Forward Solver Research Articles

Some Remarks About Forward and Inverse Modelling in Hydrology, Within a General Conceptual Framework

The solution to inverse problems is crucial for model calibration and to provide a good basis for model results to be reliable. This paper is based on a recently proposed conceptual framework for the development and application of mathematical models that require the solution of forward and inverse problems. The focus of this paper is on the discussion of some terminology related to the results of forward problems and their reanalysis, on the use of the proposed framework to revise and generalise some methods of solutions of the inverse problem, and to provide a non-standard insight in some aspects about the Bayesian approach to model calibration.

Amplitude variation with azimuth direct inversion for effective pressure-sensitivity parameters in horizontal transversely isotropic media

Effective pressure prediction is of great significance for fracturability evaluation in unconventional reservoirs. Quantitative characterization of effective pressure remains a great challenge in the inversion prediction of subsurface media. Our goal is to develop a novel Bayesian amplitude varying with azimuth (AVAZ) inversion method to estimate fracture weaknesses and effective pressure for a horizontal transversely isotropic (HTI) medium. Using Schoenberg’s linear slip model and pore space compressibility theory, an anisotropic saturated stiffness matrix directly characterized by effective pressure is derived based on a weak-contrast interface separating two weakly anisotropic half-spaces of HTI medium. Based on the perturbation stiffness matrix and scattering theory, we derive a PP-wave reflection coefficient approximation directly characterized by fracture weaknesses and effective pressure-sensitive parameters, which establishes a quantitative relationship between seismic reflection coefficient and effective pressure. The analysis of accuracy and sensitivity is performed to verify the reliability and applicability of the constructed reflection coefficient. And then, the azimuth-amplitude-difference forward solver is constructed based on seismic reflection coefficients to eliminate the influence of isotropic background perturbations on seismic reflection coefficients and reduce the condition number of the forward solver. Finally, we present a novel Bayesian azimuth-amplitude-difference AVAZ inversion method for transversely isotropic media constrained by Cauchy and Gaussian regularization to directly estimate the fracture weaknesses and effective pressure-sensitive parameters. The application of synthetic data and field data verifies that the inversion results are in good agreement with the actual data, indicating that the method can effectively realize the stable prediction of anisotropic parameters and effective pressure-sensitive parameters of complex media.

Improving EEG Forward Modeling Using High-Resolution Five-Layer BEM-FMM Head Models: Effect on Source Reconstruction Accuracy

Electroencephalographic (EEG) source localization is a fundamental tool for clinical diagnoses and brain-computer interfaces. We investigate the impact of model complexity on reconstruction accuracy by comparing the widely used three-layer boundary element method (BEM) as an inverse method against a five-layer BEM accelerated by the fast multipole method (BEM-FMM) and coupled with adaptive mesh refinement (AMR) as forward solver. Modern BEM-FMM with AMR can solve high-resolution multi-tissue models efficiently and accurately. We generated noiseless 256-channel EEG data from 15 subjects in the Connectome Young Adult dataset, using four anatomically relevant dipole positions, three conductivity sets, and two head segmentations; we mapped localization errors across the entire grey matter from 4,000 dipole positions. The average location error among our four selected dipoles is ∼5mm (±2mm) with an orientation error of ∼12∘ (±7∘). The average source localization error across the entire grey matter is ∼9mm (±4mm), with a tendency for smaller errors on the occipital lobe. Our findings indicate that while three-layer models are robust under noiseless conditions, substantial localization errors (10–20mm) are common. Therefore, models of five or more layers may be needed for accurate source reconstruction in critical applications involving noisy EEG data.

Seismic traveltime tomography based on ensemble Kalman inversion

SUMMARY In this paper, we present a new seismic traveltime tomography approach that combines ensemble Kalman inversion (EKI) with neural networks (NNs) to facilitate the inference of complex underground velocity fields. Our methodology tackles the challenges of high-dimensional velocity models through an efficient NN parametrization, enabling efficient training on coarse grids and accurate output on finer grids. This unique strategy, combined with a reduced-resolution forward solver, significantly enhances computational efficiency. Leveraging the robust capabilities of EKI, our method not only achieves rapid computations but also delivers informative uncertainty quantification for the estimated results. Through extensive numerical experiments, we demonstrate the exceptional accuracy and uncertainty quantification capabilities of our EKI-NNs approach. Even in the face of challenging geological scenarios, our method consistently generates valuable initial models for full wave inversion (FWI).

