Dynamical system identification, model selection, and model uncertainty quantification by Bayesian inference.

This study presents a Bayesian maximum a posteriori (MAP) framework for dynamical system identification from time-series data. This is shown to be equivalent to a generalized Tikhonov regularization, providing a rational justification for the choice of the residual and regularization terms, respectively, from the negative logarithms of the likelihood and prior distributions. In addition to the estimation of model coefficients, the Bayesian interpretation gives access to the full apparatus for Bayesian inference, including the ranking of models, the quantification of model uncertainties, and the estimation of unknown (nuisance) hyperparameters. Two Bayesian algorithms, joint MAP and variational Bayesian approximation, are compared to the least absolute shrinkage and selection operator (LASSO), ridge regression, and the sparse identification of nonlinear dynamics (SINDy) algorithms for sparse regression by application to several dynamical systems with added Gaussian or Laplace noise. For multivariate Gaussian likelihood and prior distributions, the Bayesian formulation gives Gaussian posterior and evidence distributions, in which the numerator terms can be expressed in terms of the Mahalanobis distance or "Gaussian norm" ||y-y^||M-12=(y-y^)⊤M-1(y-y^), where y is a vector variable, y^ is its estimator, and M is the covariance matrix. The posterior Gaussian norm is shown to provide a robust metric for quantitative model selection for the different systems and noise models examined.

Dimensionless Groups by Entropic Similarity: II-Wave Phenomena and Information-Theoretic Flow Regimes.

The aim of this study is to explore the insights of the information-theoretic definition of similarity for a multitude of flow systems with wave propagation. This provides dimensionless groups of the form Πinfo=U/c, where U is a characteristic flow velocity and c is a signal velocity or wave celerity, to distinguish different information-theoretic flow regimes. Traditionally, dimensionless groups in science and engineering are defined by geometric similarity, based on ratios of length scales; kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on ratios of forces. In Part I, an additional category of entropic similarity was proposed based on ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. In this Part II, the information-theoretic definition is applied to a number of flow systems with wave phenomena, including acoustic waves, blast waves, pressure waves, surface or internal gravity waves, capillary waves, inertial waves and electromagnetic waves. These are used to define the appropriate Mach, Euler, Froude, Rossby or other dimensionless number(s)-including new groups for internal gravity, inertial and electromagnetic waves-to classify their flow regimes. For flows with wave dispersion, the coexistence of different celerities for individual waves and wave groups-each with a distinct information-theoretic group-is shown to imply the existence of more than two information-theoretic flow regimes, including for some acoustic wave systems (subsonic/mesosonic/supersonic flow) and most systems with gravity, capillary or inertial waves (subcritical/mesocritical/supercritical flow). For electromagnetic wave systems, the additional vacuum celerity implies the existence of four regimes (subluminal/mesoluminal/transluminal/superluminal flow). In addition, entropic analyses are shown to provide a more complete understanding of frictional behavior and sharp transitions in compressible and open channel flows, as well as the transport of entropy by electromagnetic radiation. The analyses significantly extend the applications of entropic similarity for the analysis of flow systems with wave propagation.

Dimensionless Groups by Entropic Similarity: I - Diffusion, Chemical Reaction and Dispersion Processes.

Since the time of Buckingham in 1914, dimensional analysis and similarity arguments based on dimensionless groups have served as powerful tools for the analysis of systems in all branches of science and engineering. Dimensionless groups are generally classified into those arising from geometric similarity, based on ratios of length scales; kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on ratios of forces. We propose an additional category of dimensionless groups based on entropic similarity, defined by ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. Since all processes involving work against friction, dissipation, diffusion, dispersion, mixing, separation, chemical reaction, gain of information or other irreversible changes are driven by (or must overcome) the second law of thermodynamics, it is appropriate to analyze them directly in terms of competing entropy-producing and transporting phenomena and the dominant entropic regime, rather than indirectly in terms of forces. In this study, entropic groups are derived for a wide variety of diffusion, chemical reaction and dispersion processes relevant to fluid mechanics, chemical engineering and environmental engineering. It is shown that many dimensionless groups traditionally derived by kinematic or dynamic similarity (including the Reynolds number) can also be recovered by entropic similarity-with a different entropic interpretation-while many new dimensionless groups can also be identified. The analyses significantly expand the scope of dimensional analysis and similarity arguments for the resolution of new and existing problems in science and engineering.

