A Rate of Convergence of Physics Informed Neural Networks for the Linear Second Order Elliptic PDEs

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the number of training samples, depth and width of the deep neural networks to achieve desired accuracy. The error of PINNs is decomposed into approximation error and statistical error, where the approximation error is given in $C^2$ norm with ReLU$^3$ networks (deep network with activation function max$\{0,x^3\}$) and the statistical error is estimated by Rademacher complexity. We derive the bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm with ReLU$^3$ network, which is of immense independent interest.     

Positive Change Following Trauma and Adversity: A Review

Empirical studies (n = 39) that documented positive change following trauma and adversity (e.g., posttraumatic growth, stress-related growth, perceived benefit, thriving; collectively described as adversarial growth) were reviewed. The review indicated that cognitive appraisal variables (threat, harm, and controllability), problem-focused, acceptance and positive reinterpretation coping, optimism, religion, cognitive processing, and positive affect were consistently associated with adversarial growth. The review revealed inconsistent associations between adversarial growth, sociodemographic variables (gender, age, education, and income), and psychological distress variables (e.g., depression, anxiety, posttraumatic stress disorder). However, the evidence showed that people who reported and maintained adversarial growth over time were less distressed subsequently. Methodological limitations and recommended future directions in adversarial growth research are discussed, and the implications of adversarial growth for clinical practice are briefly considered.     

Generative adversarial networks (GAN)-based data augmentation of rare liver cancers: The SFR 2021 Artificial Intelligence Data Challenge

PurposeThe 2021 edition of the Artificial Intelligence Data Challenge was organized by the French Society of Radiology together with the Centre National d’Études Spatiales and CentraleSupélec with the aim to implement generative adversarial networks (GANs) techniques to provide 1000 magnetic resonance imaging (MRI) cases of macrotrabecular-massive (MTM) hepatocellular carcinoma (HCC), a rare and aggressive subtype of HCC, generated from a limited number of real cases from multiple French centers.Materials and methodsA dedicated platform was used by the seven inclusion centers to securely upload their anonymized MRI examinations including all three cross-sectional images (one late arterial and one portal-venous phase T1-weighted images and one fat-saturated T2-weighted image) in compliance with general data protection regulation. The quality of the database was checked by experts and manual delineation of the lesions was performed by the expert radiologists involved in each center. Multidisciplinary teams competed between October 11^th, 2021 and February 13^th, 2022.ResultsA total of 91 MTM-HCC datasets of three images each were collected from seven French academic centers. Six teams with a total of 28 individuals participated in this challenge. Each participating team was asked to generate one thousand 3-image cases. The qualitative evaluation was performed by three radiologists using the Likert scale on ten randomly selected cases generated by each participant. A quantitative evaluation was also performed using two metrics, the Frechet inception distance and a leave-one-out accuracy of a 1-Nearest Neighbor algorithm.ConclusionThis data challenge demonstrates the ability of GANs techniques to generate a large number of images from a small sample of imaging examinations of a rare malignant tumor.     

Adaptive dynamic surface control using neural networks for hypersonic flight vehicle with input nonlinearities

This study proposes an effective adaptive dynamic surface control (DSC) method based on the radial basis function neural networks and the auxiliary system for hypersonic flight vehicle (HFV) systems in the presence of system uncertainties, external disturbances, and state variable and control input constraints. Firstly, to enhance the robustness of the system, the neural network is combined with the robust term to deal with the uncertainties and external disturbances of the system. Secondly, to prevent the deterioration of the dynamic performance of the system due to the over-adaptation of the neural networks and the robust terms caused by the state and control input constraints, the auxiliary system is added at each step in the DSC design to adjust the dynamic process of the reference signal and virtual control. Furthermore, the variable structure control is used to solve the problem of dead zone in the control input. Using the Lyapunov analysis method, all signals of the closed-loop system are semi-globally uniformly ultimate bounded. The simulation results illustrate the effectiveness of the proposed control scheme for the HFVs.     

