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DOE OSTI2022

Deep Learning for Simultaneous Inference of Hydraulic and Transport Properties

Abstract Identification of a heterogeneous conductivity field and reconstruction of a contaminant release history are key aspects of subsurface remediation. These two goals are achieved by combining model predictions with sparse and noisy hydraulic head and concentration measurements. Solution of this inverse problem is notoriously difficult due to, in part, high dimensionality of the parameter space and high computational cost of repeated forward solves. We use a convolutional adversarial autoencoder (CAAE) to parameterize a heterogeneous non‐Gaussian conductivity field via a low‐dimensional latent representation. A three‐dimensional dense convolutional encoder‐decoder (DenseED) network serves as a forward surrogate of the flow and transport model. The CAAE‐DenseED surrogate is fed into the ensemble smoother with multiple data assimilation (ESMDA) algorithm to sample from the Bayesian posterior distribution of the unknown parameters, forming a CAAE‐DenseED‐ESMDA inversion framework. The resulting CAAE‐DenseED‐ESMDA inversion strategy is used to identify a three‐dimensional contaminant source and conductivity field. A comparison of the inversion results from CAAE‐ESMDA with physical flow and transport simulator and from CAAE‐DenseED‐ESMDA shows that the latter yields accurate reconstruction results at the fraction of the computational cost of the former.

Zhou, Zitong↗

DOE OSTI2024

Seismic Tomography 2024

Seismic tomography is the most abundant source of information about the internal structure of the Earth at scales ranging from a few meters to thousands of kilometers. It constrains the properties of active volcanoes, earthquake fault zones, deep reservoirs and storage sites, glaciers and ice sheets, or the entire globe. It contributes to outstanding societal problems related to natural hazards, resource exploration, underground storage, and many more. The recent advances in seismic tomography are being translated to nondestructive testing, medical ultrasound, and helioseismology. Nearly 50 yr after its first successful applications, this article offers a snapshot of modern seismic tomography. Focused on major challenges and particularly promising research directions, it is intended to guide both Earth science professionals and early-career scientists. The individual contributions by the coauthors provide diverse perspectives on topics that may at first seem disconnected but are closely tied together by a few coherent threads: multiparameter inversion for properties related to dynamic processes, data quality, and geographic coverage, uncertainty quantification that is useful for geologic interpretation, new formulations of tomographic inverse problems that address concrete geologic questions more directly, and the presentation and quantitative comparison of tomographic models. Finally, it remains to be seen which of these problems will be considered solved, solved to some extent, or practically unsolvable over the next decade.

58 GEOSCIENCES↗

DOE OSTI2023

Feature-adjacent multi-fidelity physics-informed machine learning for partial differential equations

Physics-informed neural networks have emerged as an alternative method for solving partial differential equations. However, for complex problems, the training of such networks can still require high-fidelity data which can be expensive to generate. To reduce or even eliminate the dependency on high-fidelity data, we propose a novel multi-fidelity architecture which is based on a feature space shared by the low- and high-fidelity solutions. In the feature space, the projections of the low-fidelity and high-fidelity solutions are adjacent by constraining their relative distance. The feature space is represented with an encoder and its mapping to the original solution space is effected through a decoder. In conclusion, the proposed multi-fidelity approach is validated on forward and inverse problems for steady and unsteady problems described by partial differential equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Probabilistic constrained Bayesian inversion for transpiration cooling

Abstract To enable safe operations in applications such as rocket combustion chambers, the materials require cooling to avoid material damage. Here, transpiration cooling is a promising cooling technique. Numerous studies investigate possibilities to simulate and evaluate the complex cooling mechanism. One naturally arising question is the amount of coolant required to ensure a safe operation. To study this, we introduce an approach that determines the posterior probability distribution of the Reynolds number using an inverse problem and constraining the maximum temperature of the system under parameter uncertainties. Mathematically, this chance inequality constraint is dealt with by a generalized polynomial chaos expansion of the system. The posterior distribution will be evaluated by different Markov chain Monte Carlo based methods. A novel method for the constrained case is proposed and tested among others on two‐dimensional transpiration cooling models.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

A persistent adjoint method with dynamic time-scaling and an application to mass action kinetics

