Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models

TL;DR

This work develops a total uncertainty quantification framework for inverse PDE solutions that rely on reduced-order deep learning surrogates. By formulating the inverse problem in latent space and employing a randomized, likelihood-free MAP sampling strategy, the approach accounts for measurement, PDE, and surrogate-model uncertainties, including non-Gaussian surrogate errors. The authors compare three strategies—rI-KL-DNN, DE-KL-DNN, and IES—using a nonlinear diffusion groundwater example, showing that the randomized inverse KL-DNN (rI-KL-DNN) yields more informative posterior distributions, especially for small data or ensemble sizes, and robust predictive capability for forecasted states. The method offers a scalable alternative to full PDE-based Bayesian inference in high-dimensional settings, with practical implications for data assimilation in subsurface applications. Mathematical rigor is maintained by leveraging KL representations, differentiable surrogates, and randomized objective functions to capture total uncertainty in a computationally efficient, likelihood-free framework.

Abstract

We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models, including operator learning models. The proposed method accounts for uncertainty in the observations and PDE and surrogate models. First, we use the surrogate model to formulate a minimization problem in the reduced space for the maximum a posteriori (MAP) inverse solution. Then, we randomize the MAP objective function and obtain samples of the posterior distribution by minimizing different realizations of the objective function. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation with an unknown space-dependent diffusion coefficient. Among other problems, this equation describes groundwater flow in an unconfined aquifer. Depending on the training dataset and ensemble sizes, the proposed method provides similar or more descriptive posteriors of the parameters and states than the iterative ensemble smoother method. Deep ensembling underestimates uncertainty and provides less informative posteriors than the other two methods.

Figures (7)

• Figure 1: The discretized domain of the hypothetical "Freyberg" aquifer. The color map shows the reference log hydraulic conductivity field $y_\text{ref}(\bm{x})=\ln K_\text{ref}(\bm{x})$. Inactive or no-flow cells, the stream reach, general head boundary (GHB) cells, pumping wells, and observation wells are also shown.
• Figure 2: The reference field $u_\text{ref}$ at times $t_1=10$, $t_2= 11$, and $t_3=12$ years (first row). The point error $\bar{u}_\text{rKL-DNN}(\bm{x},t)-u_\text{ref}(\bm{x},t)$ (second row). The standard deviation $\sigma_\text{rKL-DNN}(\bm{x},t)$ (third row). The rKL-DNN model is trained with $N_\text{train}=5000$ samples. The average standard deviations of $u$ at the observation locations are 0.024 at $t=t_1$, 0.023 at $t=t_2$, and 0.022 at $t=t_3$.
• Figure 3: The estimated $y$ field with DE-KL-DNN (left column), rKL-DNN (mid column), and IES (right column). $N_\text{train} = 5000$ and $\sigma^2_{y^s} = 0.01$. The top row is the estimated $\overline{y}(\bm{x})$, the mid row is the estimated $\sigma^2_y(\bm{x})$, and the bottom row is the coverage.
• Figure 4: The estimated $y$ field with DE-KL-DNN (left column), rKL-DNN (mid column), and IES (right column). $N_\text{train} = 5000$ and $\sigma^2_{y^s} = 10^{-4}$. The top row is the estimated $\overline{y}(\bm{x})$, the mid row is the estimated $\sigma^2_y(\bm{x})$, and the bottom row is the coverage.
• Figure 5: The $\ell_2$ and $\ell_\infty$ errors, coverages and LPPs of $y(\bm{x})$ estimated by DE KL-DNN (red lines), rKL-DNN (blue lines), IES (black lines) for $\sigma^2_{y^s} = 10^{-4}$ (dashed lines) and $10^{-2}$ (solid lines).