# Single- to multi-fidelity history-dependent learning with uncertainty quantification and disentanglement: application to data-driven constitutive modeling

arXiv: https://arxiv.org/abs/2507.13416 · HTML: https://mlsanity.com/paper/2507.13416/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 62 references that are in the map.
- First posted on arXiv 17 Jul 2025.

## Builds on (most cited first, 27 of 27)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Deep Bayesian Active Learning with Image Data](https://mlsanity.com/paper/1703.02910/) (2017)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networks](https://mlsanity.com/paper/1512.07666/) (2015)
- [Position: Bayesian Deep Learning is Needed in the Age of Large-Scale AI](https://mlsanity.com/paper/2402.00809/) (2024)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [Multi-Fidelity Bayesian Optimization via Deep Neural Networks](https://mlsanity.com/paper/2007.03117/) (2020)
- [Bayesian Recurrent Neural Networks](https://mlsanity.com/paper/1704.02798/) (2017)
- [Thermodynamically Consistent Machine-Learned Internal State Variable Approach for Data-Driven Modeling of Path-Dependent Materials](https://mlsanity.com/paper/2205.00578/) (2022)
- [Multi-fidelity surrogate modeling using long short-term memory networks](https://mlsanity.com/paper/2208.03115/) (2022)
- [How to Evaluate Uncertainty Estimates in Machine Learning for Regression?](https://mlsanity.com/paper/2106.03395/) (2021)
- [A physics-informed deep neural network for surrogate modeling in classical elasto-plasticity](https://mlsanity.com/paper/2204.12088/) (2022)
- [Neural networks meet anisotropic hyperelasticity: A framework based on generalized structure tensors and isotropic tensor functions](https://mlsanity.com/paper/2410.03378/) (2024)
- [Scalable Bayesian Learning of Recurrent Neural Networks for Language Modeling](https://mlsanity.com/paper/1611.08034/) (2016)
- [Spiderweb nanomechanical resonators via Bayesian optimization: inspired by nature and guided by machine learning](https://mlsanity.com/paper/2108.04809/) (2021)
- [Cooperative data-driven modeling](https://mlsanity.com/paper/2211.12971/) (2022)
- [BARNN: A Bayesian Autoregressive and Recurrent Neural Network](https://mlsanity.com/paper/2501.18665/) (2025)
- [Automatically Differentiable Model Updating (ADiMU): conventional, hybrid, and neural network material model discovery including history-dependency](https://mlsanity.com/paper/2505.07801/) (2025)
- [Adaptivity for clustering-based reduced-order modeling of localized history-dependent phenomena](https://mlsanity.com/paper/2109.11897/) (2021)
- [Consistent machine learning for topology optimization with microstructure-dependent neural network material models](https://mlsanity.com/paper/2408.13843/) (2024)
- [A heteroencoder architecture for prediction of failure locations in porous metals using variational inference](https://mlsanity.com/paper/2202.00078/) (2022)

## Built on by (most cited first, 1 of 1)

- [Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modeling](https://mlsanity.com/paper/2607.20339/) (2026)

## Abstract (arXiv)

Data-driven learning is generalized to consider history-dependent multi-fidelity data, while quantifying epistemic uncertainty and disentangling it from data noise (aleatoric uncertainty). This generalization is hierarchical and adapts to different learning scenarios: from training the simplest single-fidelity deterministic neural networks up to the proposed multi-fidelity variance estimation Bayesian recurrent neural networks. The versatility and generality of the proposed methodology are demonstrated by applying it to different data-driven constitutive modeling scenarios that include multiple fidelities with and without aleatoric uncertainty (noise). The method accurately predicts the response and quantifies model error while also discovering the noise distribution (when present). This opens opportunities for future real-world applications in diverse scientific and engineering domains; especially, the most challenging cases involving design and analysis under uncertainty.