Deep Learning-Based Fast Full-Wave Electromagnetic Forward Solver Using Physics-Induced Loss

Deep Learning-Based Fast Full-Wave Electromagnetic Forward Solver Using Physics-Induced Loss

PC-Kriging-powered parallelizing Bayesian updating for stochastic vehicle-track dynamical system with contact force measurements and Gaussian process discrepancy model

High-precision vehicle-track coupled dynamical models play a vital role in assessing the running safety of the vehicle, tracking fatigue behavior, and establishing a digital twin for high-speed railways. This study established a high-fidelity vehicle-track coupled dynamical model as a forward solver, which was then updated in a Bayesian inference framework based on the field tests of rail irregularities and wheel/rail normal contact forces. The prediction errors corresponding to different time steps were represented through a Gaussian process (GP) model characterized by a covariance function with the Gaussian kernel to incorporate the correlation between different time steps. To cope with the considerable expense of the likelihood function calculations with large-scale loop operations and the computational burden due to the large batch of repetitive evaluations of the high-fidelity forward model involved in the stochastic sampling from the posterior probability distribution, a metamodel-powered parallelizing stochastic sampling scheme was adopted. The posterior distribution could be formulated by replacing the expensive explicit model evaluation involved in the likelihood function with a Polynomial-Chaos-Kriging (PC-Kriging) predictor providing a surrogate mapping between the wheel/rail normal contact forces and physical parameters. Finally, a parallelizing computing scheme running across multiple CPU cores on high-performance computers was realized to sample from the vectorized formula of the emulator-powered posterior distribution. Field test data from a high-speed railway in China was adopted to demonstrate that the Bayesian inference scheme can quantify the uncertainties of the core physical parameters of the high-fidelity vehicle-track coupled dynamical system by efficiently trading off accuracy and computational efficiency.

Flexible-Center Hat Complete Electrode Model for EEG Forward Problem.

This study aims to develop a more realistic electrode model by incorporating the non-uniform distribution of electrode contact conductance (ECC) and the shunting effects, to accurately solve EEG forward problem (FP). Firstly, a hat function is introduced to construct a more realistic hat-shaped distribution (HD) for ECC. Secondly, this hat function is modified by applying two parameters - offset ratio and offset direction - to account for the variability in ECC's center and to develop the flexible-center HD (FCHD). Finally, by integrating this FCHD into the complete electrode model (CEM) with the shunting effects, a novel flexible-center hat complete electrode model (FCH-CEM) is proposed and used to solve FP. Simulation experiments using a realistic head model demonstrate the necessity of FCH-CEM and its potential to improve the accuracy of the FP solution compared to current models, i.e., the point electrode model (PEM) and CEM. And compared to PEM, it has better performance under coarse mesh conditions (2 mm). Further experiments indicate the significance of considering shunting effects, as ignoring them results in larger errors than coarse mesh when the average contact conductance is large (101S/m2). The proposed FCH-CEM has better accuracy and performance than PEM and complements CEM in finer meshes, making it necessary for coarse meshes. This study proposes a novel model that enhances electrode modeling and FP accuracy, and provides new ideas and methods for future research.

Forward and Inverse Modeling of Ice Sheet Flow Using Physics‐Informed Neural Networks: Application to Helheim Glacier, Greenland

AbstractPredicting the future contribution of the ice sheets to sea level rise over the next decades presents several challenges due to a poor understanding of critical boundary conditions, such as basal sliding. Traditional numerical models often rely on data assimilation methods to infer spatially variable friction coefficients by solving an inverse problem, given an empirical friction law. However, these approaches are not versatile, as they sometimes demand extensive code development efforts when integrating new physics into the model. Furthermore, this approach makes it difficult to handle sparse data effectively. To tackle these challenges, we use the Physics‐Informed Neural Networks (PINNs) to seamlessly integrate observational data and governing equations of ice flow into a unified loss function, facilitating the solution of both forward and inverse problems within the same framework. We illustrate the versatility of this approach by applying the framework to two‐dimensional problems on the Helheim Glacier in southeast Greenland. By systematically concealing one variable (e.g., ice speed, ice thickness, etc.), we demonstrate the ability of PINNs to accurately reconstruct hidden information. Furthermore, we extend this application to address a challenging mixed inversion problem. We show how PINNs are capable of inferring the basal friction coefficient while simultaneously filling gaps in the sparsely observed ice thickness. This unified framework offers a promising avenue to enhance the predictive capabilities of ice sheet models, reducing uncertainties, and advancing our understanding of poorly constrained physical processes.

