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Randomized physics-informed machine learning for uncertainty quantification in high-dimensional inverse problems

Journal Article · · Journal of Computational Physics
 [1];  [2];  [3]
  1. University of Illinois at Urbana-Champaign, IL (United States)
  2. Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
  3. University of Illinois at Urbana-Champaign, IL (United States); Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
+ Show Author Affiliations
We propose the randomized physics-informed conditional Karhunen-Loève expansion (rPICKLE) method for uncertainty quantification in high-dimensional inverse problems. In rPICKLE, the states and parameters of the governing partial differential equation (PDE) are approximated via truncated conditional Karhunen-Loève expansions (cKLEs). Uncertainty in the inverse solution is quantified via the posterior distribution of cKLE coefficients formulated with independent standard normal priors and a likelihood containing PDE residuals evaluated over the computational domain. The maximum a posteriori (MAP) estimate of the cKLE coefficients is found by minimizing a loss function given (up to a constant) by the negative log posterior. The posterior is sampled by adding zero-mean Gaussian noises into the MAP loss function and minimizing the loss for different noise realizations. For linear and low-dimensional nonlinear problems, we show that the rPICKLE posterior converges to the true Bayesian posterior. For high-dimensional non-linear problems, we obtain rPICKLE posterior approximations with high log-predictive probability. For a low-dimensional problem, the traditional Hamiltonian Monte Carlo (HMC) and Stein Variational Gradient Descent (SVGD) methods yield similar (to rPICKLE) posteriors. However, both HMC and SVGD fail for the high-dimensional problem. These results demonstrate the advantages of rPICKLE for approximately sampling high-dimensional posterior distributions.
Research Organization:
Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
Sponsoring Organization:
National Science Foundation (NSF); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
Grant/Contract Number:
AC05-76RL01830
OSTI ID:
2440115
Alternate ID(s):
OSTI ID: 2485356
Report Number(s):
PNNL-SA--203615
Journal Information:
Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 519; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

References (31)

Uncertainty Quantification in Scale‐Dependent Models of Flow in Porous Media journal November 2017
Parameter estimation and predictive uncertainty in stochastic inverse modeling of groundwater flow: Comparing null-space Monte Carlo and multiple starting point methods: NULL-SPACE MONTE CARLO AND MULTIPLE STARTING POINT journal January 2013
Ensemble Randomized Maximum Likelihood Method as an Iterative Ensemble Smoother journal December 2011
Inverse methods in hydrogeology: Evolution and recent trends journal January 2014
On uncertainty quantification in hydrogeology and hydrogeophysics journal December 2017
Improved training of physics-informed neural networks for parabolic differential equations with sharply perturbed initial conditions journal September 2023
Bayesian reduced-order deep learning surrogate model for dynamic systems described by partial differential equations journal September 2024
A model-independent iterative ensemble smoother for efficient history-matching and uncertainty quantification in very high dimensions journal November 2018
A review of uncertainty quantification in deep learning: Techniques, applications and challenges journal December 2021
Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems journal April 2009
Gaussian process regression and conditional polynomial chaos for parameter estimation journal September 2020
Physics-informed machine learning with conditional Karhunen-Loève expansions journal February 2021
Physics-constrained bayesian neural network for fluid flow reconstruction with sparse and noisy data journal March 2020
Inverse problems: A Bayesian perspective journal May 2010
A hybrid regularized inversion methodology for highly parameterized environmental models: HYBRID REGULARIZATION METHODOLOGY journal October 2005
Predictive uncertainty analysis of a saltwater intrusion model using null-space Monte Carlo: ANALYSIS OF A SALTWATER INTRUSION MODEL journal May 2011
Physics‐Informed Deep Neural Networks for Learning Parameters and Constitutive Relationships in Subsurface Flow Problems journal May 2020
Physics‐Informed Neural Network Method for Forward and Backward Advection‐Dispersion Equations journal July 2021
Physics‐Informed Machine Learning Method for Large‐Scale Data Assimilation Problems journal May 2022
Pilot Point Methodology for Automated Calibration of an Ensemble of conditionally Simulated Transmissivity Fields: 1. Theory and Computational Experiments journal March 1995
Estimation of Aquifer Parameters Under Transient and Steady State Conditions: 2. Uniqueness, Stability, and Solution Algorithms journal February 1986
Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators journal March 2021
Variational Inference: A Review for Statisticians journal July 2016
Numerical identifications of parameters in parabolic systems journal February 1998
Hamiltonian Monte Carlo solution of tomographic inverse problems journal November 2018
Riemann manifold Langevin and Hamiltonian Monte Carlo methods: Riemann Manifold Langevin and Hamiltonian Monte Carlo Methods journal March 2011
Randomize-Then-Optimize: A Method for Sampling from Posterior Distributions in Nonlinear Inverse Problems journal January 2014
A Randomized Maximum A Posteriori Method for Posterior Sampling of High Dimensional Nonlinear Bayesian Inverse Problems journal January 2018
A Subspace, Interior, and Conjugate Gradient Method for Large-Scale Bound-Constrained Minimization Problems journal January 1999
Hamiltonian Monte Carlo in Inverse Problems. Ill-Conditioning and Multimodality journal January 2023
Optimal tuning of the hybrid Monte Carlo algorithm journal November 2013

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Related Subjects

97 MATHEMATICS AND COMPUTING
Approximate Bayesian inference
Physics-informed machine learning
Bayesian inference
Dimension reduction
Hamiltonian Monte Carlo
Inverse problems
Uncertainty quantification