# Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Data

arXiv: https://arxiv.org/abs/2303.11379 · HTML: https://mlsanity.com/paper/2303.11379/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 60 references that are in the map.
- First posted on arXiv 20 Mar 2023.

## Builds on (most cited first, 22 of 22)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [A Tutorial on Principal Component Analysis](https://mlsanity.com/paper/1404.1100/) (2014)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Dimension-independent likelihood-informed MCMC](https://mlsanity.com/paper/1411.3688/) (2014)
- [A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion](https://mlsanity.com/paper/1308.1313/) (2013)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)
- [Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference](https://mlsanity.com/paper/2110.07653/) (2021)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems](https://mlsanity.com/paper/1308.6221/) (2013)
- [GMLS-Nets: A framework for learning from unstructured data](https://mlsanity.com/paper/1909.05371/) (2019)
- [Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet](https://mlsanity.com/paper/1410.1221/) (2014)
- [Nonlinear integro-differential operator regression with neural networks](https://mlsanity.com/paper/1810.08552/) (2018)
- [Error-in-variables modelling for operator learning](https://mlsanity.com/paper/2204.10909/) (2022)
- [Estimation of the Robin coefficient field in a Poisson problem with uncertain conductivity field](https://mlsanity.com/paper/1801.03592/) (2018)

## Built on by (most cited first, 4 of 4)

- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [Stratospheric aerosol source inversion: Noise, variability, and uncertainty quantification](https://mlsanity.com/paper/2409.06846/) (2024)
- [Random Forest Regression Feature Importance for Climate Impact Pathway Detection](https://mlsanity.com/paper/2409.16609/) (2024)
- [Spatio-temporal Multivariate Cluster Evolution Analysis for Detecting and Tracking Climate Impacts](https://mlsanity.com/paper/2410.16544/) (2024)

## Abstract (arXiv)

In complex large-scale systems such as climate, important effects are caused by a combination of confounding processes that are not fully observable. The identification of sources from observations of system state is vital for attribution and prediction, which inform critical policy decisions. The difficulty of these types of inverse problems lies in the inability to isolate sources and the cost of simulating computational models. Surrogate models may enable the many-query algorithms required for source identification, but data challenges arise from high dimensionality of the state and source, limited ensembles of costly model simulations to train a surrogate model, and few and potentially noisy state observations for inversion due to measurement limitations. The influence of auxiliary processes adds an additional layer of uncertainty that further confounds source identification. We introduce a framework based on (1) calibrating deep neural network surrogates to the flow maps provided by an ensemble of simulations obtained by varying sources, and (2) using these surrogates in a Bayesian framework to identify sources from observations via optimization. Focusing on an atmospheric dispersion exemplar, we find that the expressive and computationally efficient nature of the deep neural network operator surrogates in appropriately reduced dimension allows for source identification with uncertainty quantification using limited data. Introducing a variable wind field as an auxiliary process, we find that a Bayesian approximation error approach is essential for reliable source inversion when uncertainty due to wind stresses the algorithm.