An Adaptive Combined Preconditioner with Applications in Radiation Diffusion Equations

The paper aims to develop an effective preconditioner and conduct the convergence analysis of the corresponding preconditioned GMRES for the solution of discrete problems originating from multi-group radiation diffusion equations. We firstly investigate the performances of the most widely used preconditioners (ILU(k) and AMG) and their combinations ($B_{co}$ and$\widetilde{B}_{co}$), and provide drawbacks on their feasibilities. Secondly, we reveal the underlying complementarity of ILU(k) and AMG by analyzing the features suitable for AMG using more detailed measurements on multiscale nature of matrices and the effect of ILU(k) on multiscale nature. Moreover, we present an adaptive combined preconditioner $B^α_{co}$ involving an improved ILU(0) along with its convergence constraints. Numerical results demonstrate that $B^α_{co}$-GMRES holds the best robustness and efficiency. At last, we analyze the convergence of GMRES with combined preconditioning which not only provides a persuasive support for our proposed algorithms, but also updates the existing estimation theory on condition numbers of combined preconditioned systems.     

Challenges and Technologies in Reservoir Modeling

Reservoir modeling is playing an increasingly important role in developing and producing hydrocarbon reserves. In this paper, we provide a brief overview of some main challenges in reservoir modeling, i.e., accurate and efficient modeling of complex reservoir geometry and heterogeneous reservoir properties. We then present modeling techniques we recently developed in addressing these challenges, including a method for generating constrained Voronoi grids and a generic global scale-up method. We focus on the Voronoi gridding method, which is based on a new constrained Delaunay triangulation algorithm and a rigorous method of adapting Voronoi grids to piecewise linear constraints. The global scale-up method based on generic flows is briefly described. Numerical examples are provided to demonstrate the techniques and the advantage of combining them in constructing accurate and efficient reservoir models.     

Biological constraints that limit compensation of a common skeletal trait variant lead to inequivalence of tibial function among healthy young adults

Having a better understanding of how complex systems like bone compensate for the natural variation in bone width to establish mechanical function will benefit efforts to identify traits contributing to fracture risk. Using a collection of pQCT images of the tibial diaphysis from 696 young adult women and men, we tested the hypothesis that bone cells cannot surmount the nonlinear relationship between bone width and whole bone stiffness to establish functional equivalence across a healthy population. Intrinsic cellular constraints limited the degree of compensation, leading to functional inequivalence relative to robustness, with slender tibias being as much as two to three times less stiff relative to body size compared with robust tibias. Using Path Analysis, we identified a network of compensatory trait interactions that explained 79% of the variation in whole-bone bending stiffness. Although slender tibias had significantly less cortical area relative to body size compared with robust tibias, it was the limited range in tissue modulus that was largely responsible for the functional inequivalence. Bone cells coordinately modulated mineralization as well as the cortical porosity associated with internal bone multicellular units (BMU)-based remodeling to adjust tissue modulus to compensate for robustness. Although anecdotal evidence suggests that functional inequivalence is tolerated under normal loading conditions, our concern is that the functional deficit of slender tibias may contribute to fracture susceptibility under extreme loading conditions, such as intense exercise during military training or falls in the elderly. Thus, we show the natural variation in bone robustness was associated with predictable functional deficits that were attributable to cellular constraints limiting the amount of compensation permissible in human long bone. Whether these cellular constraints can be circumvented prophylactically to better equilibrate function among individuals remains to be determined.     

Investigating Plasma Jets Behavior Using Axisymmetric Lattice Boltzmann Model Under Temperature Dependent Viscosity

This study aims to investigate turbulent plasma flow using the lattice Boltzmann (LB) method. A double population model D2Q9-D2Q4 is employed to calculate the plasma velocity and temperature fields. Along with the calculation process a conversion procedure is made between the LB and the physical unit systems, so that thermo-physical properties variation is fully accounted for and the convergence is checked in physical space. The configuration domain and the boundary condition treatment are selected based on the most cited studies in order to illustrate a realistic situation. The jet morphology analysis gives credible results by comparison with commonly published works. It was demonstrated also that accounting for the substrate as wall boundary condition modify greatly the flow and temperature structures with may affect absolutely the particles behavior during its in-flight in the hot gas.     

