# MultiAuto-DeepONet: A Multi-resolution Autoencoder DeepONet for Nonlinear Dimension Reduction, Uncertainty Quantification and Operator Learning of Forward and Inverse Stochastic Problems

arXiv: https://arxiv.org/abs/2204.03193 · HTML: https://mlsanity.com/paper/2204.03193/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 64 references that are in the map.
- First posted on arXiv 7 Apr 2022.

## Builds on (most cited first, 21 of 21)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Bayesian SegNet: Model Uncertainty in Deep Convolutional Encoder-Decoder Architectures for Scene Understanding](https://mlsanity.com/paper/1511.02680/) (2015)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Evaluating Bayesian Deep Learning Methods for Semantic Segmentation](https://mlsanity.com/paper/1811.12709/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations](https://mlsanity.com/paper/1708.02691/) (2017)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [SpinalNet: Deep Neural Network with Gradual Input](https://mlsanity.com/paper/2007.03347/) (2020)
- [A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations](https://mlsanity.com/paper/1806.02957/) (2018)
- [Neural-net-induced Gaussian process regression for function approximation and PDE solution](https://mlsanity.com/paper/1806.11187/) (2018)
- [Learning functionals via LSTM neural networks for predicting vessel dynamics in extreme sea states](https://mlsanity.com/paper/1912.13382/) (2019)
- [Bayesian Sparse learning with preconditioned stochastic gradient MCMC and its applications](https://mlsanity.com/paper/2006.16376/) (2020)
- [Enhancing Sparsity of Hermite Polynomial Expansions by Iterative Rotations](https://mlsanity.com/paper/1506.04344/) (2015)

## Built on by (most cited first, 13 of 13)

- [The Impact of Large Language Models on Scientific Discovery: a Preliminary Study using GPT-4](https://mlsanity.com/paper/2311.07361/) (2023)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [Learning in latent spaces improves the predictive accuracy of deep neural operators](https://mlsanity.com/paper/2304.07599/) (2023)
- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [Modeling Unknown Stochastic Dynamical System Subject to External Excitation](https://mlsanity.com/paper/2406.15747/) (2024)
- [Towards Size-Independent Generalization Bounds for Deep Operator Nets](https://mlsanity.com/paper/2205.11359/) (2022)
- [Size Lowerbounds for Deep Operator Networks](https://mlsanity.com/paper/2308.06338/) (2023)
- [Data-driven Effective Modeling of Multiscale Stochastic Dynamical Systems](https://mlsanity.com/paper/2408.14821/) (2024)
- [Capturing the Diffusive Behavior of the Multiscale Linear Transport Equations by Asymptotic-Preserving Convolutional DeepONets](https://mlsanity.com/paper/2306.15891/) (2023)
- [DON-LSTM: Multi-Resolution Learning with DeepONets and Long Short-Term Memory Neural Networks](https://mlsanity.com/paper/2310.02491/) (2023)
- [A numerical study into neural network surrogate model performance for uncertainty propagation](https://mlsanity.com/paper/2605.16078/) (2026)
- [Data-driven Effective Modeling of Stochastic Chemical Reaction Networks](https://mlsanity.com/paper/2608.25421/) (2026)

## Abstract (arXiv)

A new data-driven method for operator learning of stochastic differential equations(SDE) is proposed in this paper. The central goal is to solve forward and inverse stochastic problems more effectively using limited data. Deep operator network(DeepONet) has been proposed recently for operator learning. Compared to other neural networks to learn functions, it aims at the problem of learning nonlinear operators. However, it can be challenging by using the original model to learn nonlinear operators for high-dimensional stochastic problems. We propose a new multi-resolution autoencoder DeepONet model referred to as MultiAuto-DeepONet to deal with this difficulty with the aid of convolutional autoencoder. The encoder part of the network is designed to reduce the dimensionality as well as discover the hidden features of high-dimensional stochastic inputs. The decoder is designed to have a special structure, i.e. in the form of DeepONet. The first DeepONet in decoder is designed to reconstruct the input function involving randomness while the second one is used to approximate the solution of desired equations. Those two DeepONets has a common branch net and two independent trunk nets. This architecture enables us to deal with multi-resolution inputs naturally. By adding L1 regularization to our network, we found the outputs from the branch net and two trunk nets all have sparse structures. This reduces the number of trainable parameters in the neural network thus making the model more efficient. Finally, we conduct several numerical experiments to illustrate the effectiveness of our proposed MultiAuto-DeepONet model with uncertainty quantification.