Scientific machine learning (SciML) has emerged recently as an effective and powerful tool for data fusion, solving ordinary/partial differential equations (ODEs, PDEs), and learning operator mappings in various scientific and engineering disciplines. Physics-informed neural networks (PINNs) and deep operator networks (DeepONets) are two such models for solving ODEs/PDEs and learning operator mappings, respectively. Quantifying predictive uncertainties is crucial for risk-sensitive applications as well as for efficient and economical design. NeuralUQ is a Python library for uncertainty quantification in various SciML algorithms. In NeuralUQ, each UQ method is decomposed into a surrogate and an inference method for posterior estimation. NeuralUQ has included various surrogates and inference methods, i.e.,

• Surrogates
  • Bayesian Neural Networks (BNNs)
  • Deterministic Neural Networks, e.g., fully-connected neural networks (FNNs)
  • Deep Generative Models, e.g., Generative Adversarial Nets (GANs)
• Inference Methods
  • Sampling methods
    • Hamiltonian Monte Carlo (HMC)
    • Langevin Dynamics (LD)
    • No-U-Turn (NUTS)
    • Metropolis-adjusted Langevin algorithm (MALA)
  • Variational Methods
    • Mean-field Variational Inference (MFVI)
    • Monte Carlo Dropout (MCD)
  • Ensemble Methods
    • Deep ensembles (DEns)
    • Snapshot ensemble (SEns)
    • Laplace approximation (LA)

Users can refer to this paper for the design and description, as well as the examples, of the NeuralUQ library:

• NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators

Users can refer to the following papers for more details on the algorithms:

• A comprehensive review on uncertainty quantification in scientific machine learning
• UQ for physics-informed neural networks
  • B-PINNs: Bayesian Physics-informed Networks
  • Learning Functional Priors and Posteriors from Data and Physics
  • Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations
  • ...
• UQ for DeepONets
  • Learning Functional Priors and Posteriors from Data and Physics
  • Bayesian DeepONets
  • Randomized Priors
  • ...

NeuralUQ requires the following dependencies to be installed:

• Python 3.7.0
• Tensorflow 2.9.1
• TensorFlow Probability 0.17.0

Then install with python:

$ python setup.py install

For developers, you could clone the folder to your local machine via

$ git clone https://github.com/Crunch-UQ4MI/neuraluq.git

NeuralUQ for uncertainty quantification in general neural differential equations and operators:

• NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators
• Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons
• Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators

NeuralUQ for physical model misspecification and uncertainty:

• Correcting model misspecification in physics-informed neural networks (PINNs)

NeuralUQ for Biomechanical constitutive models with experimental data (inferring model parameters from known model and data; inferring functions from pre-trained GAN and data):

• A generative modeling framework for inferring families of biomechanical constitutive laws in data-sparse regimes

Extensions of NeuralUQ:

• L-HYDRA: Multi-Head Physics-Informed Neural Networks

@article{zou2024neuraluq,
  title={NeuralUQ: A Comprehensive Library for Uncertainty Quantification in Neural Differential Equations and Operators},
  author={Zou, Zongren and Meng, Xuhui and Psaros, Apostolos F and Karniadakis, George E},
  journal={SIAM Review},
  volume={66},
  number={1},
  pages={161--190},
  year={2024},
  publisher={SIAM}
}