In this article, we consider an optimization problem where the objective function is evaluated at the fixed-point of a contraction mapping parameterized by a control variable, and optimization takes place over this control variable. Since the derivative of the fixed-point with respect to the parameter can usually not be evaluated exactly, an adjoint dynamical system can be used to estimate gradients. Using this estimation procedure, the optimization algorithm alternates between derivative estimation and an approximate gradient descent step. We analyze a variant of this approach involving dynamic time-scaling, where after each parameter update the adjoint system is iterated until a convergence threshold is passed. Here, we prove that, under certain conditions, the algorithm can find approximate stationary points of the objective function. We demonstrate the approach in the settings of an inverse problem in chemical kinetics, and learning in attractor networks.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

2022 AI Testbed Expeditions Report

By exploiting the coherent properties of a light source, coherent diffraction imaging (CDI) is able to obtain the sample image at a nanoscale resolution using the measured diffraction pattern. Bragg Coherent Diffraction Imaging (BCDI) has become valuable for recovering the displacement and strain field of crystals, providing a valuable tool in material science and solid-state physics. X-ray ptychography is another emerging CDI technique that can produce a high-resolution image of the extended sample and has become popular in many research areas (e.g., materials science, biology, electronics, and optics characterization). CDI including BCDI and ptychography has become an established technique in Synchrotron Facilities including the Advanced Photon Source (APS) and will greatly benefit from the 100x coherent flux increase of the upcoming APS Upgrade (APSU). The current image formation process in CDI employs iterative phase retrieval algorithms, which is a time-consuming and computationally expensive process. Especially after APSU, the traditional iterative methods will not be able to match the experimental data acquisition speed. We employ deep learning (DL) approach to replace the iterative approaches, therefore allowing hundreds of times faster recovery of the object. We developed AutoPhaseNN, a DL-based approach which learns to solve the inverse problem without labeled data. Taking 3D BCDI as a representative technique, AutoPhaseNN has been demonstrated to be one hundred times faster than traditional iterative phase retrieval methods while providing comparable image quality. The current network is trained with 64 x 64 x 64 data size, to achieve higher resolution imaging, we will need to scale the network to input and train/infer 3D arrays of size 256 x 256 x 256 (today) and of size 2560x2560x2560 (APSU). However, the scalability of the network is restricted due to the memory-intensive training process. To perform the training for a 256 x 256 x 256 data size, the required memory exceeds the capacity of the current machine. In this project, we explore using Sambanova system to train the network for the direct data inversion for CDI.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Tackling the curse of dimensionality in fractional and tempered fractional PDEs with physics-informed neural networks

Fractional and tempered fractional partial differential equations (PDEs) are effective models of long-range interactions, anomalous diffusion, and non-local effects. Traditional numerical methods for these problems are mesh-based, thus struggling with the curse of dimensionality (CoD). Physics-informed neural networks (PINNs) offer a promising solution due to their universal approximation, generalization ability, and mesh-free training. In principle, Monte Carlo fractional PINN (MC-fPINN) estimates fractional derivatives using Monte Carlo methods and thus could lift CoD. However, this may cause significant variance and errors, hence affecting convergence; in addition, MC-fPINN is sensitive to hyperparameters. In general, numerical methods and specifically PINNs for tempered fractional PDEs are under-developed. Herein, we extend MC-fPINN to tempered fractional PDEs to address these issues, resulting in the Monte Carlo tempered fractional PINN (MC-tfPINN). To reduce possible high variance and errors from Monte Carlo sampling, we replace the one-dimensional (1D) Monte Carlo with 1D Gaussian quadrature, applicable to both MC-fPINN and MC-tfPINN. We validate our methods on various forward and inverse problems of fractional and tempered fractional PDEs, scaling up to 100,000 dimensions. Our improved MC-fPINN/MC-tfPINN using quadrature consistently outperforms the original versions in accuracy and convergence speed in very high dimensions.

42 ENGINEERING↗

DOE OSTI2024

MatCal Users Guide: Release 1.3.0

Any continuum mechanics model will require three components: (1) a discretized geometry of the boundary value problem being studied, (2) the partial differential equations to be solved, and (3) the initial conditions and boundary conditions for the problem. To describe material behavior in these computational models, material models contribute to (2) the underlying equations and, occasionally, to (3) the initial conditions for the simulation. These material models can exhibit a mathematical form that is empirically based, based on first principles, or developed from both empirical observations and known physics. In general, these models are meant to represent a class of materials with well understood behavior. As a result, material models have parameters that must be tuned or calibrated so that the model response matches characterization data available for the specific material it is intended to represent when used to simulate a specific system. For simple models, such as isotropic, linear elastic materials in solid mechanics, this calibration process can be a simple analytical calculation directly extracting the parameters from experimental measurements. For complex models that have many inputs and require many characterization datasets to adequately identify the material behavior, the model calibration process can require an inverse problem approach where an optimization is performed to tune the model parameters to the available data.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Scalable algorithms for physics-informed neural and graph networks

Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space–time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.