Comparison of DRASTIC and DRASTICL groundwater vulnerability assessments of the Burdekin Basin, Queensland, Australia.

In the Burdekin Basin, Queensland, Australia, groundwater contamination due to agricultural activities has led to concerns over its impacts on globally significant ecosystems such as the Great Barrier Reef. An appropriate method for groundwater vulnerability assessment is essential for the sustainable use of this groundwater resource and its longer-term environmental management. The aim of this study is to apply and assess the suitability of the standard DRASTIC index-based method for groundwater vulnerability assessment of the Burdekin Basin. The intrinsic groundwater vulnerability is calculated in ArcGIS, using data for the period 2010 to 2021. The results are compared to available water quality data. The calculated DRASTIC scores are found to be only weakly correlated with water quality parameters, including the nitrate concentration (R = 0.07), which should behave as a proxy measure of groundwater vulnerability. To address this, a modified DRASTICL method containing a land use parameter is also implemented, to assess the specific groundwater vulnerability. The correlation between DRASTICL scores and nitrate levels (R = 0.2) is more significant but is still relatively weak. From this study, it is recommended that alternative methods be developed to assess groundwater vulnerability in the Burdekin Basin, and other comparable aquifer systems.

Kinetic model of PFAS removal by semi-batch foam fractionation and validation by experimental data for K-PFOS.

Adsorptive bubble separation techniques such as foam fractionation have recently been applied for the extraction of per- and polyfluoroalkyl substances (PFAS) from waters at both laboratory and operational scales. However, few authors have developed mathematical models of their removal of PFAS. This study presents a theoretical framework for the kinetics of PFAS removal from fresh and monovalent saline waters by a semi-batch foam fractionation process, by the mechanisms of adsorption, entrainment and volatilization, as a function of pertinent parameters including PFAS air-water adsorption, bubble radius, electrolyte concentration and ionic strength, PFAS volatility, and flow and geometric parameters. The freshwater model is validated for the removal of potassium perfluorooctane sulfonate (K-PFOS) using published experimental data (Meng, P. et al., Chemosphere, 2018, 203, 263-270). The proposed models provide quantitative tools for process design and the optimization of individual PFAS removal by semi-batch adsorptive bubble separation techniques such as foam fractionation.

Groundwater vulnerability assessment: A review including new statistical and hybrid methods.

The concept of groundwater vulnerability was first introduced in the 1970s in France to recognize sensitive areas in which surface pollution could affect groundwater, and to enable others to develop management methods for groundwater protection against surface pollutants. Since this time, numerous methods have been developed for groundwater vulnerability assessment (GVA). These can be categorized into four groups: (i) overlay and index-based methods, (ii) process-based simulation models, (iii) statistical methods, and (iv) hybrid methods. This work provides a comprehensive review of modern GVA methods, which in contrast to previous reviews, examines the last two categories in detail. First, the concept of groundwater vulnerability is defined, then the major GVA methods are introduced and classified. This includes detailed accounts of statistical methods, which can be subdivided into orthodox statistical, data-driven and Bayesian methods, and their advantages and disadvantages, as well as modern hybrid methods. It is concluded that Bayesian inference offers many advantages compared with other GVA methods. It combines theory and data to give the posterior probabilities of different models, which can be continually updated with new data. Furthermore, using the Bayesian approach, it is possible to calculate the probability of a proposition, which is exactly what is needed to make decisions. However, despite the advantages of Bayesian inference, its applications to date have been very limited.

A Hierarchy of Probability, Fluid and Generalized Densities for the Eulerian Velocivolumetric Description of Fluid Flow, for New Families of Conservation Laws.

