Collocation polynomial neural forms and domain fragmentation for solving initial value problems

Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance, one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article, we consider time-dependent initial value problems, which require to set up the neural form framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first-order polynomials. In this work, we propose to extend the polynomial order of the neural forms. The novel collocation-type construction includes several feedforward neural networks, one for each order. Additionally, we propose the fragmentation of the computational domain into subdomains. The neural forms are solved on each subdomain, whereas the interfacing grid points overlap in order to provide initial values over the whole fragmentation. We illustrate in experiments that the combination of collocation neural forms of higher order and the domain fragmentation allows to solve initial value problems over large domains with high accuracy and reliability.

Solving surface deformation of radio telescope antenna by artificial neural network

Recent investigations have derived the relation between the near-field plane amplitude and the surface deformation of reflector antenna, namely deformation-amplitude equation (DAE), which could be used as a mathematical foundation of antenna surface measurement if an effective numerical algorithm is employed. Traditional algorithms are hard to work directly due to the complexity mathematical model. This paper presents a local approximation algorithm based on artificial neural network (ANN) to solve DAE. Length factor method is used to construct a trial solution for the deformation, which ensures the final solution always to satisfy the boundary conditions. To improve the algorithm efficiency, Adam optimizer is employed to train the network parameters. Combining the application of data normalization method proposed in this paper and a step-based learning rate, a further optimized loss function could be converged quickly. The algorithm proposed in this paper could effectively solve partial differential equations (PDEs) without boundary conditions such as DAE, which at the same time contains the first-order and the second-order partial derivatives, and constant terms. Simulation results show that compared with the original algorithm by FFT, this algorithm is more stable and accurate, which is significant for the antenna measurement method based on DAE.

Plate Structural Analysis Based on a Double Interpolation Element with Arbitrary Meshing

This paper presents the plate structural analysis based on the finite element method (FEM) using a double interpolation element with arbitrary meshing. This element used in this research is related to the first-order shear deformation theory (FSDT) and the double interpolation procedure. The first stage of the procedure is the same with the standard FEM for the quadrilateral element, but the averaged nodal gradients must be computed for the second stage of this interpolation. Shape functions established by the double interpolation procedure exhibit more continuous nodal gradients and higher-order polynomial contrast compared to the standard FEM when analysing the same mesh. Note that the total degrees of freedom (DOFs) do not increase in this procedure, and the trial solution and its derivatives are continuous across inter-element boundaries. Besides, with controlling distortion factors, the interior nodes of a plate domain are derived from a set of regular nodes. Four practical examples with good results and small errors are considered in this study for showing excellent efficiency for this element. Last but not least, this element allows us to implement the procedure in an existing FEM computer code as well as can be used for nonlinear analysis in the near future.

Computer simulation of pantograph delay differential equations

Ritz method is widely used in variational theory to search for an approximate solution. This paper suggests a Ritz-like method for integral equations with an emphasis of pantograph delay equations. The unknown parameters involved in the trial solution can be determined by balancing the fundamental terms.

Exact analytical solutions of the fractional biological population model, fractional EW and modified EW equations

In this paper, exact analytical solutions of the biological population model, the EW and the modified EW equations with a conformable derivative operator have been examined by means of the trial solution algorithm and the complete discrimination system. Dark, bright and singular traveling wave solutions of the equations have been obtained by algorithm. Also, revealed singular periodic solutions have been listed. All solutions were verified by substituting them into their corresponding equation via Mathematica package program.

The Genetic Algorithm: Using Biology to Compute Liquid Crystal Director Configurations

The genetic algorithm is an optimization routine for finding the solution to a problem that requires a function to be minimized. It accomplishes this by creating a population of solutions and then producing “offspring” solutions from this population by combining two “parental” solutions in much the way that the DNA of biological parents is combined in the DNA of offspring. Strengths of the algorithm include that it is simple to implement, no trial solution is required, and the results are fairly accurate. Weaknesses include its slow computational speed and its tendency to find a local minimum that does not represent the global minimum of the function. By minimizing the elastic, surface, and electric free energies, the genetic algorithm is used to compute the liquid crystal director configuration for a wide range of situations, including one- and two-dimensional problems with various forms of boundary conditions, with and without an applied electric field. When appropriate, comparisons are made with the exact solutions. Ways to increase the performance of the algorithm as well as how to avoid various pitfalls are discussed.

