Physics-informed and data-driven discovery of governing equations for complex phenomena in heterogeneous media.

Rapid evolution of sensor technology, advances in instrumentation, and progress in devising data-acquisition software and hardware are providing vast amounts of data for various complex phenomena that occur in heterogeneous media, ranging from those in atmospheric environment, to large-scale porous formations, and biological systems. The tremendous increase in the speed of scientific computing has also made it possible to emulate diverse multiscale and multiphysics phenomena that contain elements of stochasticity or heterogeneity, and to generate large volumes of numerical data for them. Thus, given a heterogeneous system with annealed or quenched disorder in which a complex phenomenon occurs, how should one analyze and model the system and phenomenon, explain the data, and make predictions for length and time scales much larger than those over which the data were collected? We divide such systems into three distinct classes. (i) Those for which the governing equations for the physical phenomena of interest, as well as data, are known, but solving the equations over large length scales and long times is very difficult. (ii) Those for which data are available, but the governing equations are only partially known, in the sense that they either contain various coefficients that must be evaluated based on the data, or that the number of degrees of freedom of the system is so large that deriving the complete equations is very difficult, if not impossible, as a result of which one must develop the governing equations with reduced dimensionality. (iii) In the third class are systems for which large amounts of data are available, but the governing equations for the phenomena of interest are not known. Several classes of physics-informed and data-driven approaches for analyzing and modeling of the three classes of systems have been emerging, which are based on machine learning, symbolic regression, the Koopman operator, the Mori-Zwanzig projection operator formulation, sparse identification of nonlinear dynamics, data assimilation combined with a neural network, and stochastic optimization and analysis. This perspective describes such methods and the latest developments in this highly important and rapidly expanding area and discusses possible future directions.

Percolation and conductivity in evolving disordered media.

Percolation theory and the associated conductance networks have provided deep insights into the flow and transport properties of a vast number of heterogeneous materials and media. In practically all cases, however, the conductance of the networks' bonds remains constant throughout the entire process. There are, however, many important problems in which the conductance of the bonds evolves over time and does not remain constant. Examples include clogging, dissolution and precipitation, and catalytic processes in porous materials, as well as the deformation of a porous medium by applying an external pressure or stress to it that reduces the size of its pores. We introduce two percolation models to study the evolution of the conductivity of such networks. The two models are related to natural and industrial processes involving clogging, precipitation, and dissolution processes in porous media and materials. The effective conductivity of the models is shown to follow known power laws near the percolation threshold, despite radically different behavior both away from and even close to the percolation threshold. The behavior of the networks close to the percolation threshold is described by critical exponents, yielding bounds for traditional percolation exponents. We show that one of the two models belongs to the traditional universality class of percolation conductivity, while the second model yields nonuniversal scaling exponents.

Interpreting water demands of forests and grasslands within a new Budyko formulation of evapotranspiration using percolation theory.

The relationship between carbon cycle and water demand is key to understanding global climate change, vegetation productivity, and predicting the future of water resources. The water balance, which enumerates the relative fractions of precipitation P that run off, Q, or are returned to the atmosphere through evapotranspiration, ET, links drawdown of atmospheric carbon with the water cycle through plant transpiration. Our theoretical description based on percolation theory proposes that dominant ecosystems tend to maximize drawdown of atmospheric carbon in the process of growth and reproduction, thus providing a link between carbon and water cycles. In this framework, the only parameter is the fractal dimensionality df of the root system. Values of df appear to relate to the relative roles of nutrient and water accessibility. Larger values of df lead to higher ET values. Known ranges of grassland root fractal dimensions predict reasonably the range of ET(P) in such ecosystems as a function of aridity index. Forests with shallower root systems, should be characterized by a smaller df and, therefore, ET that is a smaller fraction of P. The prediction of ET/P using the 3D percolation value of df matches rather closely results deemed typical for forests based on a phenomenology already in common use. We test predictions of Q with P against data and data summaries for sclerophyll forests in southeastern Australia and the southeastern USA. Applying PET data from a nearby site constrains the data from the USA to lie between our ET predictions for 2D and 3D root systems. For the Australian site, equating cited "losses" with PET underpredicts ET. This discrepancy is mostly removed by referring to mapped values of PET in that region. Missing in both cases is local PET variability, more important for reducing data scatter in southeastern Australia, due to the greater relief.

