# Multi-fidelity Bayesian Neural Networks: Algorithms and Applications

arXiv: https://arxiv.org/abs/2012.13294 · HTML: https://mlsanity.com/paper/2012.13294/

## Key facts

- Cited by 59 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2020.

## Built on by (most cited first, 30 of 59)

- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [Learning Functional Priors and Posteriors from Data and Physics](https://mlsanity.com/paper/2106.05863/) (2021)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities](https://mlsanity.com/paper/2102.13403/) (2021)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (2022)
- [Rethinking materials simulations: Blending direct numerical simulations with neural operators](https://mlsanity.com/paper/2312.05410/) (2023)
- [Multi-fidelity surrogate modeling using long short-term memory networks](https://mlsanity.com/paper/2208.03115/) (2022)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Multi-fidelity Hierarchical Neural Processes](https://mlsanity.com/paper/2206.04872/) (2022)
- [A DeepONet multi-fidelity approach for residual learning in reduced order modeling](https://mlsanity.com/paper/2302.12682/) (2023)
- [Multi-Fidelity Residual Neural Processes for Scalable Surrogate Modeling](https://mlsanity.com/paper/2402.18846/) (2024)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)
- [Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2205.06948/) (2022)
- [Disentangled Multi-Fidelity Deep Bayesian Active Learning](https://mlsanity.com/paper/2305.04392/) (2023)
- [A Latent Variable Approach for Non-Hierarchical Multi-Fidelity Adaptive Sampling](https://mlsanity.com/paper/2310.03298/) (2023)
- [Multifidelity Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2410.14764/) (2024)
- [Active Learning with Multifidelity Modeling for Efficient Rare Event Simulation](https://mlsanity.com/paper/2106.13790/) (2021)
- [Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations](https://mlsanity.com/paper/2303.11577/) (2023)
- [Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systems](https://mlsanity.com/paper/2109.00538/) (2021)
- [Multifidelity data fusion in convolutional encoder/decoder networks](https://mlsanity.com/paper/2205.05187/) (2022)
- [Residual Multi-Fidelity Neural Network Computing](https://mlsanity.com/paper/2310.03572/) (2023)
- [A Framework for Strategic Discovery of Credible Neural Network Surrogate Models under Uncertainty](https://mlsanity.com/paper/2403.08901/) (2024)
- [Multi-fidelity climate model parameterization for better generalization and extrapolation](https://mlsanity.com/paper/2309.10231/) (2023)
- [Practical multi-fidelity machine learning: fusion of deterministic and Bayesian models](https://mlsanity.com/paper/2407.15110/) (2024)
- [E-PINNs: Epistemic Physics-Informed Neural Networks](https://mlsanity.com/paper/2503.19333/) (2025)
- [Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensembles](https://mlsanity.com/paper/2604.17156/) (2026)
- [Towards Reliable Uncertainty Quantification via Deep Ensembles in Multi-output Regression Task](https://mlsanity.com/paper/2303.16210/) (2023)

## Abstract (arXiv)

We propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a posteriori probability (MAP) method to fit the low-fidelity data; the second is a Bayesian neural network employed to capture the cross-correlation with uncertainty quantification between the low- and high-fidelity data; and the last one is the physics-informed neural network, which encodes the physical laws described by PDEs. For the training of the last two neural networks, we use the Hamiltonian Monte Carlo method to estimate accurately the posterior distributions for the corresponding hyperparameters. We demonstrate the accuracy of the present method using synthetic data as well as real measurements. Specifically, we first approximate a one- and four-dimensional function, and then infer the reaction rates in one- and two-dimensional diffusion-reaction systems. Moreover, we infer the sea surface temperature (SST) in the Massachusetts and Cape Cod Bays using satellite images and in-situ measurements. Taken together, our results demonstrate that the present method can capture both linear and nonlinear correlation between the low- and high-fideilty data adaptively, identify unknown parameters in PDEs, and quantify uncertainties in predictions, given a few scattered noisy high-fidelity data. Finally, we demonstrate that we can effectively and efficiently reduce the uncertainties and hence enhance the prediction accuracy with an active learning approach, using as examples a specific one-dimensional function approximation and an inverse PDE problem.