42 ENGINEERING↗

DOE OSTI2020

Confronting the Challenge of Modeling Cloud and Precipitation Microphysics

In the atmosphere, microphysics refers to the microscale processes that affect cloud and precipitation particles and is a key linkage among the various components of Earth’s atmospheric water and energy cycles. The representation of microphysical processes in models continues to pose a major challenge leading to uncertainty in numerical weather forecasts and climate simulations. In this paper, the problem of treating microphysics in models is divided into two parts: i) how to represent the population of cloud and precipitation particles, given the impossibility of simulating all particles individually within a cloud, and ii) uncertainties in the microphysical process rates owing to fundamental gaps in knowledge of cloud physics. The recently-developed Lagrangian particle-based method is advocated as a way to address several conceptual and practical challenges of representing particle populations using traditional bulk and bin microphysics parameterization schemes. For addressing critical gaps in cloud physics knowledge, sustained investment for observational advances from laboratory experiments, new probe development, and next-generation instruments in space is needed. Greater emphasis on laboratory work, which has apparently declined over the past several decades relative to other areas of cloud physics research, is argued to be an essential ingredient for improving process-level understanding. More systematic use of natural cloud and precipitation observations to constrain microphysics schemes is also advocated. Because it is generally difficult to quantify individual microphysical process rates from these observations directly, this presents an inverse problem that can be viewed from the standpoint of Bayesian statistics. Following this idea, a probabilistic framework is proposed that combines elements from statistical and physical modeling. Besides providing rigorous constraint of schemes, there is an added benefit of quantifying uncertainty systematically. Finally, a broader hierarchical approach is proposed to accelerate improvements in microphysics schemes, leveraging the advances described in this paper related to process modeling (using Lagrangian particle-based schemes), laboratory experimentation, cloud and precipitation observations, and statistical methods.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Joint state-parameter estimation for the reduced fracture model via the united filter

Here, in this paper, we introduce an effective United Filter method for jointly estimating the solution state and physical parameters in flow and transport problems within fractured porous media. Fluid flow and transport in fractured porous media are critical in subsurface hydrology, geophysics, and reservoir geomechanics. Reduced fracture models, which represent fractures as lower-dimensional interfaces, enable efficient multi-scale simulations. However, reduced fracture models also face accuracy challenges due to modeling errors and uncertainties in physical parameters such as permeability and fracture geometry. To address these challenges, we propose a United Filter method, which integrates the Ensemble Score Filter (EnSF) for state estimation with the Direct Filter for parameter estimation. EnSF, based on a score-based diffusion model framework, produces ensemble representations of the state distribution without deep learning. Meanwhile, the Direct Filter, a recursive Bayesian inference method, estimates parameters directly from state observations. The United Filter combines these methods iteratively: EnSF estimates are used to refine parameter values, which are then fed back to improve state estimation. Numerical experiments demonstrate that the United Filter method surpasses the state-of-the-art Augmented Ensemble Kalman Filter, delivering more accurate state and parameter estimation for reduced fracture models. This framework also provides a robust and efficient solution for PDE-constrained inverse problems with uncertainties and sparse observations.

Bayesian inference↗

DOE OSTI2023

1-D coupled surface flow and transport equations revisited via the physics-informed neural network approach

The de Saint-Venant equation (SVE) and advection–diffusion equation (ADE) are commonly employed to solve solute transport problems in surface water. In this work, we propose a mesh-free method based on the physics-informed neural network (PINN) to solve the one dimensional (1-D) SVE, ADE, and the coupled SVE and ADE (SVE-ADE) under various initial and boundary conditions. The PINN model extends the architecture of deep neural networks (DNNs) with implementation of loss function, which are additionally subject to constraints imposed by the physical laws of SVE and ADE, along with their initial and boundary conditions. In such a manner, PINNs can be quickly steered to the true solution while obeying the physical laws. The results of PINN model are compared with the analytical and/or numerical solutions under various conditions to investigate its accuracy and efficiency in solving the SVE, ADE, and SVE-ADE. Our results indicate PINN can accurately simulate the shock wave morphology and avoid numerical dissipation in unsteady flow condition. The PINN method outweighed traditional numerical methods in several aspects, including its ability to function with small amounts of data, no grid discretization, and random selection of sampling points, etc. Additionally, the PINN method is also suitable for solving inverse problems with sparse and noisy data. With 1% noise and 2000 initial and boundary condition points (N u ), the errors of the estimated flow rate (v) and diffusion coefficient (D) are 0.003% and 0.105%, respectively, which indicate the accuracy and robustness of the proposed method. Finally, our results indicate the capability and robustness of the proposed PINN methodology for solving multi-physics problems, irrespective of the presence of sparse and noisy data in the training dataset.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Lossy checkpoint compression in full waveform inversion: a case study with ZFPv0.5.5 and the overthrust model

