Stochastic Heterogeneity Research Articles

Modeling fluid flow in heterogeneous porous media with physics-informed neural networks: Weighting strategies for the mixed pressure head-velocity formulation

Physics-informed neural networks (PINNs) are receiving increased attention in modeling flow in porous media because they can surpass purely data-driven approaches. However, in heterogeneous domains, PINNs often face convergence challenges due to discontinuities in rock properties. A promising alternative is the mixed formulation of PINNs, which utilizes pressure head and velocity fields as primary variables. This formulation introduces a multi-term loss function whose terms must be carefully balanced to ensure effective convergence during training. The main goal of this work is to identify the most suitable weighting technique to overcome convergence issues and enhance the applicability of the mixed formulation of PINNs for modeling flow in heterogeneous porous media. Thus, we implement and adapt different global and local weighting techniques and evaluate their performance through multiple test scenarios, involving stochastic and block heterogeneity. The results reveal that the most appropriate weighting strategy is the max-average technique. In the case of stochastic heterogeneity, this technique allows for improving the convergence of the training algorithm. In the case of discontinuous heterogeneity, the max-average method is the only strategy that achieved convergence, highlighting its robustness. The results also show that under high heterogeneity, using an appropriate weighting technique becomes imperative because baseline PINN failed to converge. Implementing an optimal weighting strategy can improve convergence and yield accurate solutions with fewer learnable parameters, thereby enhancing overall model performance.

Spatially Dependent Seismic Wavefield Scattering from an Underground Chemical Explosion: Analysis of the Source Physics Experiment Dry Alluvium Geology Large-N Array

ABSTRACT Explosion sources have been observed to generate significant shear-wave energy despite their isotropic nature. To investigate this phenomenon, we conduct an analysis of the seismic data collected as part of the Source Physics Experiment (SPE): Dry Alluvium Geology (DAG) and investigate the generation of shear-wave energy via scattering. The data were produced by three underground chemical explosions and consist of three-component seismograms, which were recorded by the DAG Large-N array. Synthetic tests suggest that for the DAG experiments, small-scale stochastic heterogeneities, defined as features with correlation lengths of 10–100s of meters, are more effective than large-scale geologic structure (scales >1–10 km) at reproducing the scattering of explosion generated wavefields observed at DAG. We analyze the seismic data for spatially variable ratios between transversely and radially polarized seismic energy, and then estimate the mean free path of P and S waves. All analyses are conducted within a frequency band of 5–50 Hz. The ratio of transversely to radially polarized energy is the highest in the east and west portion of the Large-N array. In addition, the magnitude of the estimated S-wave mean free path is shorter in the eastern portion of the Large-N array. This variation indicates that the eastern area of the DAG array is where more scattering is occurring, suggesting azimuthal dependence of P-to-P and P-to-S scattering. This azimuthal dependence of P-to-S scattering can have implications for explosion discrimination based on spectral ratios of seismic wave types, because the general assumption is that explosions do not generate shear-wave energy. Synthetic tests modeling only larger-scale geologic structure had lower transversely polarized energy (only four stations showing a transversely to radially polarized energy ratio greater than 1) and fewer stations (<10) displaying shorter (<300 m) mean free paths than what was observed in the DAG data results.

A thermodynamic model for predicting the stress–strain relation of stochastic heterogeneous materials with experimental verification

Materials are intrinsic stochastic heterogeneity regardless of their natural or artificial origin. Traditional models for predicting mechanical properties of materials typically involve homogenization, which exhibits certain limitations in practical applications, such as investigating the size effect in stress–strain relationships. In this study, a random field thermodynamic theory for predicting the mechanical properties of stochastic heterogeneous materials was proposed, and its applicability and reliability were verified by testing hard-drawn 304 stainless-steel wires with gauge lengths of 10–500 mm. In these wires composed of the lamellar martensite with a thickness of ∼47.1 nm, it is found that failure strains decreased significantly with increasing gauge length, whereas fracture stresses show little dependence on the gauge length leading to a great increase of the effective modulus. Digital image correlation investigations revealed a significant strain heterogeneity during the tensile process, even at the elastic deformation stage. Nanoindentation tests show the non-uniformly distributed reduced modulus and hardness, which can be attributed to the intrinsic stochastic heterogeneity of materials. Taking Young's modulus as a one-dimensional stationary random field along the longitudinal direction, our random field model can well demonstrate the variation of failure strain and effective modulus with changing gauge lengths. The size effect on the macroscopic mechanical properties mainly originated from the microscopic mechanical property correlation of different points within the materials. These findings could provide new insights into the size effect on the mechanical properties of materials.

