# Physics-Informed Machine Learning Method for Large-Scale Data Assimilation Problems

arXiv: https://arxiv.org/abs/2108.00037 · HTML: https://mlsanity.com/paper/2108.00037/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 43 references that are in the map.
- First posted on arXiv 30 Jul 2021.

## Builds on (most cited first, 8 of 8)

- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks](https://mlsanity.com/paper/2105.13136/) (2021)
- [Physics-Informed CoKriging: A Gaussian-Process-Regression-Based Multifidelity Method for Data-Model Convergence](https://mlsanity.com/paper/1811.09757/) (2018)
- [Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs](https://mlsanity.com/paper/1910.13444/) (2019)
- [Approximate Bayesian Model Inversion for PDEs with Heterogeneous and State-Dependent Coefficients](https://mlsanity.com/paper/1902.06718/) (2019)
- [Conditional Karhunen-Lo`eve expansion for uncertainty quantification and active learning in partial differential equation models](https://mlsanity.com/paper/1904.08069/) (2019)
- [Learning to regularize with a variational autoencoder for hydrologic inverse analysis](https://mlsanity.com/paper/1906.02401/) (2019)

## Built on by (most cited first, 6 of 6)

- [Score-based Data Assimilation](https://mlsanity.com/paper/2306.10574/) (2023)
- [Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions](https://mlsanity.com/paper/2208.08635/) (2022)
- [Randomized Physics-Informed Machine Learning for Uncertainty Quantification in High-Dimensional Inverse Problems](https://mlsanity.com/paper/2312.06177/) (2023)
- [Gaussian process regression and conditional Karhunen-Lo'eve models for data assimilation in inverse problems](https://mlsanity.com/paper/2301.11279/) (2023)
- [Physics-informed solution reconstruction in elasticity and heat transfer using the explicit constraint force method](https://mlsanity.com/paper/2505.04875/) (2025)
- [Conditional Korhunen-Lo'eve regression model with Basis Adaptation for high-dimensional problems: uncertainty quantification and inverse modeling](https://mlsanity.com/paper/2307.02572/) (2023)

## Abstract (arXiv)

We develop a physics-informed machine learning approach for large-scale data assimilation and parameter estimation and apply it for estimating transmissivity and hydraulic head in the two-dimensional steady-state subsurface flow model of the Hanford Site given synthetic measurements of said variables. In our approach, we extend the physics-informed conditional Karhunen-Lo'eve expansion (PICKLE) method for modeling subsurface flow with unknown flux (Neumann) and varying head (Dirichlet) boundary conditions. We demonstrate that the PICKLE method is comparable in accuracy with the standard maximum a posteriori (MAP) method, but is significantly faster than MAP for large-scale problems. Both methods use a mesh to discretize the computational domain. In MAP, the parameters and states are discretized on the mesh; therefore, the size of the MAP parameter estimation problem directly depends on the mesh size. In PICKLE, the mesh is used to evaluate the residuals of the governing equation, while the parameters and states are approximated by the truncated conditional Karhunen-Lo'eve expansions with the number of parameters controlled by the smoothness of the parameter and state fields, and not by the mesh size. For a considered example, we demonstrate that the computational cost of PICKLE increases near linearly (as NFV1.15) with the number of grid points NFV, while that of MAP increases much faster as NFV3.28. We demonstrated that once trained for one set of Dirichlet boundary conditions (i.e., one river stage), the PICKLE method provides accurate estimates of the hydraulic head for any value of the Dirichlet boundary conditions (i.e., for any river stage).