A reinforcement learning‐based scheme for direct adaptive optimal control of linear stochastic systems

Reinforcement learning where decision‐making agents learn optimal policies through environmental interactions is an attractive paradigm for model‐free, adaptive controller design. However, results for systems with continuous state and action variables are rare. In this paper, we present convergence results for optimal linear quadratic control of discrete‐time linear stochastic systems. This work can be viewed as a generalization of a previous work on deterministic linear systems. Key differences between the algorithms for deterministic and stochastic systems are highlighted through examples. The usefulness of the algorithm is demonstrated through a nonlinear chemostat bioreactor case study. Copyright © 2009 John Wiley & Sons, Ltd.     

Convergence Rate Analysis for Deep Ritz Method

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) [47] for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with ${\rm ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bound on the approximation error of deep ${\rm ReLU}^2$ network in $C^1$ norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ${\rm ReLU}^2$ network, both of which are of independent interest.     

On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification

In this work, we concern with the numerical comparison between different kinds of design points in least square (LS) approach on polynomial spaces. Such a topic is motivated by uncertainty quantification (UQ). Three kinds of design points are considered, which are the Sparse Grid (SG) points, the Monte Carlo (MC) points and the Quasi Monte Carlo (QMC) points. We focus on three aspects during the comparison: (i) the convergence properties; (ii) the stability, i.e. the properties of the resulting condition number of the design matrix; (iii) the robustness when numerical noises are present in function values. Several classical high dimensional functions together with a random ODE model are tested. It is shown numerically that (i) neither the MC sampling nor the QMC sampling introduces the low convergence rate, namely, the approach achieves high order convergence rate for all cases provided that the underlying functions admit certain regularity and enough design points are used; (ii)The use of SG points admits better convergence properties only for very low dimensional problems (say d ≤ 2); (iii)The QMC points, being deterministic, seem to be a good choice for higher dimensional problems not only for better convergence properties but also in the stability point of view.     

A Pressure-Correction Scheme for Rotational Navier-Stokes Equations and Its Application to Rotating Turbulent Flows

The rotational incremental pressure-correction (RIPC) scheme, described in Timmermans et al. [Int. J. Numer. Methods. Fluids., 22 (1996)] and Shen et al. [Math. Comput., 73 (2003)] for non-rotational Navier-Stokes equations, is extended to rotating incompressible flows. The method is implemented in the context of a pseudo Fourier-spectral code and applied to several rotating laminar and turbulent flows. The performance of the scheme and the computational results are compared to the so-called diagonalization method (DM) developed by Morinishi et al. [Int. J. Heat. Fluid. Flow., 22 (2001)]. The RIPC predictions are in excellent agreement with the DM predictions, while being simpler to implement and computationally more efficient. The RIPC scheme is not in anyway limited to implementation in a pseudo-spectral code or periodic boundary conditions, and can be used in complex geometries and with other suitable boundary conditions.     

Reduced Basis Approximation and Error Bounds for Potential Flows in Parametrized Geometries

In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for potential flows in affinely parametrized geometries. We review the essential ingredients: i) a Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold" in order to get a dimension reduction; ii) an efficient and effective greedy sampling method for identification of optimal and numerically stable approximations to have a rapid convergence; iii) an a posteriori error estimation procedure: rigorous and sharp bounds for the linear-functional outputs of interest and over the potential solution or related quantities of interest like velocity and/or pressure; iv) an Offline-Online computational decomposition strategies to achieve a minimum marginal computational cost for high performance in the real-time and many-query (e.g., design and optimization) contexts. We present three illustrative results for inviscid potential flows in parametrized geometries representing a Venturi channel, a circular bend and an added mass problem.     