Cell population growth kinetics in the presence of stochastic heterogeneity of cell phenotype

Recent studies at individual cell resolution have revealed phenotypic heterogeneity in nominally clonal tumor cell populations. The heterogeneity affects cell growth behaviors, which can result in departure from the idealized uniform exponential growth of the cell population. Here we measured the stochastic time courses of growth of an ensemble of populations of HL60 leukemia cells in cultures, starting with distinct initial cell numbers to capture a departure from the uniform exponential growth model for the initial growth (“take-off”). Despite being derived from the same cell clone, we observed significant variations in the early growth patterns of individual cultures with statistically significant differences in growth dynamics, which could be explained by the presence of inter-converting subpopulations with different growth rates, and which could last for many generations. Based on the hypothesis of existence of multiple subpopulations, we developed a branching process model that was consistent with the experimental observations.

Regression analysis of data based on the method of least absolute deviations in dynamic estimation problems

The use of regression analysis in dynamic problems of system estimation requires a high-speed algorithm of model parameter determination. Moreover, the original data may have stochastic heterogeneity which entails the necessity of the estimates of model parameters be resistant to various data anomalies. However, stable estimation methods, including the least absolute deviations method, are significantly inferior to the parametric ones. The goal of the study is to describe a computationally efficient algorithm for implementing the method of least absolute deviations for dynamic estimation of regression models and to study its capabilities for solving practical problems. This algorithm is based on descending along nodal lines. In this case, instead of the values of the objective function, its derivative in the direction of descent is considered. The computational complexity of the algorithm is also reduced due to the use of the solution of the problem at the previous step as a starting point and efficient updating of observations in the current data sample. The external performance of the proposed dynamic version of the algorithm of gradient descent along nodal lines has been compared with the static version and with the least squares method. It is shown that the dynamic version of the algorithm of gradient descent along the nodal lines make it possible to bring the speed close to that of the least squares method for common practical situations and to use the proposed version in dynamic estimation problems for a wide class of systems.

Household heterogeneity in macroeconomic models: A historical perspective

In this paper, we trace the rise of heterogeneous household models in mainstream macroeconomics from the turn of the 1980s to the early 2000s, when these models evolved into an identifiable and consistent literature. We show that different communities across the US and Europe considered heterogeneous agents for various reasons and developed models that differed in their theoretical and empirical strategies. Minnesota economists primarily focused on incorporating stochastic heterogeneity into general equilibrium models. Other researchers refined growth models or tried to find alternatives to the permanent income hypothesis, leading them to explore more structural heterogeneity. We also document the computational challenges that some of these communities faced, how they gradually became aware of each other's work, and how they faced criticisms from macro- and microeconomists, many of them trained in European countries and dissatisfied with the theoretical and empirical aggregation strategies underlying these models.

Mapping faults in the laboratory with seismic scattering 2: the modelling perspective

