# Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations

arXiv: https://arxiv.org/abs/2303.11577 · HTML: https://mlsanity.com/paper/2303.11577/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 53 references that are in the map.
- First posted on arXiv 21 Mar 2023.

## Builds on (most cited first, 26 of 26)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Frequency Bias in Neural Networks for Input of Non-Uniform Density](https://mlsanity.com/paper/2003.04560/) (2020)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecture](https://mlsanity.com/paper/2005.00615/) (2020)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers](https://mlsanity.com/paper/2008.09768/) (2020)
- [Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities](https://mlsanity.com/paper/2102.13403/) (2021)
- [Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONets](https://mlsanity.com/paper/2204.00997/) (2022)
- [On transfer learning of neural networks using bi-fidelity data for uncertainty propagation](https://mlsanity.com/paper/2002.04495/) (2020)
- [Transfer Learning on Multi-Fidelity Data](https://mlsanity.com/paper/2105.00856/) (2021)
- [Neural Network Training Using ℓ1-Regularization and Bi-fidelity Data](https://mlsanity.com/paper/2105.13011/) (2021)
- [Use of Multifidelity Training Data and Transfer Learning for Efficient Construction of Subsurface Flow Surrogate Models](https://mlsanity.com/paper/2204.11138/) (2022)
- [A Data-driven Multi-fidelity Physics-informed Learning Framework for Smart Manufacturing: A Composites Processing Case Study](https://mlsanity.com/paper/2202.06139/) (2022)

## Built on by (most cited first, 8 of 8)

- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks](https://mlsanity.com/paper/2407.01613/) (2024)
- [Multifidelity Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.14764/) (2024)
- [Bi-fidelity Variational Auto-encoder for Uncertainty Quantification](https://mlsanity.com/paper/2305.16530/) (2023)
- [Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation](https://mlsanity.com/paper/2512.09165/) (2025)
- [Transfer Learning on Multi-Dimensional Data: A Novel Approach to Neural Network-Based Surrogate Modeling](https://mlsanity.com/paper/2410.12241/) (2024)
- [PC-SRGAN: Physically Consistent Super-Resolution Generative Adversarial Network for General Transient Simulations](https://mlsanity.com/paper/2505.06502/) (2025)
- [Bridging quantum and classical computing for partial differential equations through multifidelity machine learning](https://mlsanity.com/paper/2512.05241/) (2025)

## Abstract (arXiv)

Physics-informed neural networks have emerged as an alternative method for solving partial differential equations. However, for complex problems, the training of such networks can still require high-fidelity data which can be expensive to generate. To reduce or even eliminate the dependency on high-fidelity data, we propose a novel multi-fidelity architecture which is based on a feature space shared by the low- and high-fidelity solutions. In the feature space, the projections of the low-fidelity and high-fidelity solutions are adjacent by constraining their relative distance. The feature space is represented with an encoder and its mapping to the original solution space is effected through a decoder. The proposed multi-fidelity approach is validated on forward and inverse problems for steady and unsteady problems described by partial differential equations.