Data-driven discovery of the governing equations for transport in heterogeneous media by symbolic regression and stochastic optimization.

With advances in instrumentation and the tremendous increase in computational power, vast amounts of data are becoming available for many complex phenomena in macroscopically heterogeneous media, particularly those that involve flow and transport processes, which are problems of fundamental interest that occur in a wide variety of physical systems. The absence of a length scale beyond which such systems can be considered as homogeneous implies that the traditional volume or ensemble averaging of the equations of continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equations for flow and transport processes is an open question. We propose a data-driven approach that uses stochastic optimization and symbolic regression to discover the governing equations for flow and transport processes in heterogeneous media. The data could be experimental or obtained by microscopic simulation. As an example, we discover the governing equation for anomalous diffusion on the critical percolation cluster at the percolation threshold, which is in the form of a fractional partial differential equation, and agrees with what has been proposed previously.

Retraction: Phase transitions, percolation, fracture of materials, and deep learning [Phys. Rev. E 102, 011001(R) (2020)].

Retraction of DOI: 10.1103/PhysRevE.102.011001.

Graphyne-3: a highly efficient candidate for separation of small gas molecules from gaseous mixtures.

Two-dimensional nanosheets, such as the general family of graphenes have attracted considerable attention over the past decade, due to their excellent thermal, mechanical, and electrical properties. We report on the result of a study of separation of gaseous mixtures by a model graphyne-3 membrane, using extensive molecular dynamics simulations and density functional theory. Four binary and one ternary mixtures of H[Formula: see text], CO[Formula: see text], CH[Formula: see text] and C[Formula: see text]H[Formula: see text] were studied. The results indicate the excellence of graphyne-3 for separation of small gas molecules from the mixtures. In particular, the H[Formula: see text] permeance through the membrane is on the order of [Formula: see text] gas permeation unit, by far much larger than those in other membranes, and in particular in graphene. To gain deeper insights into the phenomenon, we also computed the density profiles and the residence times of the gases near the graphyne-3 surface, as well as their interaction energies with the membrane. The results indicate clearly the tendency of H[Formula: see text] to pass through the membrane at high rates, leaving behind C[Formula: see text]H[Formula: see text] and larger molecules on the surface. In addition, the possibility of chemisorption is clearly ruled out. These results, together with the very good mechanical properties of graphyne-3, confirm that it is an excellent candidate for separating small gas molecules from gaseous mixtures, hence opening the way for its industrial use.

Elastic moduli of body-centered cubic lattice near rigidity percolation threshold: Finite-size effects and evidence for first-order phase transition.

Extensive numerical simulations of rigidity percolation with only central forces in large three-dimensional lattices have indicated that many of their topological properties undergo a first-order phase transition at the rigidity percolation threshold p_{ce}. In contrast with such properties, past numerical calculations of the elastic moduli of the same lattices had provided evidence for a second-order phase transition. In this paper we present the results of extensive simulation of rigidity percolation in large body-centered cubic (bcc) lattices, and show that as the linear size L of the lattice increases, the elastic moduli close to p_{ce} decrease in a stepwise, discontinuous manner, a feature that is absent in lattices with L<30. The number and size of such steps increase with L. As p_{ce} is approached, long-range, nondecaying orientational correlations are built up, giving rise to compact, nonfractal clusters. As a result, we find that the backbone of the lattice at p_{ce} is compact with a fractal dimension D_{bb}≈3. The absence of fractal, scale-invariant clusters, the hallmark of second-order phase transitions, together with the stairwise behavior of the elastic moduli, provide strong evidence that, at least in bcc lattices, many of the topological properties of rigidity percolation as well as its elastic moduli may undergo a first-order phase transition at p_{ce}. In relatively small lattices, however, the boundary effects interfere with the nonlocal nature of the rigidity percolation. As a result, only when such effects diminish in large lattices does the true nature of the phase transition emerge.

Molecular Dynamics Study of Structure, Folding, and Aggregation of Poly-PR and Poly-GR Proteins.