SUMMARY Peak delays of acoustic emission (AE) data from rock deformation laboratory experiments are sensitive to both sample heterogeneities and deformation-induced impedance contrasts inside the sample. However, the relative importance of stochastic heterogeneity and discontinuities is uncertain, as is the relationship between peak delays and applied stress and strain. In the companion paper, we presented and analysed peak delay data from AE recorded in a sandstone sample that was triaxially deformed to failure. Here, we simulate P–SV waveforms of dominant frequency 200 kHz in a 2-D isotropic, layered medium using realistic parameters derived from the laboratory experiments previously analysed. Our aim is to provide a physical interpretation of the laboratory findings and constrain the role of a proxy of the evolving fault zone on peak delays. We consider a 2-D fault zone embedded in a host material that simulates the fracture plane as a more compliant layer and allows us to numerically investigate variations in peak delay. Measurements of background parameters, including isotropic velocity and fault thickness were optimized using laboratory data via an evolutionary algorithm. Our simulations clarify that near-source peak delay observations are sensitive to the heterogeneity within zones of intense strain even when far-field approximations are not valid. This sensitivity manifests through the arrival of trapped waves within the layer that is coupled with multiple reflections from the sample boundaries. Substantial uncertainties remain on the possibility of inverting sample parameters with 2-D simulations and such complex physics. Our combined experimental and modelling study suggests that peak delays and coda parameters are sensitive to the heterogeneity caused by faulting and strain variations at different stages of fault-inducing slow deformation.

The role of three-dimensional fault interactions in creating complex seismic sequences

A physics-based earthquake simulator should reproduce first-order empirical power-law behaviors of magnitudes and clustering. These laws have emerged spontaneously in either discrete or low-dimension continuum simulations without power-law or stochastic heterogeneity. We show that the same emergence can occur in 3-D continuum simulations with fault interactions and rate-and-state friction. Our model approximates a strike-slip fault system as three en echelon faults. Simulations show spatio-temporally clustered earthquake sequences exhibiting characteristic Gutenberg-Richter scaling as well as empirical inter-event time distribution. The Gutenberg-Richter scaling occurs only in partial ruptures that result from fault interactions. With fault interactions, partial ruptures emerge when seismogenic width W over characteristic nucleation length L∞ is larger than 16.24, but none occur without fault interaction. The mainshock recurrence times of individual faults remain quasi-periodic. The system mainshock recurrence time is a combination of short-term Omori-type decay and Brownian passage time. Higher W/L∞ increase short-term clustering probability to at most 30%. These results indicate that physics-based multi-cycle models adequately reflect observed statistical signatures and show practical potential for long-term hazard assessment and medium-term forecasting.

Influence of process-based, stochastic and deterministic methods for representing heterogeneity in fluvial geothermal systems

Focus is on comparing stochastic, process-based and deterministic methods for modelling heterogeneity in hydraulic properties of fluvial geothermal reservoirs. Models are considered a generalized representation of a fluvial sequence in the upper part of the Gassum Formation in northern Denmark. Two ensemble realizations of process-based and stochastic heterogeneity were simulated using the state-of-the-art numerical modelling software, Delft Advanced Research Terra Simulator (DARTS), to assess differences on a statistically relevant sample size. Simulator settings were optimized to achieve two orders of magnitude improvement in simulation time. Our general findings show that the stochastic and process-based methods produce nearly identical results in terms of predicted breakthrough time and production temperature decline for high net-to-gross ratios (N/G). Simple homogenous and layered models overestimate breakthrough and underestimate temperature decline. More complex representation of facies in process-based models show smaller variance in results and stay within the limits of ensemble runs compared to simpler facies representation. Results indicate that a multivariate Gaussian based stochastic representation of heterogeneity provides comparable thermal response to a process-based model in fluvial systems of similar quality.

Insurance pricing in an equilibrium model

We develop a dynamic equilibrium model of insurance pricing in a competitive market consisting of heterogeneous insurance companies. The insurers have different beliefs on expected loss rate of an underlying risk process and the belief divergences are stochastic. The insurers select optimal insurance market shares to maximize their individual utilities. The equilibrium insurance price is formulated when the insurance market is cleared. We provide a general equilibrium framework with a continuum of insurers in the market and then solve for the equilibrium insurance price explicitly in the case of N insurers. We find that the stochastic heterogeneity brings extra volatility to insurance price and makes it dynamic. The mean-reverting divergences of insurers may explain cycles of insurance business documented by empirical studies. Compared to the previous literature of optimal insurance, this paper introduces an asset pricing framework of general equilibrium to the research of insurance pricing.

Stochastic Spatial Heterogeneity in Activities of H+-ATP-Ases in Electrically Connected Plant Cells Decreases Threshold for Cooling-Induced Electrical Responses.