The Reynolds transport theorem occupies a central place in continuum mechanics, providing a generalized integral conservation equation for the transport of any conserved quantity within a fluid or material volume, which can be connected to its corresponding differential equation. Recently, a more generalized framework was presented for this theorem, enabling parametric transformations between positions on a manifold or in any generalized coordinate space, exploiting the underlying continuous multivariate (Lie) symmetries of a vector or tensor field associated with a conserved quantity. We explore the implications of this framework for fluid flow systems, based on an Eulerian velocivolumetric (position-velocity) description of fluid flow. The analysis invokes a hierarchy of five probability density functions, which by convolution are used to define five fluid densities and generalized densities relevant to this description. We derive 11 formulations of the generalized Reynolds transport theorem for different choices of the coordinate space, parameter space and density, only the first of which is commonly known. These are used to generate a table of integral and differential conservation laws applicable to each formulation, for eight important conserved quantities (fluid mass, species mass, linear momentum, angular momentum, energy, charge, entropy and probability). The findings substantially expand the set of conservation laws for the analysis of fluid flow and dynamical systems.

Invariance Properties of the Entropy Production, and the Entropic Pairing of Inertial Frames of Reference by Shear-Flow Systems.

This study examines the invariance properties of the thermodynamic entropy production in its global (integral), local (differential), bilinear, and macroscopic formulations, including dimensional scaling, invariance to fixed displacements, rotations or reflections of the coordinates, time antisymmetry, Galilean invariance, and Lie point symmetry. The Lie invariance is shown to be the most general, encompassing the other invariances. In a shear-flow system involving fluid flow relative to a solid boundary at steady state, the Galilean invariance property is then shown to preference a unique pair of inertial frames of reference-here termed an entropic pair-respectively moving with the solid or the mean fluid flow. This challenges the Newtonian viewpoint that all inertial frames of reference are equivalent. Furthermore, the existence of a shear flow subsystem with an entropic pair different to that of the surrounding system, or a subsystem with one or more changing entropic pair(s), requires a source of negentropy-a power source scaled by an absolute temperature-to drive the subsystem. Through the analysis of different shear flow subsystems, we present a series of governing principles to describe their entropic pairing properties and sources of negentropy. These are unaffected by Galilean transformations, and so can be understood to "lie above" the Galilean inertial framework of Newtonian mechanics. The analyses provide a new perspective into the field of entropic mechanics, the study of the relative motions of objects with friction.

Unification of surface tension isotherms of PFOA or GenX salts in electrolyte solutions by mean ionic activity.

The surface tension isotherms of soluble salts of per- and polyfluoroalkyl substances (PFAS) in electrolyte solutions are typically reported as functions of the PFAS concentration. However, for univalent salts and electrolytes, the Langmuir-Szyszkowski equation is a function of the mean ionic activity a*. Using previously reported data, we show that for salts of perfluorooctanoic acid (PFOA) or hexafluoropropylene oxide dimer acid (GenX™), use of a* rather than concentration provides a unified surface tension isotherm, independent of the electrolyte concentration. This suggests that the electrolyte dependence of the isotherm arises purely from its effect on PFAS activity, rather than an intrinsic surface property. This finding has important implications for the understanding of PFAS retention in saline unsaturated soils, and for PFAS extraction from saline waters by foam fractionation.

Maximum Entropy Analysis of Flow Networks: Theoretical Foundation and Applications.

The concept of a "flow network"-a set of nodes and links which carries one or more flows-unites many different disciplines, including pipe flow, fluid flow, electrical, chemical reaction, ecological, epidemiological, neurological, communications, transportation, financial, economic and human social networks. This Feature Paper presents a generalized maximum entropy framework to infer the state of a flow network, including its flow rates and other properties, in probabilistic form. In this method, the network uncertainty is represented by a joint probability function over its unknowns, subject to all that is known. This gives a relative entropy function which is maximized, subject to the constraints, to determine the most probable or most representative state of the network. The constraints can include "observable" constraints on various parameters, "physical" constraints such as conservation laws and frictional properties, and "graphical" constraints arising from uncertainty in the network structure itself. Since the method is probabilistic, it enables the prediction of network properties when there is insufficient information to obtain a deterministic solution. The derived framework can incorporate nonlinear constraints or nonlinear interdependencies between variables, at the cost of requiring numerical solution. The theoretical foundations of the method are first presented, followed by its application to a variety of flow networks.

