# DiffHybrid-UQ: Uncertainty Quantification for Differentiable Hybrid Neural Modeling

arXiv: https://arxiv.org/abs/2401.00161 · HTML: https://mlsanity.com/paper/2401.00161/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 43 references that are in the map.
- First posted on arXiv 30 Dec 2023.

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

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Snapshot Ensembles: Train 1, get M for free](https://mlsanity.com/paper/1704.00109/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [A Differentiable Programming System to Bridge Machine Learning and Scientific Computing](https://mlsanity.com/paper/1907.07587/) (2019)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [Learned Turbulence Modelling with Differentiable Fluid Solvers: Physics-based Loss-functions and Optimisation Horizons](https://mlsanity.com/paper/2202.06988/) (2022)
- [Differentiable hybrid neural modeling for fluid-structure interaction](https://mlsanity.com/paper/2303.12971/) (2023)
- [DiffGCN: Graph Convolutional Networks via Differential Operators and Algebraic Multigrid Pooling](https://mlsanity.com/paper/2006.04115/) (2020)
- [AMAGOLD: Amortized Metropolis Adjustment for Efficient Stochastic Gradient MCMC](https://mlsanity.com/paper/2003.00193/) (2020)
- [Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty](https://mlsanity.com/paper/2210.08095/) (2022)
- [Sequential Bayesian Neural Subnetwork Ensembles](https://mlsanity.com/paper/2206.00794/) (2022)
- [Prediction of Reynolds Stresses in High-Mach-Number Turbulent Boundary Layers using Physics-Informed Machine Learning](https://mlsanity.com/paper/1808.07752/) (2018)

## Built on by (most cited first, 5 of 5)

- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Assessment of Uncertainty Quantification in Universal Differential Equations](https://mlsanity.com/paper/2406.08853/) (2024)
- [Uncertainty in Supply Chain Digital Twins: A Quantum-Classical Hybrid Approach](https://mlsanity.com/paper/2411.10254/) (2024)
- [Implicit Neural Differential Model for Spatiotemporal Dynamics](https://mlsanity.com/paper/2504.02260/) (2025)
- [LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process](https://mlsanity.com/paper/2507.22493/) (2025)

## Abstract (arXiv)

The hybrid neural differentiable models mark a significant advancement in the field of scientific machine learning. These models, integrating numerical representations of known physics into deep neural networks, offer enhanced predictive capabilities and show great potential for data-driven modeling of complex physical systems. However, a critical and yet unaddressed challenge lies in the quantification of inherent uncertainties stemming from multiple sources. Addressing this gap, we introduce a novel method, DiffHybrid-UQ, for effective and efficient uncertainty propagation and estimation in hybrid neural differentiable models, leveraging the strengths of deep ensemble Bayesian learning and nonlinear transformations. Specifically, our approach effectively discerns and quantifies both aleatoric uncertainties, arising from data noise, and epistemic uncertainties, resulting from model-form discrepancies and data sparsity. This is achieved within a Bayesian model averaging framework, where aleatoric uncertainties are modeled through hybrid neural models. The unscented transformation plays a pivotal role in enabling the flow of these uncertainties through the nonlinear functions within the hybrid model. In contrast, epistemic uncertainties are estimated using an ensemble of stochastic gradient descent (SGD) trajectories. This approach offers a practical approximation to the posterior distribution of both the network parameters and the physical parameters. Notably, the DiffHybrid-UQ framework is designed for simplicity in implementation and high scalability, making it suitable for parallel computing environments. The merits of the proposed method have been demonstrated through problems governed by both ordinary and partial differentiable equations.