H+-ATP-ases, which support proton efflux through the plasma membrane, are key molecular transporters for electrogenesis in cells of higher plants. Initial activities of the transporters can influence the thresholds of generation of electrical responses induced by stressors and modify other parameters of these responses. Previously, it was theoretically shown that the stochastic heterogeneity of individual cell thresholds for electrical responses in a system of electrically connected neuronal cells can decrease the total threshold of the system (“diversity-induced resonance”, DIR). In the current work, we tested a hypothesis about decreasing the thresholds of generation of cooling-induced electrical responses in a system of electrically connected plant cells with increasing stochastic spatial heterogeny in the initial activities of H+-ATP-ases in these cells. A two-dimensional model of the system of electrically connected excitable cells (simple imitation of plant leaf), which was based on a model previously developed in our works, was used for the present investigation. Simulation showed that increasing dispersion in the distribution of initial activities of H+-ATP-ases between cells decreased the thresholds of generation of cooling-induced electrical responses. In addition, the increasing weakly influenced the amplitudes of electrical responses. Additional analysis showed two different mechanisms of the revealed effect. The increasing spatial heterogeneity in activities of H+-ATP-ases induced a weak positive shift of the membrane potential at rest. The shift decreased the threshold of electrical response generation. However, the decreased threshold induced by increasing the H+-ATP-ase activity heterogeneity was also observed after the elimination of the positive shift. The result showed that the “DIR-like” mechanism also participated in the revealed effect. Finally, we showed that the standard deviation of the membrane potentials before the induction of action potentials could be used for the estimation of thresholds of cooling-induced plant electrical responses. Thus, spatial heterogeneity in the initial activities of H+-ATP-ases can be a new regulatory mechanism influencing the generation of electrical responses in plants under actions of stressors.

Virus transport from drywells under constant head conditions: A modeling study

Many arid and semi-arid regions of the world face challenges in maintaining the water quantity and quality needs of growing populations. A drywell is an engineered vadose zone infiltration device widely used for stormwater capture and managed aquifer recharge. To our knowledge, no prior studies have quantitatively examined virus transport from a drywell, especially in the presence of subsurface heterogeneity. Axisymmetric numerical experiments were conducted to systematically study virus fate from a drywell for various virus removal and subsurface heterogeneity scenarios under steady-state flow conditions from a constant head reservoir. Subsurface domains were homogeneous or had stochastic heterogeneity with selected standard deviation (σ) of lognormal distribution in saturated hydraulic conductivity and horizontal (X) and vertical (Z) correlation lengths. Low levels of virus concentration tailing can occur even at a separation distance of 22 m from the bottom of the drywell, and 6-log10 virus removal was not achieved when a small detachment rate (kd1=1 × 10⁻⁵ min⁻¹) is present in a homogeneous domain. Improved virus removal was achieved at a depth of 22 m in the presence of horizontal lenses (e.g., X=10 m, Z=0.1 m, σ=1) that enhanced the lateral movement and distribution of the virus. In contrast, faster downward movement of the virus with an early arrival time at a depth of 22 m occurred when considering a vertical correlation in permeability (X=1 m, Z=2 m, σ=1). Therefore, the general assumption of a 1.5–12 m separation distance to protect water quality may not be adequate in some instances, and site-specific microbial risk assessment is essential to minimize risk. Microbial water quality can potentially be improved by using an in situ soil treatment with iron oxides to increase irreversible attachment and solid-phase inactivation.