Uncertainty propagation for MAPoD in eddy current NDT by polynomial chaos–kriging

ABSTRACT In this article, the Polynomial Chaos–Kriging (PCK) metamodel is proposed to surrogate the physical model for model-assisted probability of detection (MAPoD) analysis in eddy current non-destructive testing (ECNDT). Two popular non-intrusive metamodels, the Polynomial Chaos Expansion (PCE) and Kriging, are combined properly to replace the time-consuming boundary element method-based ECNDT forward solver for accelerating the uncertainty propagation within MAPoD analysis in NDT. In PCK, the global behaviour of the model is estimated by the trend function of PCE, and the remaining local changes of it are captured by Kriging. Both the accuracy and efficiency of the proposed PCK metamodel are demonstrated on 3D ECNDT MAPoD cases involving surface flaws on the conducting thick plate using the detecting coil with rectangular cross section. Numerical results show improved performance of PCK metamodel over PCE and Kriging metamodels, respectively, while maintaining at least two digits of accuracy of the PoD metrics.

Deep-Learning-Based Simulation and Inversion of Transient Electromagnetic Sounding Signals in Permafrost Monitoring Problem

Abstract —This article presents a novel approach to addressing the challenges in permafrost monitoring through the integration of deep-learning techniques with conventional electromagnetic sounding methods. Our methodology comprises a forward finite element method (FEM) solver, augmented with the Sumudu transform, and an artificial neural network (ANN) solver trained on FEM-generated data. Remarkably, the ANN solver demonstrates similar accuracy to the FEM solver but operates at a superior speed that is nearly 10,000 times faster. Furthermore, we introduce an inverse problem solution drawing on the PARS algorithm. In addition, we present an ANN-based inverse solver, where the input and output roles are inverted. The ANN inverse solver is trained on the same data, thereby offering an alternative approach to solving the inverse problem. In a computational experiment, we compare the numerical inversion results using the PARS algorithm with those obtained from the ANN forward solver, ANN inversion, and a linear combination of these solutions. This comprehensive analysis sheds light on the effectiveness of our deep-learning-based approach in permafrost monitoring, providing insights for future applications in geophysics and environmental science.

Image Reconstruction Method Based on Iterative Linear Back Projection and Logistic Forward Solver for Electrical Capacitance Tomography Measurement System

Image Reconstruction Method Based on Iterative Linear Back Projection and Logistic Forward Solver for Electrical Capacitance Tomography Measurement System

StocIPNet: A novel probabilistic interpretable network with affine-embedded reparameterization layer for high-dimensional stochastic inverse problems

The stochastic inverse problem (StocIP), which aims to align push-forward and observed output distributions by estimating probability distributions of unknown system inputs, often faces optimization challenges and the curse of dimensionality. A novel deep network called StocIPNet which comprises an affine-embedded reparameterization subnetwork (ReparNet) and a complex system metamodeling subnetwork (MetaNet) is proposed to alleviate these issues. The ReparNet subnetwork embeds the affine transformation to convert the statistical parameters of the physical random vector into learnable weights and biases, effectively implementing the reparameterization trick by separating random and deterministic elements in the stochastic sampling operation to preserve differentiability. In parallel, the MetaNet subnetwork offers a computationally efficient alternative to time-consuming forward solvers, facilitating the generation of push-forward distributions. The entire StocIPNet utilizes the kernel maximum mean discrepancy (MMD) as a distribution-free loss function, quantifying the discrepancy between push-forward and observed output distributions. By leveraging the ReparNet’s advantage of reformulating the sampling process as a differentiable transformation and combining two subnetworks seamlessly, the StocIP is reconfigured into a pure network training paradigm preserving differentiability perfectly, which allows for direct modeling and efficient inference of uncertainty within the network using automatic differentiation, backpropagation and gradient-based optimization methods, enabling ease of scaling to high-dimensional problems. The proposed framework has been theoretically demonstrated to be equivalent to the maximum likelihood method, ensuring its solid probabilistic interpretable foundation. The proposed framework is applied to perform stochastic model updating on a numerical and an experimental structure, which effectively demonstrates the framework’s remarkable effectiveness and high efficiency in treating high-dimensional StocIPs.

