Source-linked AI summary
Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identification
Shaoxing Mo, Nicholas Zabaras, Xiaoqing Shi, Jichun Wu
TL;DR
Jointly identifying a groundwater contaminant source and heterogeneous hydraulic conductivity creates a high-dimensional inverse problem that challenges existing surrogate methods. The paper combines an autoregressive convolutional surrogate with ILUES, achieving accurate forward approximations and inversion results with substantially fewer forward-model evaluations.
Problem
Joint identification of contaminant source characteristics and highly heterogeneous conductivity is a high-dimensional inverse problem, while existing surrogate methods suffer from the curse of dimensionality.
Method
The method uses a deep autoregressive neural network with dense convolutional encoder-decoder image-to-image regression, embedded within surrogate-based ILUES.
Results
The surrogate accurately approximates high-dimensional contaminant-transport mappings and surrogate-based ILUES achieves comparably accurate inversion and predictive uncertainty estimates with much fewer forward-model evaluations.
Takeaways & Limitations
The approach provides an efficient surrogate-based route for solving the demonstrated 686-dimensional groundwater contaminant source identification problem.
Takeaways & Limitations
Existing surrogate methods had previously been applied mainly to problems with fewer than 110 uncertain parameters or relatively simple flow models.
Abstract
from arXiv · showhide
Identification of a groundwater contaminant source simultaneously with the hydraulic conductivity in highly-heterogeneous media often results in a high-dimensional inverse problem. In this study, a deep autoregressive neural network-based surrogate method is developed for the forward model to allow us to solve efficiently such high-dimensional inverse problems. The surrogate is trained using limited evaluations of the forward model. Since the relationship between the time-varying inputs and outputs of the forward transport model is complex, we propose an autoregressive strategy, which treats the output at the previous time step as input to the network for predicting the output at the current time step. We employ a dense convolutional encoder-decoder network architecture in which the high-dimensional input and output fields of the model are treated as images to leverage the robust capability of convolutional networks in image-like data processing. An iterative local updating ensemble smoother (ILUES) algorithm is used as the inversion framework. The proposed method is evaluated using a synthetic contaminant source identification problem with 686 uncertain input parameters. Results indicate that, with relatively limited training data, the deep autoregressive neural network consisting of 27 convolutional layers is capable of providing an accurate approximation for the high-dimensional model input-output relationship. The autoregressive strategy substantially improves the network's accuracy and computational efficiency. The application of the surrogate-based ILUES in solving the inverse problem shows that it can achieve accurate inversion results and predictive uncertainty estimates.
Plain Language Summary
The paper develops a deep autoregressive neural-network surrogate for efficiently solving high-dimensional inverse problems with time-varying forward-model inputs and outputs.
- High-dimensional inverse problems are computationally expensive because they require many forward-model evaluations.
- The proposed surrogate uses an autoregressive strategy to predict current outputs from previous outputs and time-varying inputs.
- A dense convolutional encoder-decoder treats high-dimensional model fields as images for image-to-image regression.
- Nonlinear projections into low-dimensional latent spaces address the curse of dimensionality during surrogate construction.
1 Introduction
The study addresses computationally difficult joint identification of contaminant sources and heterogeneous conductivity fields by combining deep autoregressive surrogates with ILUES. The approach targets limitations of conventional surrogate construction in high-dimensional, nonlinear groundwater problems.
- Jointly identifying contaminant sources and heterogeneous conductivity fields produces a high-dimensional inverse problem.
- High-dimensional inverse problems require many forward-model evaluations, creating substantial computational expense.
- ILUES can require large ensembles and many iterations when uncertain inputs are numerous and input-output relationships are complex.
- Most existing surrogate methods face exponentially increasing construction costs as input dimensionality rises, while adaptive sampling offers limited improvement for high-dimensional problems.
- The proposed method combines a dense convolutional encoder-decoder with an autoregressive network that uses previous outputs to predict current time-dependent outputs.
- The integrated methodology jointly estimates the contaminant source and highly heterogeneous conductivity field in a synthetic problem with 686 unknown input parameters.
2 Governing Equations and Problem Formulation
The paper formulates contaminant source identification as an inverse problem using concentration and hydraulic-head measurements under steady groundwater flow. The forward model couples Darcy flow with advection-dispersion transport.
- The groundwater transport system assumes steady-state flow with advection and dispersion as the transport mechanisms.
- Darcy’s law computes pore-space velocity from hydraulic conductivity, effective porosity, and hydraulic head.
- MODFLOW solves the groundwater-flow equation, and the resulting velocity supplies the advection-dispersion transport model.
- MT3DMS solves the contaminant transport equation for concentration.
- The inverse problem simultaneously identifies source location, time-varying release strength, and the random hydraulic-conductivity field from concentration and hydraulic-head measurements.
3 Methods
The method combines ILUES with a deep convolutional surrogate and an autoregressive formulation to handle high-dimensional, time-varying contaminant transport efficiently. The network predicts each concentration field from conductivity, the current source term, and the previous output state.
- Iterative Local Updating Ensemble Smoother: ILUES iteratively updates each parameter sample using a local ensemble selected from model-response and parameter-distance criteria.The local ensemble is formed from samples with small combined distance, then updated through an ensemble-smoother step.
