# Multi-Fidelity Delayed Acceptance: hierarchical MCMC sampling for Bayesian inverse problems combining multiple solvers through deep neural networks

arXiv: https://arxiv.org/abs/2512.16430 · HTML: https://mlsanity.com/paper/2512.16430/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 72 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Builds on (most cited first, 19 of 19)

- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Multi-fidelity regression using artificial neural networks: efficient approximation of parameter-dependent output quantities](https://mlsanity.com/paper/2102.13403/) (2021)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2403.08220/) (2024)
- [Multilevel Delayed Acceptance MCMC](https://mlsanity.com/paper/2202.03876/) (2022)
- [Multifidelity Simulation-based Inference for Computationally Expensive Simulators](https://mlsanity.com/paper/2502.08416/) (2025)
- [Online structural health monitoring by model order reduction and deep learning algorithms](https://mlsanity.com/paper/2103.14328/) (2021)
- [Deep Kernel Learning of Dynamical Models from High-Dimensional Noisy Data](https://mlsanity.com/paper/2208.12975/) (2022)
- [EKF-SINDy: Empowering the extended Kalman filter with sparse identification of nonlinear dynamics](https://mlsanity.com/paper/2404.07536/) (2024)
- [A Delayed Acceptance Auxiliary Variable MCMC for Spatial Models with Intractable Likelihood Function](https://mlsanity.com/paper/2504.17147/) (2025)
- [Progressive multi-fidelity learning with neural networks for physical system predictions](https://mlsanity.com/paper/2510.13762/) (2025)
- [Bayesian Calibration of MEMS Accelerometers](https://mlsanity.com/paper/2306.06144/) (2023)

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

- [Multi-Fidelity SINDy: Sparse Discovery of Nonlinear Dynamical Systems with Fidelity-Weighted Measurements](https://mlsanity.com/paper/2606.15690/) (2026)

## Abstract (arXiv)

Inverse uncertainty quantification (UQ) tasks such as parameter estimation are computationally demanding whenever dealing with physics-based models, and typically require repeated evaluations of complex numerical solvers. When partial differential equations are involved, full-order models such as those based on the Finite Element Method can make traditional sampling approaches like Markov Chain Monte Carlo (MCMC) computationally infeasible. Although data-driven surrogate models may help reduce evaluation costs, their utility is often limited by the expense of generating high-fidelity data. In contrast, low-fidelity data can be produced more efficiently, although relying on them alone may degrade the accuracy of the inverse UQ solution. To address these challenges, we propose a Multi-Fidelity Delayed Acceptance scheme for Bayesian inverse problems. Extending the Multi-Level Delayed Acceptance framework, the method introduces multi-fidelity neural networks that combine the predictions of solvers of varying fidelity, with high fidelity evaluations restricted to an offline training stage. During the online phase, likelihood evaluations are obtained by evaluating the coarse solvers and passing their outputs to the trained neural networks, thereby avoiding additional high-fidelity simulations. This construction allows heterogeneous coarse solvers to be incorporated consistently within the hierarchy, providing greater flexibility than standard Multi-Level Delayed Acceptance. The proposed approach improves the approximation accuracy of the low fidelity solvers, leading to longer sub-chain lengths, better mixing, and accelerated posterior inference. The effectiveness of the strategy is demonstrated on two benchmark inverse problems involving (i) steady isotropic groundwater flow, (ii) an unsteady reaction-diffusion system, for which substantial computational savings are obtained.