A Modified Crank-Nicolson Numerical Scheme for the Flory-Huggins Cahn-Hilliard Model

In this paper we propose and analyze a second order accurate numerical scheme for the Cahn-Hilliard equation with logarithmic Flory Huggins energy potential. A modified Crank-Nicolson approximation is applied to the logarithmic nonlinear term, while the expansive term is updated by an explicit second order Adams-Bashforth extrapolation, and an alternate temporal stencil is used for the surface diffusion term. A nonlinear artificial regularization term is added in the numerical scheme, which ensures the positivity-preserving property, i.e., the numerical value of the phase variable is always between -1 and 1 at a point-wise level. Furthermore, an unconditional energy stability of the numerical scheme is derived, leveraging the special form of the logarithmic approximation term. In addition, an optimal rate convergence estimate is provided for the proposed numerical scheme, with the help of linearized stability analysis. A few numerical results, including both the constant-mobility and solution-dependent mobility flows, are presented to validate the robustness of the proposed numerical scheme.     

Telementoring for minimally invasive surgical training by wireless robot

Hands-on training courses with local mentoring are excellent educational tools in laparoscopic surgery; however, the need for the physical presence of specialized instructors represents a limitation because of costs, time, and geographic constraints. Remote robotic telementoring using a wireless videoconferencing mobile robot could represent an alternative to local instruction. The authors compare local active and passive mentoring with remote robotic telementoring using the wireless RP-6 Robot that worked through a WiFi 802.11b connection during a hands-on laparoscopic training session. Surgeons were mentored once in France from the United States. Robot mentoring was well received and appreciated (assessment score of 2.65; scale, 0 to 4). There was no statistical difference in the different mentoring sessions (active, passive, and remote). Mobile wireless robot is a valuable tool in laparoscopic telementoring. Robotic-assisted telementoring may not replace onsite mentoring, but it may enhance educational opportunities and the quality of hands-on training courses by implementing tutoring with expert assistance from remote locations.     

Conversion of Laparoscopic Cholecystectomy to Open Cholecystectomy in Acute Cholecystitis: Artificial Neural Networks Improve the Prediction of Conversion

Laparoscopic cholecystectomy is now also performed for acute cholecystitis. In the presence of inflammatory conditions, technical difficulties leading to conversion to open cholecystectomy may occur and overshadow the advantages of the laparoscopic approach. Factors associated with these undue events combined with techniques capable of learning from them may help in determining when to completely avoid the laparoscopic procedure. In this study we determined predictors of conversion in acute cholecystitis and tested their predictive ability by means of statistical multivariate analysis and artificial neural networks. Between January 1994 and February 1997, 225 patients underwent laparoscopic cholecystectomy for acute cholecystitis. Preoperative and operative data were prospectively collected on standardized forms. The first 180 laparoscopically approached cases entered the training set, which was learned by both the statistical and the artificial neural networks methods. Conversion was first studied in relation to a set of preoperative data. Prediction models were then fitted by both of these methods. The last 45 operated cases, which remained unknown to the learning systems, served for testing the fitted models. The forward stepwise logistic regression technique, the forward stepwise linear discriminant analysis, and the artificial neural networks method enabled positive prediction of conversion in 0%, 27%, and 100% of the cases, and a negative prediction in 80%, 85.5%, and 97% respectively, in the training set. A positive prediction of conversion in 0%, 25%, and 67% of the cases, and a negative prediction in 82%, 88%, and 94%, respectively, in the untrained, validation set of patients. An artificial neural networks based model provides a practical tool for the prediction of successful laparoscopic cholecystectomies and their conversion. The high degree of certainty of prediction in untrained cases reveals its potential, and justifies, under appropriate conditions, the complete avoidance of laparoscopy and turning directly to open cholecystectomy.     

Physical therapy rehabilitation of the ankle

Physical therapy intervention following an ankle injury is crucial and essential to returning a patient to his/her prior level of function. Following a physical therapy evaluation, a physical therapy diagnosis is established by relating the physical impairments found (e.g., limitations in range of motion, strength, proprioception, etc.) to functional limitations. The goal of physical therapy intervention is to improve these physical impairments, thereby restoring a patient's normal function. The physical therapist can administer such treatment as joint mobilization, strength training, proprioceptive training, and patient education. Since individuals vary in the extent and severity of physical impairments, physical therapy intervention will also vary on a patient-by-patient basis. Therefore, the purpose of this article is not to serve as a protocol for physical therapy intervention but as a review of evidence-based treatment that is relevant for the impairments found after completing a physical therapy evaluation.     