Poly-proline-arginine (poly-PR) and poly-glycine-arginine (poly-GR) proteins are believed to be the most toxic dipeptide repeat (DPR) proteins that are expressed by the hexanucleotide repeat expansion mutation in C9ORF72, which are associated with amyotrophic lateral sclerosis (ALS) and frontotemporal dementia (FTD) diseases. Their structural information and mechanisms of toxicity remain incomplete, however. Using molecular dynamics simulation and all-atom model of proteins, we study folding and aggregation of both poly-PR and poly-GR. The results indicate formation of double-helix structure during the aggregation of poly-PR into dimers, whereas no stable aggregate is formed during the aggregation of poly-GR; the latter only folds into α-helix and double-helix structures that are similar to those formed in the folding of poly-glycine-alanine (poly-GA) protein. Our findings are consistent with the experimental data indicating that poly-PR and poly-GR are less likely to aggregate because of the hydrophilic arginine residues within their structures. Such characteristics could, however, in some respect facilitate migration of the DPR proteins between and within cells and, at the same time, give proline residues the benefits of activating the receptors that regulate ionotropic effect in neurons, resulting in death or malfunction of neurons because of the abnormal increase or decrease of the ion transmission. This may explain the neurotoxicities of poly-PR and poly-GR associated with many neurodegenerative diseases. To our knowledge, this is the first molecular dynamics simulation of the phenomena involving poly-PR and poly-GR proteins.

Molecular origin of sliding friction and flash heating in rock and heterogeneous materials.

It is generally believed that earthquakes occur when faults weaken with increasing slip rates. An important factor contributing to this phenomenon is the faults' dynamic friction, which may be reduced during earthquakes with high slip rates, a process known as slip-rate weakening. It has been hypothesized that the weakening phenomenon during fault slip may be activated by thermal pressurization of pores' fluid and flash heating, a microscopic phenomenon in which heat is generated at asperity contacts due to high shear slip rates. Due to low thermal conductivity of rock, the heat generated at the contact points or surfaces cannot diffuse fast enough, thus concentrating at the contacts, increasing the local contact temperature, and reducing its frictional shear strength. We report the results of what we believe to be the first molecular scale study of the decay of the interfacial friction force in rock, observed in experiemntal studies and attributed to flash heating. The magnitude of the reduction in the shear stress and the local friction coefficients have been computed over a wide range of shear velocities V. The molecular simulations indicate that as the interfacial temperature increases, bonds between the atoms begin to break, giving rise to molecular-scale fracture that eventually produces the flash heating effect. The frequency of flash heating events increases with increasing sliding velocity, leaving increasingly shorter times for the material to relax, hence contributing to the increased interfacial temperature. If the material is thin, the heat quickly diffuses away from the interface, resulting in sharp decrease in the temperature immediately after flash heating. The rate of heat transfer is reduced significantly with increasing thickness, keeping most of the heat close to the interface and producing weakened material. The weakening behavior is demonstrated by computing the stress-strain diagram. For small strain rates there the frictional stress is essentially independent of the materials' thickness. As the strain rate increases, however, the dependence becomes stronger. Specifically, the stress-strain diagrams at lower velocities V manifest a pronounced strength decrease over small distances, whereas they exhibit progressive increase in the shear stress at higher V, which is reminiscent of a transition from ductile behavior at high velocities to brittle response at low velocities.

Formation of a Stable Bridge between Two Disjoint Nanotubes with Single-File Chains of Water.

It was recently demonstrated that stable water bridges can form between two relatively large disjoint nanochannels, such as carbon nanotubes (CNTs), under an applied pressure drop. Such bridges are relevant to fabrication of nanostructured materials, drug delivery, water desalination devices, hydrogen fuel cells, dip-pen nanolithography, and several other applications. If the nanotubes are small enough, however, then one has only single-file hydrogen-bonded chains of water molecules. The distribution of water in such nanotubes manifests unusual physical properties that are attributed to the low number of hydrogen bonds (HBs) formed in the channel since, on average, each water molecule in a single-file chain forms only 1.7 HBs, almost half of the value for bulk water. Using extensive molecular dynamics simulations, we demonstrate that stable bridges can form even between two small disjoint CNTs that contain single-file chains of water. The structure, stability, and properties of such bridges and their dependence on the applied pressure drop and the length of the gap between the two CNTs are studied in detail, as is the distribution of the HBs. We demonstrate, in particular, that the efficiency of flow through the bridge is at maximum at a specific pressure difference.

