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In the intensive care unit, the monitored variables collected from sensors may have different behaviors among patients with different clinical basic information. Giving prior information of the monitored variables based on their specific basic information as soon as the patient is admitted will support the clinicians with better decisions during the surgery. Instead of black box models, the explainable hidden Markov model is proposed, which can estimate the possible distribution parameters of the monitored variables under different clinical basic information. A Student’s t-test or correlation test is conducted further to test whether the parameters have a significant relationship with the basic variables. The specific relationship is explored by using a conditional inference tree, which is an explainable model giving deciding rules. Instead of point estimation, interval forecast is chosen as the performance metrics including coverage rate and relative interval width, which provide more reliable results. By applying the methods to an intensive care unit data set with more than 20 thousand patients, the model has good performance with an area under the ROC Curve value of 0.75, which means the hidden states can generally be correctly labelled. The significant test shows that only a few combinations of the basic and monitored variables are not significant under the 0.01 significant level. The tree model based on different quantile intervals provides different coverage and width combination choices. A coverage rate around 0.8 is suggested, which has a relative interval width of 0.77.

Although stochastic fractional partial differential equations have received increasing attention in the last decade, the parameter estimation of these equations has been seldom reported in literature. In this paper, we propose a pseudo-likelihood approach to estimating the parameters of stochastic time-fractional diffusion equations, whose forward solver has been investigated very recently by Gunzburger, Li, and Wang (2019). Our approach can accurately recover the fractional order, diffusion coefficient, as well as noise magnitude given the discrete observation data corresponding to only one realization of driving noise. When only partial data is available, our approach can also attain acceptable results for intermediate sparsity of observation.

To characterize the visco–elasto-plastic behavior of metals and alloys we propose a new constitutive equation based on a time–space fractional derivative. The rheological representative of the model can be analogous to that of the Bingham–Maxwell model, while the dashpot element and sliding friction element are replaced by the corresponding fractional elements. The model is applied to describe the constant strain rate, stress relaxation and creep tests of different metals and alloys. The results suggest that the proposed simple model can describe the main characteristics of the experimental observations. More importantly, the model can also provide more accurate predictions than the classic Bingham–Maxwell model and the Bingham–Norton model.

Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integral equations with sparse, noisy, unstructured, and multi-fidelity data. PINNs incorporate all available information into a loss function, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal operator, which converges to the classical Laplacian as one of the operator parameters, the nonlocal interaction radius $\delta$ goes to zero, and to the fractional Laplacian as $\delta$ goes to infinity. This universal operator forms a super-set of classical Laplacian and fractional Laplacian operators and, thus, has the potential to fit a broad spectrum of data sets. We provide theoretical convergence rates with respect to $\delta$ and verify them via numerical experiments. Second, we use nPINNs to estimate the two parameters, $\delta$ and $\alpha$. The strong non-convexity of the loss function yielding multiple (good) local minima reveals the occurrence of the operator mimicking phenomenon: different pairs of estimated parameters could produce multiple solutions of comparable accuracy. Third, we propose another nonlocal operator with spatially variable order $\alpha(y)$, which is more suitable for modeling turbulent Couette flow. Our results show that nPINNs can jointly infer this function as well as $\delta$. Also, these parameters exhibit a universal behavior with respect to the Reynolds number, a finding that contributes to our understanding of nonlocal interactions in wall-bounded turbulence., Comment: 31 pages, 20 figures

We review and compare physics-informed learning models built upon Gaussian processes and deep neural networks for solving forward and inverse problems governed by linear and nonlinear partial differential equations. We define a unified data model on which Gaussian processes, physics-informed Gaussian processes, neural networks, and physics-informed neural networks are based. We develop continuous-time and discrete-time models to facilitate different application scenarios. We present a connection between a Gaussian process and an infinitely wide neural network, which enables us to obtain a “best” kernel, which is determined directly by the data. We demonstrate the implementation of physics-informed Gaussian processes and physics-informed neural networks using a pedagogical example. Additionally, we compare physics-informed Gaussian processes and physics-informed neural networks for two nonlinear partial differential equations, i.e. the 1D Burgers’ equation and the 2D Navier–Stokes, and provide guidance in choosing the proper machine learning model according to the problem type, i.e. forward or inverse problem, and the availability of data. These new methods for solving partial differential equations governing multi-physics problems do not require any grid, and they are simple to implement, and agnostic to specific application. Hence, we expect that variants and proper extensions of these methods will find broad applicability in the near future across different scientific disciplines but also in industrial applications.