Extension of Near-Wall Domain Decomposition to Modeling Flows with Laminar-Turbulent Transition

The near-wall domain decomposition method (NDD) has proved to be very efficient for modeling near-wall fully turbulent flows. In this paper the NDD is extended to non-equilibrium regimes with laminar-turbulent transition (LTT) for the first time. The LTT is identified with the use of the $e^N$-method which is applied to both incompressible and compressible flows. The NDD is modified to take into account LTT in an efficient way. In addition, implementation of the intermittency expands the capabilities of NDD to model non-equilibrium turbulent flows with transition. Performance of the modified NDD approach is demonstrated on various test problems of subsonic and supersonic flows past a flat plate, a supersonic flow over a compression corner and a planar shock wave impinging on a turbulent boundary layer. The results of modeling with and without decomposition are compared in terms of wall friction and show good agreement with each other while NDD significantly reducing computational resources needed. It turns out that the NDD can reduce the computational time as much as three times while retaining practically the same accuracy of prediction.     

Theoretical neuroanatomy: relating anatomical and functional connectivity in graphs and cortical connection matrices

Neuroanatomy places critical constraints on the functional connectivity of the cerebral cortex. To analyze these constraints we have examined the relationship between structural features of networks (expressed as graphs) and the patterns of functional connectivity to which they give rise when implemented as dynamical systems. We selected among structurally varying graphs using as selective criteria a number of global information-theoretical measures that characterize functional connectivity. We selected graphs separately for increases in measures of entropy (capturing statistical independence of graph elements), integration (capturing their statistical dependence) and complexity (capturing the interplay between their functional segregation and integration). We found that dynamics with high complexity were supported by graphs whose units were organized into densely linked groups that were sparsely and reciprocally interconnected. Connection matrices based on actual neuroanatomical data describing areas and pathways of the macaque visual cortex and the cat cortex showed structural characteristics that coincided best with those of such complex graphs, revealing the presence of distinct but interconnected anatomical groupings of areas. Moreover, when implemented as dynamical systems, these cortical connection matrices generated functional connectivity with high complexity, characterized by the presence of highly coherent functional clusters. We also found that selection of graphs as they responded to input or produced output led to increases in the complexity of their dynamics. We hypothesize that adaptation to rich sensory environments and motor demands requires complex dynamics and that these dynamics are supported by neuroanatomical motifs that are characteristic of the cerebral cortex.     

Sleep and general anesthesia

Purpose

The mechanisms through which general anesthetics cause reversible loss of consciousness are characterized poorly. In this review, we examine the evidence that anesthetic-induced loss of consciousness may be caused by actions on the neuronal pathways that produce natural sleep.

Principal findings

It is clear that many general anesthetics produce effects in the brain (detected on electroencephalogram recordings) that are similar to those seen during non-rapid eye movement non-(REM) sleep. Gamma aminobutyric acid (GABA)ergic hypnogenic neurons are thought to be critical for generating non-REM sleep through their inhibitory projections to wake-active regions of the brain. The postsynaptic GABA_A receptor is a major molecular target of many anesthetics and thus may be a point of convergence between natural sleep and anesthesia. Furthermore, we also present growing evidence in this review that modulating wake-active neurotransmitter (e.g., acetylcholine, histamine) release can impact on anesthesia, supporting the idea that this point of convergence is at the level of the brain arousal systems.

Conclusions

While it is clear that general anesthetics can have effects at various points in the sleep-wake circuitry, it remains to be seen which points are true anesthetic targets. It will be challenging to separate non-specific effects on baseline arousal from a causal mechanism. Sophisticated experimental approaches are necessary to address basic mechanisms of sleep and anesthesia and should advance our understanding in both of these fields.