This paper proposes a new method that combines checkpointing methods with error-controlled lossy compression for large-scale high-performance full-waveform inversion (FWI), an inverse problem commonly used in geophysical exploration. This combination can significantly reduce data movement, allowing a reduction in run time as well as peak memory. In the exascale computing era, frequent data transfer (e.g., memory bandwidth, PCIe bandwidth for GPUs, or network) is the performance bottleneck rather than the peak FLOPS of the processing unit. Like many other adjoint-based optimization problems, FWI is costly in terms of the number of floating-point operations, large memory footprint during backpropagation, and data transfer overheads. Past work for adjoint methods has developed checkpointing methods that reduce the peak memory requirements during backpropagation at the cost of additional floating-point computations. Combining this traditional checkpointing with error-controlled lossy compression, we explore the three-way tradeoff between memory, precision, and time to solution. We investigate how approximation errors introduced by lossy compression of the forward solution impact the objective function gradient and final inverted solution. Empirical results from these numerical experiments indicate that high lossy-compression rates (compression factors ranging up to 100) have a relatively minor impact on convergence rates and the quality of the final solution.

58 GEOSCIENCES↗

DOE OSTI2025

Calibration of RAFM Micromechanical Model for Creep Using Bayesian Optimization for Functional Output

A Bayesian optimization procedure is presented for calibrating a multimechanism micromechanical model for creep to experimental data of F82H steel. Reduced activation ferritic martensitic (RAFM) steels based on Fe(8–9)%Cr are the most promising candidates for some fusion reactor structures. Although there are indications that RAFM steel could be viable for fusion applications at temperatures up to 600°C, the maximum operating temperature will be determined by the creep properties of the structural material and the breeder material compatibility with the structural material. Due to the relative paucity of available creep data on F82H steel compared to other alloys such as Grade 91 steel, micromechanical models are sought for simulating creep based on relevant deformation mechanisms. As a point of departure, this work recalibrates a model form that was previously proposed for Grade 91 steel to match creep curves for F82H steel. Due to the large number of parameters (9) and cost of the nonlinear simulations, an automated approach for tuning the parameters is pursued using a recently developed Bayesian optimization for functional output (BOFO) framework (Huang et al., 2021, “Bayesian optimization of functional output in inverse problems,” Optim. Eng., 22, pp. 2553–2574). Incorporating extensions such as batch sequencing and weighted experimental load cases into BOFO, a reasonably small error between experimental and simulated creep curves at two load levels is achieved in a reasonable number of iterations. In conclusion, validation with an additional creep curve provides confidence in the fitted parameters obtained from the automated calibration procedure to describe the creep behavior of F82H steel.

42 ENGINEERING↗

DOE OSTI2023

Robust scalable initialization for Bayesian variational inference with multi-modal Laplace approximations

Predictive modeling typically relies on Bayesian model calibration to provide uncertainty quantification. Variational inference utilizing fully independent (“mean-field”) Gaussian distributions are often used as approximate probability density functions. This simplification is attractive since the number of variational parameters grows only linearly with the number of unknown model parameters. However, the resulting diagonal covariance structure and unimodal behavior can be too restrictive to provide useful approximations of intractable Bayesian posteriors that exhibit highly non-Gaussian behavior, including multimodality. High-fidelity surrogate posteriors for these problems can be obtained by considering the family of Gaussian mixtures. Gaussian mixtures are capable of capturing multiple modes and approximating any distribution to an arbitrary degree of accuracy, while maintaining some analytical tractability. Unfortunately, variational inference using Gaussian mixtures with full-covariance structures suffers from a quadratic growth in variational parameters with the number of model parameters. The existence of multiple local minima due to strong nonconvex trends in the loss functions often associated with variational inference present additional complications, These challenges motivate the need for robust initialization procedures to improve the performance and computational scalability of variational inference with mixture models. In this work, we propose a method for constructing an initial Gaussian mixture model approximation that can be used to warm-start the iterative solvers for variational inference. The procedure begins with a global optimization stage in model parameter space. In this step, local gradient-based optimization, globalized through multistart, is used to determine a set of local maxima, which we take to approximate the mixture component centers. Around each mode, a local Gaussian approximation is constructed via the Laplace approximation. Finally, the mixture weights are determined through constrained least squares regression. The robustness and scalability of the proposed methodology is demonstrated through application to an ensemble of synthetic tests using high-dimensional, multimodal probability density functions. Here, the practical aspects of the approach are demonstrated with inversion problems in structural dynamics.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