The paper makes a comparative study of the finite element method (FEM) and the finite difference method (FDM) for two-dimensional fractional advection-dispersion equation (FADE) which has recently been considered a promising tool in modeling non-Fickian solute transport in groundwater. Due to the non-local property of integro-differential operator of the space-fractional derivative, numerical solution of FADE is very challenging and little has been reported in literature, especially for high-dimensional case. In order to effectively apply the FEM and the FDM to the FADE on a rectangular domain, a backward-distance algorithm is presented to extend the triangular elements to generic polygon elements in the finite element analysis, and a variable-step vector Grünwald formula is proposed to improve the solution accuracy of the conventional finite difference scheme. Numerical investigation shows that the FEM compares favorably with the FDM in terms of accuracy and convergence rate whereas the latter enjoys less computational effort.

The paper presents a symmetric formulation of the singular boundary method (SBM) for potential problems, whose resultant interpolation matrix is symmetric irrespective of boundary conditions. The SBM is a recent strong-form boundary discretization numerical technique, which is meshless, integration-free, and easy-to-implement. In stark contrast to the method of fundamental solution that requires the fictitious boundary outside physical domain to avoid the singularity of fundamental solution, the SBM can place source points, which coincide with collocation points, on the real physical boundary via the concept of origin intensity factor. The proposed symmetric formulation preserves all the merits of the SBM. The novel idea behind this study is that the unknown solution is represented by both the single and double layer potentials compared with the single layer potential in the original SBM. Numerical solution of three-dimensional potential and two-dimensional fluid flow and heat transfer problems demonstrates the utility and validity of the present SBM symmetric formulation.

Physics-informed neural networks (PINNs) are effective in solving integer-order partial differential equations (PDEs) based on scattered and noisy data. PINNs employ standard feedforward neural networks (NNs) with the PDEs explicitly encoded into the NN using automatic differentiation, while the sum of the mean-squared PDE-residuals and the mean-squared error in initial/boundary conditions is minimized with respect to the NN parameters. We extend PINNs to fractional PINNs (fPINNs) to solve space-time fractional advection-diffusion equations (fractional ADEs), and we demonstrate their accuracy and effectiveness in solving multi-dimensional forward and inverse problems with forcing terms whose values are only known at randomly scattered spatio-temporal coordinates (black-box forcing terms). A novel element of the fPINNs is the hybrid approach that we introduce for constructing the residual in the loss function using both automatic differentiation for the integer-order operators and numerical discretization for the fractional operators. We consider 1D time-dependent fractional ADEs and compare white-box (WB) and black-box (BB) forcing. We observe that for the BB forcing fPINNs outperform FDM. Subsequently, we consider multi-dimensional time-, space-, and space-time-fractional ADEs using the directional fractional Laplacian and we observe relative errors of $10^{-4}$. Finally, we solve several inverse problems in 1D, 2D, and 3D to identify the fractional orders, diffusion coefficients, and transport velocities and obtain accurate results even in the presence of significant noise., Comment: 29 pages, 15 figures, 5 tables

This paper proposes a fractional biharmonic operator equation model in the time-space domain to describe scattering attenuation of acoustic waves in heterogeneous media. Compared with the existing models, the proposed fractional model is able to describe arbitrary frequency-dependent scattering attenuation, which typically obeys an empirical power law with an exponent ranging from 0 to 4. In stark contrast to an extensive and rapidly increasing application of the fractional derivative models for wave absorption attenuation in the literature, little has been reported on frequency-dependent scattering attenuation. This is largely because the order of the fractional Laplacian is from 0 to 2 and is infeasible for scattering attenuation. In this study, the definition of the fractional biharmonic operator in space with an order varying from 0 to 4 is proposed, as well as a fractional biharmonic operator equation model of scattering attenuation which is consistent with arbitrary frequency power-law dependency and obeys the causal relation under the smallness approximation. Finally, the correlation between the fractional order and the ratio of wavelength to the diameter of the scattering heterogeneity is investigated and an expression on exponential form is also provided. © 2017 Acoustical Society of America

Anomalous diffusion is a common phenomenon observed in underground solute transport, soil water infiltration and sediment movement, etc. Time and space fractional derivative advection-dispersion equation (FADE) has been widely employed as the governing equation to characterize above mentioned anomalous diffusion related processes. However, a main problem in application of time and space FADE model to describe the real-world mass transport processes, is its low computation efficiency for long-time range and large irregular domain cases. This study offers a new algorithm in which the Kansa method is used for vector space fractional derivative term discretization and then analytical approach for resulted time fractional ordinary system. The influence of node distribution mode and node numbers on accuracy and convergence rate are analysed through the numerical examples in one and two dimensional cases. To test the application potentials of present methods, we offer the numerical results of two dimensional time and space FADE models in continuous and discrete cases. It shows that the solute plumes in heterogeneous and anisotropic media, can be well simulated by using present method compared with the previous time-consumed particle Monte-Carlo methods., 32 pages, 20 figures