Unobservable characteristics of board directors and the performance of financial services firms in Nigeria

This paper examines the intricate link between unobservable characteristics of directors on the corporate board and firm performance. It aims to extend the literature on corporate governance and firm strategic performance from the perspective of emerging African economies. A mix of performance measures were used (Tobin Q, return on assets, and share price) and unobservable characteristics were captured as a stochastic element or heterogeneity of observable board characteristics (board activity, gender diversity, size, and independence). The study applied non-linear generalized auto-regressive conditional heteroscedasticity model to examine the data set consisting of 299 firm-year observations from 23 financial firms listed on the Nigerian Stock Exchange from 2006 to 2018. Positive skewness and leptokurtic distribution were found for all the variables. Correlation matrix revealed no multicollinearity, as the highest value was 0.2386. Empirical results suggest that unobservable characteristics significantly and positively influence firm performance as measured by return on assets and share price. This is because the coefficient of the lagged-value of the variance scaling parameter is positive and significant at the 1% level. However, with respect to Tobin Q measure, the result was positive but not significant at the 5% level. Implicitly, the result is sensitive to performance proxies. Accordingly, this study concludes that unobservable characteristics drive firm performance. It is recommended that boards and regulators should pay attention to unobservable characteristics.

TISCH: a comprehensive web resource enabling interactive single-cell transcriptome visualization of tumor microenvironment.

Cancer immunotherapy targeting co-inhibitory pathways by checkpoint blockade shows remarkable efficacy in a variety of cancer types. However, only a minority of patients respond to treatment due to the stochastic heterogeneity of tumor microenvironment (TME). Recent advances in single-cell RNA-seq technologies enabled comprehensive characterization of the immune system heterogeneity in tumors but posed computational challenges on integrating and utilizing the massive published datasets to inform immunotherapy. Here, we present Tumor Immune Single Cell Hub (TISCH, http://tisch.comp-genomics.org), a large-scale curated database that integrates single-cell transcriptomic profiles of nearly 2 million cells from 76 high-quality tumor datasets across 27 cancer types. All the data were uniformly processed with a standardized workflow, including quality control, batch effect removal, clustering, cell-type annotation, malignant cell classification, differential expression analysis and functional enrichment analysis. TISCH provides interactive gene expression visualization across multiple datasets at the single-cell level or cluster level, allowing systematic comparison between different cell-types, patients, tissue origins, treatment and response groups, and even different cancer-types. In summary, TISCH provides a user-friendly interface for systematically visualizing, searching and downloading gene expression atlas in the TME from multiple cancer types, enabling fast, flexible and comprehensive exploration of the TME.

The effects of earth model uncertainty on the inversion of seismic data for seismic source functions

SUMMARY We use Monte Carlo simulations to explore the effects of earth model uncertainty on the estimation of the seismic source time functions that correspond to the six independent components of the point source seismic moment tensor. Specifically, we invert synthetic data using Green’s functions estimated from a suite of earth models that contain stochastic density and seismic wave-speed heterogeneities. We find that the primary effect of earth model uncertainty on the data is that the amplitude of the first-arriving seismic energy is reduced, and that this amplitude reduction is proportional to the magnitude of the stochastic heterogeneities. Also, we find that the amplitude of the estimated seismic source functions can be under- or overestimated, depending on the stochastic earth model used to create the data. This effect is totally unpredictable, meaning that uncertainty in the earth model can lead to unpredictable biases in the amplitude of the estimated seismic source functions.

Finite element analysis of natural fiber composites using a self-updating model

The aim of the current work was to illustrate the effect of the fibre area correction factor on the results of modelling natural fibre-reinforced composites. A mesoscopic approach is adopted to represent the stochastic heterogeneity of the composite, i.e. a meso-structural numerical model was prototyped using the finite element method including quasi-unidirectional discrete fibre elements embedded in a matrix. The model was verified by the experimental results from previous work on jute fibres but is extendable to every natural fibre with cross-sectional non-uniformity. A correction factor was suggested to fine-tune both the analytical and numerical models. Moreover, a model updating technique for considering the size-effect of fibres is introduced and its implementation was automated by means of FORTRAN subroutines and Python scripts. It was shown that correcting and updating the fibre strength is critical to obtain accurate macroscopic response of the composite when discrete modelling of fibres is intended. Based on the current study, it is found that consideration of the effect of flaws on the strength of natural fibres and inclusion of the fibre area correction factor are crucial to obtain realistic results.

X-ray Micro-Computed Tomography of Polymer Electrolyte Fuel Cells: What is the Representative Elementary Area?