A multichannel deconvolution method to retrieve source–time functions: application to the regional Lg wave

SUMMARY The retrieval of high-frequency seismic source–time functions (STFs) of similar earthquakes tends to be an ill-posed problem, causing unstable solutions. This is particularly true when waveforms are complex and band-limited, such as the regional phase Lg. We present a new procedure implementing the multichannel deconvolution (MCD) method to retrieve robust and objective STF solutions. The procedure relies on well-developed geophysical inverse theory to obtain stable STF solutions that jointly minimize the residual misfit, model roughness and data underfitting. MCD is formulated as a least-squares inverse problem with a Tikhonov regularization. The problem is solved using a convex optimization algorithm which rapidly converges to the global minimum while accommodating physical solution constraints including positivity, causality, finiteness and known seismic moments. We construct two L-shaped curves showing how the solution residual and roughness vary with trial solution durations. The optimal damping is chosen when the curves have acceptable levels while exhibiting no oscillations caused by solution instability. The optimal solution duration is chosen to avoid a rapidly decaying segment of the residual curve caused by parameter underfitting. We apply the MCD method to synthetic Lg data constructed by convolving a real Lg waveform with five pairs of simulated STFs. Four pairs consist of single triangular or parabolic pulses. The remaining pair consists of multipulse STFs with a complex, four-spike large STF. Without noise, the larger STFs in all single-pulse cases are well-recovered with Tikhonov regularization. Shape distortions are minor and duration errors are within 5 per cent. The multipulse case is a rare well-posed problem for which the true STFs are recovered without regularization. When a noise of ∼20 per cent is added to the synthetic data, the MCD method retrieves large single-pulse STFs with minor shape distortions and small duration errors (from 0 to 18 per cent). For the multipulse case, the retrieved large STF is overly smeared, losing details in the later portion. The small STF solutions for all cases are less resilient. Finally, we apply the MCD method to Lg data from two pairs of moderate earthquakes in central Asia. The problem becomes more ill-posed owing to lower signal-to-noise ratios (as low as 3) and non-identical Green's functions. A shape constraint of the small STF is needed. For the larger events with M5.7 and 5.8, the retrieved STFs are asymmetric, rising sharply and lasting about 2.0 and 2.5 s. We estimate radiated energies of 2.47 × 1013 and 2.53 × 1013 J and apparent stresses of 1.4 and 1.9 MPa, which are very reasonable. Our results are very consistent with those obtained in a previous study that used a very different, less objective ‘Landweber deconvolution’ method and a pre-fixed small STF duration. Novel improvements made by our new procedure include the application of a convex algorithm rather than a Newton-like method, a procedure for simultaneously optimizing regularization and solution duration parameters, a shape constraint for the smaller STF, and application to the complex Lg wave.

IMPROVED MACHINE LEARNING TECHNIQUE FOR SOLVING HAUSDORFF DERIVATIVE DIFFUSION EQUATIONS

This study aims at combining the machine learning technique with the Hausdorff derivative to solve one-dimensional Hausdorff derivative diffusion equations. In the proposed artificial neural network method, the multilayer feed-forward neural network is chosen and improved by using the Hausdorff derivative to the activation function of hidden layers. A trial solution is a combination of the boundary and initial condition terms and the network output, which can approximate the analytical solution. To transform the original Hausdorff derivative equation into a minimization problem, an error function is defined, where the coefficients are approximated by using the gradient descent algorithm in the back-propagation process. Two numerical examples are given to illustrate the accuracy and the robustness of the proposed method. The obtained results show that the improved machine learning technique is efficient in computing the Hausdorff derivative diffusion equations both from computational accuracy and stability.

Solution of Ruin Probability for Continuous Time Model Based on Block Trigonometric Exponential Neural Network

The ruin probability is used to determine the overall operating risk of an insurance company. Modeling risks through the characteristics of the historical data of an insurance business, such as premium income, dividends and reinvestments, can usually produce an integral differential equation that is satisfied by the ruin probability. However, the distribution function of the claim inter-arrival times is more complicated, which makes it difficult to find an analytical solution of the ruin probability. Therefore, based on the principles of artificial intelligence and machine learning, we propose a novel numerical method for solving the ruin probability equation. The initial asset u is used as the input vector and the ruin probability as the only output. A trigonometric exponential function is proposed as the projection mapping in the hidden layer, then a block trigonometric exponential neural network (BTENN) model with a symmetrical structure is established. Trial solution is set to meet the initial value condition, simultaneously, connection weights are optimized by solving a linear system using the extreme learning machine (ELM) algorithm. Three numerical experiments were carried out by Python. The results show that the BTENN model can obtain the approximate solution of the ruin probability under the classical risk model and the Erlang(2) risk model at any time point. Comparing with existing methods such as Legendre neural networks (LNN) and trigonometric neural networks (TNN), the proposed BTENN model has a higher stability and lower deviation, which proves that it is feasible and superior to use a BTENN model to estimate the ruin probability.

Strategy variability in numerosity comparison task: a study in young and older adults

We investigated strategies used by young and older adults in dot comparison tasks to further our understanding of mechanisms underlying numerosity discrimination and age-related differences therein. The participants were shown a series of two dot collections and asked to select the largest collection. Analyses of verbal protocols collected on each trial, solution times, and percentages of errors documented the strategy repertoire and strategy distribution in young and older adults. Based on visual features of dot collections, both young and older adults used a set of 9 strategies and selected strategies on a trial-by-trial basis. The findings also documented age-related differences (i.e., strategy preferences) and similarities (e.g., number of strategies used by individuals) in strategies and performance. Strategy variability found here has important implications for understanding numerosity comparison and contrasts with previous findings suggesting that participants use a single strategy when they compare dot collections.