# Novel Deep neural networks for solving Bayesian statistical inverse

arXiv: https://arxiv.org/abs/2102.03974 · HTML: https://mlsanity.com/paper/2102.03974/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 53 references that are in the map.
- First posted on arXiv 8 Feb 2021.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Layer-Parallel Training of Deep Residual Neural Networks](https://mlsanity.com/paper/1812.04352/) (2018)
- [Deep learning as optimal control problems: models and numerical methods](https://mlsanity.com/paper/1904.05657/) (2019)
- [Fractional Deep Neural Network via Constrained Optimization](https://mlsanity.com/paper/2004.00719/) (2020)
- [Machine Learning of Space-Fractional Differential Equations](https://mlsanity.com/paper/1808.00931/) (2018)

## Built on by (most cited first, 10 of 10)

- [Novel DNNs for Stiff ODEs with Applications to Chemically Reacting Flows](https://mlsanity.com/paper/2104.01914/) (2021)
- [Efficient Bayesian Physics Informed Neural Networks for Inverse Problems via Ensemble Kalman Inversion](https://mlsanity.com/paper/2303.07392/) (2023)
- [An Optimal Time Variable Learning Framework for Deep Neural Networks](https://mlsanity.com/paper/2204.08528/) (2022)
- [NINNs: Nudging Induced Neural Networks](https://mlsanity.com/paper/2203.07947/) (2022)
- [Enhanced BPINN Training Convergence in Solving General and Multi-scale Elliptic PDEs with Noise](https://mlsanity.com/paper/2408.09340/) (2024)
- [Neural Network Representation of Time Integrators](https://mlsanity.com/paper/2211.17039/) (2022)
- [A Note on Dimensionality Reduction in Deep Neural Networks using Empirical Interpolation Method](https://mlsanity.com/paper/2305.09842/) (2023)
- [Epistemic Modeling Uncertainty of Rapid Neural Network Ensembles for Adaptive Learning](https://mlsanity.com/paper/2309.06628/) (2023)
- [Time Regularization in Optimal Time Variable Learning](https://mlsanity.com/paper/2306.16111/) (2023)
- [OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case](https://mlsanity.com/paper/2509.08169/) (2025)

## Abstract (arXiv)

We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require several thousands of forward PDE solves. The goal of this paper is to introduce a fractional deep neural network based approach for the forward solves within an MCMC routine. Moreover, we discuss some approximation error estimates and illustrate the efficiency of our approach via several numerical examples.