# Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)

arXiv: https://arxiv.org/abs/2106.13361 · HTML: https://mlsanity.com/paper/2106.13361/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 46 references that are in the map.
- First posted on arXiv 25 Jun 2021.

## Builds on (most cited first, 16 of 16)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [On the number of response regions of deep feed forward networks with piece-wise linear activations](https://mlsanity.com/paper/1312.6098/) (2013)
- [Representation Benefits of Deep Feedforward Networks](https://mlsanity.com/paper/1509.08101/) (2015)
- [nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications](https://mlsanity.com/paper/2004.04276/) (2020)
- [A Radial Basis Function (RBF)-Finite Difference Method for the Simulation of Reaction-Diffusion Equations on Stationary Platelets within the Augmented Forcing Method](https://mlsanity.com/paper/1310.5207/) (2013)

## Built on by (most cited first, 25 of 25)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Finite basis Kolmogorov-Arnold networks: domain decomposition for data-driven and physics-informed problems](https://mlsanity.com/paper/2406.19662/) (2024)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning](https://mlsanity.com/paper/2202.06416/) (2022)
- [Physics-Informed Deep Learning For Traffic State Estimation: A Survey and the Outlook](https://mlsanity.com/paper/2303.02063/) (2023)
- [Multifidelity Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.14764/) (2024)
- [Combining physics-based and data-driven techniques for reliable hybrid analysis and modeling using the corrective source term approach](https://mlsanity.com/paper/2206.03451/) (2022)
- [Cost-effective Framework for Gradual Domain Adaptation with Multifidelity](https://mlsanity.com/paper/2202.04359/) (2022)
- [Towards deployment-centric multimodal AI beyond vision and language](https://mlsanity.com/paper/2504.03603/) (2025)
- [Active Learning with Bayesian Multi-Fidelity Laplace Neural Operators for Oscillatory Parametric PDEs](https://mlsanity.com/paper/2502.00550/) (2025)
- [Equilibrium Conserving Neural Operators for Super-Resolution Learning](https://mlsanity.com/paper/2504.13422/) (2025)
- [Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metrics](https://mlsanity.com/paper/2402.11126/) (2024)
- [Graph Laplacian-based Bayesian Multi-fidelity Modeling](https://mlsanity.com/paper/2409.08211/) (2024)
- [Hybrid Adaptive Modeling in Process Monitoring: Leveraging Sequence Encoders and Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.14252/) (2025)
- [Bridging quantum and classical computing for partial differential equations through multifidelity machine learning](https://mlsanity.com/paper/2512.05241/) (2025)
- [On the Role of Consistency Between Physics and Data in Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.10611/) (2026)
- [Recursive Flow Matching](https://mlsanity.com/paper/2605.26535/) (2026)
- [Multi-Fidelity SINDy: Sparse Discovery of Nonlinear Dynamical Systems with Fidelity-Weighted Measurements](https://mlsanity.com/paper/2606.15690/) (2026)
- [An iterative multi-fidelity approach for model order reduction of multi-dimensional input parametric PDE systems](https://mlsanity.com/paper/2301.09483/) (2023)
- [A few-shot graph Laplacian-based approach for improving the accuracy of low-fidelity data](https://mlsanity.com/paper/2304.04862/) (2023)
- [Improving the accuracy of physics-informed neural networks via last-layer retraining](https://mlsanity.com/paper/2603.04672/) (2026)
- [Evolutionary Two-Stage Hyperparameter Optimization Strategies for Physics-Informed Neural Networks](https://mlsanity.com/paper/2606.20442/) (2026)

## 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.