- Surrogate Modeling: The surrogate replaces expensive forward-model executions in ILUES with a fast-to-evaluate deep autoregressive neural network.This substitution targets the computational burden caused by the large ensemble sizes and iteration counts required in high-dimensional inverse problems.
- Deep Convolutional Neural Networks: The surrogate treats high-dimensional model inputs and outputs as image-like data for convolutional image-to-image regression.A fully convolutional encoder-decoder uses coarse feature extraction followed by decoder-based reconstruction, avoiding fully connected layers.
- Deep Autoregressive Neural Networks for Time-Varying Processes: The autoregressive model predicts yj from conductivity K, current source term Sj, and previous output yj−1, with y0 as the known initial state.This representation captures the dependence of each current concentration field on past source inputs and states.
- Deep Autoregressive Neural Networks for Time-Varying Processes: A single forward-model execution is reorganized into nt training samples, and predictions are generated sequentially without additional forward-model evaluations.The network first predicts y1 from (K, S1, y0), then repeatedly uses each predicted output with the next source term.
4 Application
The application evaluates the proposed surrogate and weighted-loss strategies on a synthetic, highly heterogeneous contaminant-transport inverse problem with 686 uncertain parameters. It compares nonautoregressive, autoregressive, and weighted-loss autoregressive networks using limited forward-model evaluations and test metrics.
- 4.1 Contaminant Transport Model: The synthetic two-dimensional domain spans 10 [L] × 20 [L] and is discretized into 41×81 cells with time-varying contaminant release.The model uses fixed-head left and right boundaries and no-flow lateral boundaries.
- 4.1 Contaminant Transport Model: The inverse problem contains 686 uncertain parameters, combining seven source parameters with 679 KLE conductivity coefficients.The retained KLE terms preserve approximately 95% of the total conductivity-field variance.
- 4.1 Contaminant Transport Model: The network generates conductivity fields from KLE coefficients and uses source-term images to predict hydraulic-head and concentration output fields.This image-based representation supplies the trained network with spatially organized conductivity and time-varying source information.
- 4.2 A Weighted Loss to Improve the Approximation of the Concentration Field: A weighted loss emphasizes the source pixel and its surrounding eight pixels because small dispersivity creates large concentration gradients near release locations.The additional weight is applied to the five concentration images corresponding to the contaminant-release time segments.
5 Results and Discussion
The autoregressive networks substantially improve surrogate accuracy for time-varying transport, while weighted loss further sharpens predictions near the source. The selected AR-Net-WL surrogate enables comparably accurate inversion with far fewer forward-model evaluations.
- Approximation Accuracy Assessment: Small wc values of 1.0, 3.0, 5.0, and 10.0 outperform large values of 30.0 and 50.0 for AR-Net-WL.Large weights focus training heavily on local refinement near the source-release region.
- Approximation Accuracy Assessment: With N = 400 evaluations, AR-Net and AR-Net-WL achieve RMSE values of 0.0762 and 0.0621, versus 0.3177 for Net.Their corresponding R2 values are 0.9749 and 0.9783, versus 0.8581 for Net.
- Approximation Accuracy Assessment: Weighted loss with wc = 5.0 improves sharp-concentration characterization near the source without degrading overall approximation accuracy.AR-Net-WL also achieves slightly lower RMSEs and higher R2 scores than AR-Net on the same training datasets.
- Inversion Results: The surrogate-based ILUES captures conductivity-field structure and source parameters comparably accurately while using 1500 rather than 126,000 forward-model runs.No additional forward-model executions are required during surrogate-based inversion.
- Inversion Results: The ILUES and surrogate-based ILUES approaches provide ensemble-based predictive uncertainty estimates for concentration and hydraulic head fields.The estimates are computed from outputs generated by the corresponding forward or surrogate model.
6 Conclusions
The study addresses joint contaminant-source and heterogeneous-conductivity identification with a deep autoregressive neural-network surrogate embedded in ILUES. In a synthetic 686-dimensional problem, the method accurately approximates transport and produces comparably accurate inversion and uncertainty estimates with much lower evaluation cost.
- Conclusions: The proposed method combines a deep autoregressive neural-network surrogate with ILUES for joint identification of contaminant source and hydraulic conductivity.High-dimensional input and output fields are treated as images in a convolutional encoder-decoder architecture.
- Conclusions: The autoregressive model predicts the current output from the uncertain input and the previous time-step output, improving approximation of time-varying transport.The formulation is yi = f(xi, yi−1).
- Conclusions: In a synthetic 686-dimensional inverse problem, limited forward-model evaluations yield accurate input-output approximation and improved representation of large concentration gradients.An additional loss term targets concentrations near the source-release location.
- Conclusions: Surrogate-based ILUES achieves comparably accurate inversion results and predictive uncertainty estimates while requiring much fewer forward-model evaluations than original ILUES.The reported comparison is between surrogate-based ILUES and ILUES without surrogate modeling.
- Conclusions: The case study does not evaluate combining the surrogate with design of experiments or applying it to inversion and uncertainty methods beyond groundwater solute transport.Those uses remain potential extensions rather than demonstrated results.