Phase transitions, percolation, fracture of materials, and deep learning.

Percolation and fracture propagation in disordered solids represent two important problems in science and engineering that are characterized by phase transitions: loss of macroscopic connectivity at the percolation threshold p_{c} and formation of a macroscopic fracture network at the incipient fracture point (IFP). Percolation also represents the fracture problem in the limit of very strong disorder. An important unsolved problem is accurate prediction of physical properties of systems undergoing such transitions, given limited data far from the transition point. There is currently no theoretical method that can use limited data for a region far from a transition point p_{c} or the IFP and predict the physical properties all the way to that point, including their location. We present a deep neural network (DNN) for predicting such properties of two- and three-dimensional systems and in particular their percolation probability, the threshold p_{c}, the elastic moduli, and the universal Poisson ratio at p_{c}. All the predictions are in excellent agreement with the data. In particular, the DNN predicts correctly p_{c}, even though the training data were for the state of the systems far from p_{c}. This opens up the possibility of using the DNN for predicting physical properties of many types of disordered materials that undergo phase transformation, for which limited data are available for only far from the transition point.

Quantifying accuracy of stochastic methods of reconstructing complex materials by deep learning.

Time and cost are two main hurdles to acquiring a large number of digital image I of the microstructure of materials. Thus, use of stochastic methods for producing plausible realizations of materials' morphology based on one or very few images has become an increasingly common practice in their modeling. The accuracy of the realizations is often evaluated using two-point microstructural descriptors or physics-based modeling of certain phenomena in the materials, such as transport processes or fluid flow. In many cases, however, two-point correlation functions do not provide accurate evaluation of the realizations, as they are usually unable to distinguish between high- and low-quality reconstructed models. Calculating flow and transport properties of the realization is an accurate way of checking the quality of the realizations, but it is computationally expensive. In this paper a method based on machine learning is proposed for evaluating stochastic approaches for reconstruction of materials, which is applicable to any of such methods. The method reduces the dimensionality of the realizations using an unsupervised deep-learning algorithm by compressing images and realizations of materials. Two criteria for evaluating the accuracy of a reconstruction algorithm are then introduced. One, referred to as the internal uncertainty space, is based on the recognition that for a reconstruction method to be effective, the differences between the realizations that it produces must be reasonably wide, so that they faithfully represent all the possible spatial variations in the materials' microstructure. The second criterion recognizes that the realizations must be close to the original I and, thus, it quantifies the similarity based on an external uncertainty space. Finally, the ratio of two uncertainty indices associated with the two criteria is considered as the final score of the accuracy of a stochastic algorithm, which provides a quantitative basis for comparing various realizations and the approaches that produce them. The proposed method is tested with images of three types of heterogeneous materials in order to evaluate four stochastic reconstruction algorithms.

Morphology and kinetics of random sequential adsorption of superballs: From hexapods to cubes.

Superballs represent a class of particles whose shapes are defined by the domain |x|^{2p}+|y|^{2p}+|z|^{2p}≤R^{2p}, with p∈(0,∞) being the deformation parameter. 01 represent, respectively, families of convex octahedral-like and cubelike particles, with p=1,0.5, and ∞ representing spheres, octahedra, and cubes. Colloidal zeolite suspensions, catalysis, and adsorption, as well as biomedical magnetic nanoparticles are but a few of the applications of packing of superballs. We introduce a universal method for simulating random sequential adsorption of superballs, which we refer to as the low-entropy algorithm, which is about two orders of magnitude faster than the conventional algorithms that represent high-entropy methods. The two algorithms yield, respectively, precise estimates of the jamming fraction ϕ_{∞}(p) and ν(p), the exponent that characterizes the kinetics of adsorption at long times t, ϕ_{∞}(p)-ϕ(p,t)∼t^{-ν(p)}. Precise estimates of ϕ_{∞}(p) and ν(p) are obtained and shown to be in agreement with the existing analytical and numerical results for certain types of superballs.