Prediction of dynamical systems by symbolic regression.

We study the modeling and prediction of dynamical systems based on conventional models derived from measurements. Such algorithms are highly desirable in situations where the underlying dynamics are hard to model from physical principles or simplified models need to be found. We focus on symbolic regression methods as a part of machine learning. These algorithms are capable of learning an analytically tractable model from data, a highly valuable property. Symbolic regression methods can be considered as generalized regression methods. We investigate two particular algorithms, the so-called fast function extraction which is a generalized linear regression algorithm, and genetic programming which is a very general method. Both are able to combine functions in a certain way such that a good model for the prediction of the temporal evolution of a dynamical system can be identified. We illustrate the algorithms by finding a prediction for the evolution of a harmonic oscillator based on measurements, by detecting an arriving front in an excitable system, and as a real-world application, the prediction of solar power production based on energy production observations at a given site together with the weather forecast.

Remobilization of residual non-aqueous phase liquid in porous media by freeze-thaw cycles.

The pore-scale behavior of a nonaqueous phase liquid (NAPL) trapped as residual contamination in a porous medium, subject to freeze-thaw cycles, was investigated by X-ray microcomputed tomography. It is shown that freeze-thaw cycles cause significant NAPL remobilization in the direction of the freezing front, due to the rupture and transport of a significant proportion of (supposedly entrapped) larger multipore NAPL ganglia. Significant NAPL remains in place, however, due to substantial ganglion fragmentation into single- and subpore ganglia. The contraction of branched ganglia into more rounded forms, especially near the top surface, is also observed. Three freezing-induced mechanisms are proposed to explain the results. The findings have important implications for NAPL contamination in cold regions, and for the behavior of water-hydrocarbon systems on the Earth and other planets.

Minimization of a free-energy-like potential for non-equilibrium flow systems at steady state.

This study examines a new formulation of non-equilibrium thermodynamics, which gives a conditional derivation of the 'maximum entropy production' (MEP) principle for flow and/or chemical reaction systems at steady state. The analysis uses a dimensionless potential function (st) for non-equilibrium systems, analogous to the free energy concept of equilibrium thermodynamics. Spontaneous reductions in (st) arise from increases in the 'flux entropy' of the system--a measure of the variability of the fluxes--or in the local entropy production; conditionally, depending on the behaviour of the flux entropy, the formulation reduces to the MEP principle. The inferred steady state is also shown to exhibit high variability in its instantaneous fluxes and rates, consistent with the observed behaviour of turbulent fluid flow, heat convection and biological systems; one consequence is the coexistence of energy producers and consumers in ecological systems. The different paths for attaining steady state are also classified.

Steady state of a dissipative flow-controlled system and the maximum entropy production principle.

A theory to predict the steady-state position of a dissipative flow-controlled system, as defined by a control volume, is developed based on the maximum entropy principle of Jaynes, involving minimization of a generalized free-energy-like potential. The analysis provides a theoretical justification of a local, conditional form of the maximum entropy production principle, which successfully predicts the observable properties of many such systems. The analysis reveals a very different manifestation of the second law of thermodynamics in steady-state flow systems, which provides a driving force for the formation of complex systems, including life.

Mobilization and rupture of LNAPL ganglia during freeze-thaw: two-dimensional cell experiments.

Experiments were conducted on dodecane at residual saturation (21-26%) in a two-dimensional water-saturated glass bead cell (0.5 mm diameter)--to simulate light nonaqueous phase liquid (LNAPL) trapped below the water table--subject to controlled freeze-thaw cycles. The experiments reveal substantial remobilization and rupture of LNAPL ganglia during freeze-thaw, especially during the first few cycles. This includes the detachment and upward mobilization of LNAPL from larger ganglia during upward propagation of the freezing front; the formation of numerous subsinglet ganglia during this transport process, and their entrapment in ice; and the coalescence of such small ganglia during thawing, to form larger singlets. Theoretical calculations suggest that the LNAPL redistribution is caused by large freezing-induced pressure gradients, of up to 6 orders of magnitude higher than the water-LNAPL interfacial (capillary) pressure. The findings have important implications for the understanding and remediation of LNAPLs in cold climate regions.