Fast and Accurate EEG/MEG BEM-Based Forward Problem Solution for High-Resolution Head Models.

A BEM (boundary element method) based approach is developed to accurately solve an EEG/MEG forward problem for a modern high-resolution head model in approximately 60 seconds using a common workstation. The method utilizes a charge-based BEM with fast multipole acceleration (BEM-FMM) and a "smart" mesh pre-refinement (called b-refinement) close to the singular source(s). No costly matrix-filling or direct solution steps typical for the standard BEM are required; the method generates on-skin voltages as well as MEG magnetic fields for high-resolution head models in approximately 60 seconds after initial model assembly. The method is verified both theoretically and experimentally.

Transient forward harmonic adjoint sensitivity analysis

AbstractThis paper presents a transient forward harmonic adjoint sensitivity analysis (TFHA), which is a combination of a transient forward circuit analysis with a harmonic balance-based adjoint sensitivity analysis. TFHA provides sensitivities of quantities of interest from time-periodic problems with many design parameters, as used in the design process of power-electronics devices. The TFHA shows advantages in applications where the harmonic balance-based adjoint sensitivity analysis or finite difference approaches for sensitivity analysis perform poorly. In contrast to existing methods, the TFHA can be used in combination with arbitrary forward solvers, i.e., general transient solvers.

A replica exchange preconditioned Crank-Nicolson Langevin dynamic MCMC method with multi-variance strategy for Bayesian inverse problems

The preconditioned Crank-Nicolson Markov Chain Monte Carlo (pCN-MCMC) method is an efficient approach for large-scale Bayesian inference problems. It is constructed by discretizing a stochastic partial differential equation (SPDE). When the gradient information is coupled in this SPDE, the corresponding SPDE is also referred to as the Langevin diffusion. In this work, we present the replica exchange preconditioned Crank-Nicolson Langevin dynamic (repCNLD) Monte Carlo method to speed up the convergence of the pCN-MCMC and tackle the challenge of multimodal distribution. We use the Crank-Nicolson scheme to discretize the repLD and set the preconditioned matrix as the covariance matrix of the Gaussian prior. The discretization error is analyzed. Moreover, we use the multi-variance strategy (m-repCNLD) to improve computational efficiency. Perturbations are introduced into the scheme due to the likelihood estimation error, leading to the biased swapping rate. To correct the bias, we provide an unbiased estimator derived from some assumptions on the forward solvers. The accuracy and efficiency of the proposed method are demonstrated by sampling Gaussian mixture distribution and solving PDE-based nonlinear inverse problems numerically.

Assessing and Improving the Robustness of Bayesian Evidential Learning in One Dimension for Inverting Time-Domain Electromagnetic Data: Introducing a New Threshold Procedure

Understanding the subsurface is of prime importance for many geological and hydrogeological applications. Geophysical methods offer an economical alternative for investigating the subsurface compared to costly borehole investigations. However, geophysical results are commonly obtained through deterministic inversion of data whose solution is non-unique. Alternatively, stochastic inversions investigate the full uncertainty range of the obtained models, yet are computationally more expensive. In this research, we investigate the robustness of the recently introduced Bayesian evidential learning in one dimension (BEL1D) for the stochastic inversion of time-domain electromagnetic data (TDEM). First, we analyse the impact of the accuracy of the numerical forward solver on the posterior distribution, and derive a compromise between accuracy and computational time. We also introduce a threshold-rejection method based on the data misfit after the first iteration, circumventing the need for further BEL1D iterations. Moreover, we analyse the impact of the prior-model space on the results. We apply the new BEL1D with a threshold approach on field data collected in the Luy River catchment (Vietnam) to delineate saltwater intrusions. Our results show that the proper selection of time and space discretization is essential for limiting the computational cost while maintaining the accuracy of the posterior estimation. The selection of the prior distribution has a direct impact on fitting the observed data and is crucial for a realistic uncertainty quantification. The application of BEL1D for stochastic TDEM inversion is an efficient approach, as it allows us to estimate the uncertainty at a limited cost.

