13 more
  1. 2025Towards deployment-centric multimodal AI beyond vision and languagecited by 2
  2. 2025Equilibrium Conserving Neural Operators for Super-Resolution Learningcited by 2
  3. 2024Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metricscited by 1
  4. 2024Graph Laplacian-based Bayesian Multi-fidelity Modelingcited by 1
  5. 2025Hybrid Adaptive Modeling in Process Monitoring: Leveraging Sequence Encoders and Physics-Informed Neural Networkscited by 1
  6. 2025Bridging quantum and classical computing for partial differential equations through multifidelity machine learningcited by 1
  7. 2026Multi-Fidelity SINDy: Sparse Discovery of Nonlinear Dynamical Systems with Fidelity-Weighted Measurementscited by 1
  8. 2023An iterative multi-fidelity approach for model order reduction of multi-dimensional input parametric PDE systemscited by 0
  9. 2023A few-shot graph Laplacian-based approach for improving the accuracy of low-fidelity datacited by 0
  10. 2026On the Role of Consistency Between Physics and Data in Physics-Informed Neural Networkscited by 0
  11. 2026Improving the accuracy of physics-informed neural networks via last-layer retrainingcited by 0
  12. 2026Recursive Flow Matchingcited by 0
  13. 2026Evolutionary Two-Stage Hyperparameter Optimization Strategies for Physics-Informed Neural Networkscited by 0