MTUQ: a framework for estimating moment tensors, point forces, and their uncertainties

SUMMARY We introduce MTUQ, an open-source Python package for seismic source estimation and uncertainty quantification, emphasizing flexibility and operational scalability. MTUQ provides MPI-parallelized grid search and global optimization capabilities, compatibility with 1-D and 3-D Green’s function database formats, customizable data processing, C-accelerated waveform and first-motion polarity misfit functions, and utilities for plotting seismic waveforms and visualizing misfit and likelihood surfaces. Applicability to a range of full- and constrained-moment tensor, point force, and centroid inversion problems is possible via a documented application programming interface, accompanied by example scripts and integration tests. We demonstrate the software using three different types of seismic events: (1) a 2009 intraslab earthquake near Anchorage, Alaska; (2) an episode of the 2021 Barry Arm landslide in Alaska; and (3) the 2017 Democratic People’s Republic of Korea underground nuclear test. With these events, we illustrate the well-known complementary character of body waves, surface waves, and polarities for constraining source parameters. We also convey the distinct misfit patterns that arise from each individual data type, the importance of uncertainty quantification for detecting multimodal or otherwise poorly constrained solutions, and the software’s flexible, modular design.

58 GEOSCIENCES↗

DOE OSTI2022

Panoramic Mapping of Phonon Transport from Ultrafast Electron Diffraction and Scientific Machine Learning

Abstract One central challenge in understanding phonon thermal transport is a lack of experimental tools to investigate frequency‐resolved phonon transport. Although recent advances in computation lead to frequency‐resolved information, it is hindered by unknown defects in bulk regions and at interfaces. Here, a framework that can uncover microscopic phonon transport information in heterostructures is presented, integrating state‐of‐the‐art ultrafast electron diffraction (UED) with advanced scientific machine learning (SciML). Taking advantage of the dual temporal and reciprocal‐space resolution in UED, and the ability of SciML to solve inverse problems involving coupled Boltzmann transport equations, the frequency‐dependent interfacial transmittance and frequency‐dependent relaxation times of the heterostructure from the diffraction patterns are reliably recovered. The framework is applied to experimental Au/Si UED data, and a transport pattern beyond the diffuse mismatch model is revealed, which further enables a direct reconstruction of real‐space, real‐time, frequency‐resolved phonon dynamics across the interface. The work provides a new pathway to probe interfacial phonon transport mechanisms with unprecedented details.

36 MATERIALS SCIENCE↗

DOE OSTI2021

An initial exploration of Bayesian model calibration for estimating the composition of rocks and soils on Mars

The Mars Curiosity rover carries an instrument, ChemCam, designed to measure the composition of surface rocks and soil using laser-induced breakdown spectroscopy (LIBS). The measured spectra from this instrument must be analyzed to identify the component elements in the target sample, as well as their relative proportions. This process, which we call disaggregation, is complicated by so-called matrix effects, which describe nonlinear changes in the relative heights of emission lines as an unknown function of composition due to atomic interactions within the LIBS plasma. In this work, we explore the use of the plasma physics code ATOMIC, developed at Los Alamos National Laboratory, for the disaggregation task. ATOMIC has recently been used to model LIBS spectra and can robustly reproduce matrix effects from first principles. The ability of ATOMIC to predict LIBS spectra presents an exciting opportunity to perform disaggregation in a manner not yet tried in the LIBS community, namely via Bayesian model calibration. However, using it directly to solve our inverse problem is computationally intractable due to the large parameter space and the computation time required to produce a single output. Therefore, we also explore the use of emulators as a fast solution for this analysis. We discuss a proof of concept Gaussian process emulator for disaggregating two-element compounds of sodium and copper. The training and test datasets were simulated with ATOMIC using a Latin hypercube design. After testing the performance of the emulator, we successfully recover the composition of 25 test spectra with Bayesian model calibration.

97 MATHEMATICS AND COMPUTING↗