Exact enumeration approach to first-passage time distribution of non-Markov random walks.

We propose an analytical approach to study non-Markov random walks by employing an exact enumeration method. Using the method, we derive an exact expansion for the first-passage time (FPT) distribution of any continuous differentiable non-Markov random walk with Gaussian or non-Gaussian multivariate distribution. As an example, we study the FPT distribution of the fractional Brownian motion with a Hurst exponent H∈(1/2,1) that describes numerous non-Markov stochastic phenomena in physics, biology, and geology and for which the limit H=1/2 represents a Markov process.

Regulation of migration of chemotactic tumor cells by the spatial distribution of collagen fiber orientation.

Collagen fibers, an important component of the extracellular matrix (ECM), can both inhibit and promote cellular migration. In vitro studies have revealed that the fibers' orientations are crucial to cellular invasion, while in vivo investigations have led to the development of tumor-associated collagen signatures (TACS) as an important prognostic factor. Studying biophysical regulation of cell invasion and the effect of the fibers' orientation not only deepens our understanding of the phenomenon, but also helps classify the TACSs precisely, which is currently lacking. We present a stochastic model for random or chemotactic migration of cells in fibrous ECM, and study the role of the various factors in it. The model provides a framework for quantitative classification of the TACSs, and reproduces quantitatively recent experimental data for cell motility. It also indicates that the spatial distribution of the fibers' orientations and extended correlations between them, hitherto ignored, as well as dynamics of cellular motion all contribute to regulation of the cells' invasion length, which represents a measure of metastatic risk. Although the fibers' orientations trivially affect randomly moving cells, their effect on chemotactic cells is completely nontrivial and unexplored, which we study in this paper.

Efficient Transport Between Disjoint Nanochannels by a Water Bridge.

Water channels are important to new purification systems, osmotic power harvesting in salinity gradients, hydroelectric voltage conversion, signal transmission, drug delivery, and many other applications. To be effective, water channels must have structures more complex than a single tube. One way of building such structures is through a water bridge between two disjoint channels that are not physically connected. We report on the results of extensive molecular dynamics simulation of water transport through such bridges between two carbon nanotubes separated by a nanogap. We show that not only can pressurized water be transported across a stable bridge, but also that (i) for a range of the gap's width l_{g} the bridge's hydraulic conductance G_{b} does not depend on l_{g}, (ii) the overall shape of the bridge is not cylindrical, and (iii) the dependence of G_{b} on the angle between the axes of two nonaligned nanochannels may be used to tune the flow rate between the two.

Enhancing images of shale formations by a hybrid stochastic and deep learning algorithm.

Accounting for the morphology of shale formations, which represent highly heterogeneous porous media, is a difficult problem. Although two- or three-dimensional images of such formations may be obtained and analyzed, they either do not capture the nanoscale features of the porous media, or they are too small to be an accurate representative of the media, or both. Increasing the resolution of such images is also costly. While high-resolution images may be used to train a deep-learning network in order to increase the quality of low-resolution images, an important obstacle is the lack of a large number of images for the training, as the accuracy of the network's predictions depends on the extent of the training data. Generating a large number of high-resolution images by experimental means is, however, very time consuming and costly, hence limiting the application of deep-learning algorithms to such an important class of problems. To address the issue we propose a novel hybrid algorithm by which a stochastic reconstruction method is used to generate a large number of plausible images of a shale formation, using very few input images at very low cost, and then train a deep-learning convolutional network by the stochastic realizations. We refer to the method as hybrid stochastic deep-learning (HSDL) algorithm. The results indicate promising improvement in the quality of the images, the accuracy of which is confirmed by visual, as well as quantitative comparison between several of their statistical properties. The results are also compared with those obtained by the regular deep learning algorithm without using an enriched and large dataset for training, as well as with those generated by bicubic interpolation.

Molecular dynamics study of structure, folding, and aggregation of poly-glycine-alanine (Poly-GA).