Determination of Timewise-Source Coefficient in Time-Fractional Reaction-Diffusion Equation from First Order Heat Moment

This article aims to determine the time-dependent heat coefficient together with the temperature solution for a type of semi-linear time-fractional inverse source problem by applying a method based on the finite difference scheme and Tikhonov regularization. An unconditionally stable implicit finite difference scheme is used as a direct (forward) solver. While by the MATLAB routine lsqnonlin from the optimization toolbox, the inverse problem is reformulated as nonlinear least square minimization and solved efficiently. Since the problem is generally incorrect or ill-posed that means any error inclusion in the input data will produce a large error in the output data. Therefore, the Tikhonov regularization technique is applied to obtain stable and accurate results. Finally, to demonstrate the accuracy and effectiveness of our scheme, two benchmark test problems have been considered, and its good working with different noise levels.

Physics-informed neural network for solution of forward and inverse kinematic wave problems

The one-dimensional kinematic wave (KW) model (KWM) is widely used in surface water and water quality hydrology as well as for modeling the movement of traffic on long highways. The KW model involves a first-order nonlinear partial differential equation which is solved numerically for realistic geometric representation, input, and initial and boundary conditions. This paper developed a method to solve the partial differential equation using a neural network structure, called the physics-informed neural network (PINN). The PINN was applied to solve the one-dimensional nonlinear KWM for forward and inverse problems. To enhance the accuracy and convergence and to suppress the spurious oscillation at the steep front, a technique with redistributed collocation points as an improved algorithm of PINN was employed. To deal with the fractional order of the nonlinear advection term, a stop-gradient technique was proposed. For validation, the KWM for steady and unsteady overland flow, and open channel flow without lateral inflow was analyzed for the forward problem. For the inverse problem, the unknown Manning coefficient was correctly computed, demonstrating the capability of the proposed algorithm for the identification of KWM parameters.

Multifidelity adaptive sequential Monte Carlo for geophysical inversion

SUMMARY In the context of Bayesian inversion, we consider sequential Monte Carlo (SMC) methods that provide an approximation of the posterior probability density function and the evidence (marginal likelihood). These particle approaches build a sequence of importance sampling steps between gradually tempered distributions evolving from the prior to the posterior PDF. To automate the definition of the tempering schedule, adaptive SMC (ASMC) allows tuning the temperature increments on-the-go. One general challenge in Bayesian inversions is the computational burden associated with expensive, high-fidelity forward solvers. Lower-fidelity surrogate models are interesting in this context as they can emulate the response of expensive forward solvers at a fraction of their cost. We consider surrogate modelling within ASMC and introduce first an approach involving surrogate modelling only, in which either prior samples are used to train the surrogate, or the surrogate model is retrained by updating the training set during the inversion. In our implementation, we rely on polynomial chaos expansions for surrogate modelling, principal component analysis for model parametrization and a ground-penetrating radar cross-hole tomography problem with either an eikonal or finite-difference time-domain solver as high-fidelity solver. We find that the method based on retraining the surrogate during the inversion outperforms the results obtained when only considering prior samples. We then introduce a computationally more expensive multifidelity approach including a transition to the high-fidelity forward solver at the end of the surrogate-based ASMC run leading to even more accurate results. Both methods result in speed-ups that are larger than one order of magnitude compared to standard high-fidelity ASMC inversion.

Iteration-Based Temporal Subgridding Method for the Finite-Difference Time-Domain Algorithm

A novel temporal subgridding technique is proposed for the finite-difference time-domain (FDTD) method to solve two-dimensional Maxwell’s equations of electrodynamics in the TEz mode. Based on the subgridding FDTD algorithm with a separated spatial and temporal interface, our method focuses on the temporal subgridding region, as it is the main source of late-time instability. Different from other subgridding algorithms that work on the interpolation between coarse and fine meshes, our method stabilizes the solution by using iterative updating equations on the temporal coarse–fine mesh interface. This new method presents an alternative approach aimed at improving the stability of the subgridding technique without modifying the interpolation formulas. We numerically study the stability of the proposed algorithm via eigenvalue tests and by performing long-term simulations. We employ a refinement ratio of 2:1 in our study. Our findings indicate the stability of the conventional temporal subgridding FDTD algorithm with a magnetic field (Hz) interpolation. However, when electric fields (Ex and Ey) are utilized in interpolation, late-time instability occurs. In contrast, the proposed iteration-based method with an electric field interpolation appears to be stable. We further employ our method as the forward problem solver in the Through-the-Wall Radar (TWR) imaging application.