With the growing use of X-ray computed tomography (X-ray CT) datasets for modelling of transport properties, comes the need to define the representative elementary volume (REV) if considering three dimensions or the representative elementary area (REA) if considering two dimensions. The resolution used for imaging must be suited to the features of interest in the sample and the region-of-interest must be sufficiently large to capture key information. Polymer electrolyte fuel cells have a hierarchical structure, with materials spanning multiple length scales. The work presented here examines the nature of the REA throughout a 25 cm2 membrane electrode assembly (MEA), focusing specifically on the micron length scale. Studies were carried out to investigate key structural (volume fraction, layer and penetration thickness, pore diameters) and transport (effective diffusivity) properties. Furthermore, the limiting current density of the nine regions was modelled. Stochastic heterogeneity throughout the sample results in local variations throughout. Finally, effects of resolution were probed by imaging using a range of optical magnifications (4× and 20×). The correlated and competing effects of voxel resolution and sampling size were found to cause difficulties where loss of clarity in the boundaries between phases occurs with larger imaging volumes.

Correction to: The Influence of Soil Stochastic Heterogeneity and Facility Dimensions on Stormwater Infiltration Facilities Performance

The original version of this article unfortunately contains mistakes introduced during the publishing process. The mistakes and corrections are described in the following list, the main error lies in Table 1, page 2402.

Distribution and origination of zinc contamination in newly reclaimed heterogeneous dredger fills: Field investigation and numerical simulation.

Heavy metal elements, including Zn, Cd, As, Ni, Cu, Pb and Cr, were detected in soils (no deeper than 75 m) from newly reclaimed zones of Shanghai, China. The Zn concentration exceeded soil quality limits. The Zn contamination was tested in both dredger fills and sedimentary layers (①3-3, ②3, ④ and ⑤1-1). However, it was not detected in layer ⑤1-2-⑨. PCA and HCA analysis show that exogenous Zn probably was the contaminant source of dredger fills before the fills were dredged from the neighboring waters. Stochastic heterogeneity of the dredger fills affects the Zn-depollution remarkably. Numerical simulations show both acid precipitation and widespread drainage channels in the zones contributed to Zn-decrease in the dredger fills no deeper than 1.2 m. Acid rainstorms work better than acid constant precipitation in Zn-remediation for layers below 0.4 m. To remove Zn contamination in deep dredger fills, un-consolidation of the fills should be utilized.

IL-2 Signals Program the Fate of Exhausted CD8 T cells

Abstract The high affinity IL-2 receptor (IL-2Ra, CD25) is a key activation marker, which is transiently and heterogeneously expressed during the priming . While IL-2 signals are deemed critical for expansion of antigen-specific CD8 T cells, the role of IL-2 in programing the fate of exhausted CD8 T cells in vivo is poorly understood. Exhausted CD8 T cells are broadly divided into 2 subsets: one with brightest levels of inhibitory receptors such as PD-1, and lower mitochondrial spare respiratory capacity (SRC) – these cells are terminally exhausted and are largely unresponsive to PD-1 checkpoint blockade immunotherapy. The other subset has intermediate levels of inhibitory receptors and retains higher mitochondrial SRC – these are the clinically relevant cells that are highly responsive to PD-1 blockade. Based on the ability of IL-2 signals to drive glycolytic burnout in T cells, we hypothesized that the stochastic heterogeneity in in vivo assimilation of IL-2 signals during priming drives distinct CD8 T cell fates late during exhaustion. We sorted in vivo primed CD8 T cells based on their heterogeneity in CD25 expression, and adoptively transferred them into infection-matched chronically infected recipients. We observed that the exhausted CD8 T cells generated from CD25Hi donor cells phenotypically and functionally resembled terminally exhausted cells. Furthermore, knocking down the expression of CD25 or Special AT-rich sequence binding protein 1 (SATB1), which is known to regulate CD25 expression, resulted in better survival and function of exhausted CD8 T cells. Our data underscores the critical role of IL-2 signals in the exhaustion of CD8 T cells and has clinical significance in current combination checkpoint blockade immunotherapies.