Poly-glycine-alanine (poly-GA) proteins are widely believed to be one of the main toxic dipeptide repeat molecules associated with amyotrophic lateral sclerosis (ALS) and frontotemporal dementia diseases. Using discontinuous molecular dynamics simulation and an all-atom model of the proteins, we study folding, stability, and aggregation of poly-GA. The results demonstrate that poly-GA is an aggregation-prone protein that, after a long enough time, forms ß-sheet-rich aggregates that match recent experiment data and that two unique helical structures are formed very frequently, namely, ß-helix and double-helix. The details of the two structures are analyzed. The analysis indicates that such helical structures are stable and share the characteristics of both α-helices and ß-sheets. Molecular simulations indicate that identical phenomena also occur in the aggregation of poly-glycine-arginine (poly-GR). Therefore, we hypothesize that proteins of type (GX)n in which X may be any non-glycine amino acid and n is the repeat length may share the same folding structures of ß-helix and double-helix and that it is the glycine in the repeat that contributes the most to this characteristic. Molecular dynamics simulation with continuous interaction potentials and explicit water molecules as the solvent supports the hypothesis. To our knowledge, this is the first molecular dynamics simulation of the phenomena involving poly-GA and poly-GR proteins.

Effect of heterogeneity and spatial correlations on the structure of a tumor invasion front in cellular environments.

Analysis of invasion front has been widely used to decipher biological properties, as well as the growth dynamics of the corresponding populations. Likewise, the invasion front of tumors has been investigated, from which insights into the biological mechanisms of tumor growth have been gained. We develop a model to study how tumors' invasion front depends on the relevant properties of a cellular environment. To do so, we develop a model based on a nonlinear reaction-diffusion equation, the Fisher-Kolmogorov-Petrovsky-Piskunov equation, to model tumor growth. Our study aims to understand how heterogeneity in the cellular environment's stiffness, as well as spatial correlations in its morphology, the existence of both of which has been demonstrated by experiments, affects the properties of tumor invasion front. It is demonstrated that three important factors affect the properties of the front, namely the spatial distribution of the local diffusion coefficients, the spatial correlations between them, and the ratio of the cells' duplication rate and their average diffusion coefficient. Analyzing the scaling properties of tumor invasion front computed by solving the governing equation, we show that, contrary to several previous claims, the invasion front of tumors and cancerous cell colonies cannot be described by the well-known models of kinetic growth, such as the Kardar-Parisi-Zhang equation.

Effect of the geometry of confining media on the stability and folding rate of α -helix proteins.

Protein folding in confined media has attracted wide attention over the past 15 years due to its importance to both in vivo and in vitro applications. It is generally believed that protein stability increases by decreasing the size of the confining medium, if the medium's walls are repulsive, and that the maximum folding temperature in confinement is in a pore whose size D 0 is only slightly larger than the smallest dimension of a protein's folded state. Until recently, the stability of proteins in pores with a size very close to that of the folded state has not received the attention it deserves. In a previous paper [L. Javidpour and M. Sahimi, J. Chem. Phys. 135, 125101 (2011)], we showed that, contrary to the current theoretical predictions, the maximum folding temperature occurs in larger pores for smaller α-helices. Moreover, in very tight pores, the free energy surface becomes rough, giving rise to a new barrier for protein folding close to the unfolded state. In contrast to unbounded domains, in small nanopores proteins with an α-helical native state that contain the ß structures are entropically stabilized implying that folding rates decrease notably and that the free energy surface becomes rougher. In view of the potential significance of such results to interpretation of many sets of experimental data that could not be explained by the current theories, particularly the reported anomalously low rates of folding and the importance of entropic effects on proteins' misfolded states in highly confined environments, we address the following question in the present paper: To what extent the geometry of a confined medium affects the stability and folding rates of proteins? Using millisecond-long molecular dynamics simulations, we study the problem in three types of confining media, namely, cylindrical and slit pores and spherical cavities. Most importantly, we find that the prediction of the previous theories that the dependence of the maximum folding temperature T f on the size D of a confined medium occurs in larger media for larger proteins is correct only in spherical geometry, whereas the opposite is true in the two other geometries that we study. Also studied is the effect of the strength of the interaction between the confined media's walls and the proteins. If the walls are only weakly or moderately attractive, a complex behavior emerges that depends on the size of the confining medium.