In this paper,we propose a symmetric difference data enhancement physics-informed neural network(SDE-PINN)to study soliton solutions for discrete nonlinear lattice equations(NLEs).By considering known and unknown symm...In this paper,we propose a symmetric difference data enhancement physics-informed neural network(SDE-PINN)to study soliton solutions for discrete nonlinear lattice equations(NLEs).By considering known and unknown symmetric points,numerical simulations are conducted to one-soliton and two-soliton solutions of a discrete KdV equation,as well as a one-soliton solution of a discrete Toda lattice equation.Compared with the existing discrete deep learning approach,the numerical results reveal that within the specified spatiotemporal domain,the prediction accuracy by SDE-PINN is excellent regardless of the interior or extrapolation prediction,with a significant reduction in training time.The proposed data enhancement technique and symmetric structure development provides a new perspective for the deep learning approach to solve discrete NLEs.The newly proposed SDE-PINN can also be applied to solve continuous nonlinear equations and other discrete NLEs numerically.展开更多
Laser spot thermography is a novel technique for the detection of surface cracks with a laser to heat sample locally and with an IR camera to record the surface temperature distribution. Common methods to characterize...Laser spot thermography is a novel technique for the detection of surface cracks with a laser to heat sample locally and with an IR camera to record the surface temperature distribution. Common methods to characterize cracks are only suitable for the situation that the laser scanning path is vertical to the crack. But due to the randomness of cracks,when the scanning path is parallel to the crack,surface cracks cannot be detected by these methods. To tackle this problem,a method is presented which is suitable for the situation that the scanning path is parallel to crack. The main idea is to evaluate the crack-caused asymmetries of the surface temperature distribution. The effect of temperature gradient and the maximum scanning interval are analyzed by a 2D simulation. A new crack imaging technique is presented that is based on delayed temperature difference at symmetric points to characterize the crack in the thermal image. Compared well with those obtained by the spatial first derivative method,experimental results are shown to efficiently prove this method.展开更多
For a univariate function given by its Taylor series expansion, a continued fraction expansion can be obtained with the Viscovatov＇s algorithm, as the limiting value of a Thiele interpolating continued fraction or by...For a univariate function given by its Taylor series expansion, a continued fraction expansion can be obtained with the Viscovatov＇s algorithm, as the limiting value of a Thiele interpolating continued fraction or by means of the determinantal formulas for inverse and reciprocal differences with coincident data points. In this paper, both Viscovatov-like algorithms and Taylor-like expansions are incorporated to yield bivariate blending continued expansions which are computed as the limiting value of bivariate blending rational interpolants, which are constructed based on symmetric blending differences. Numerical examples are given to show the effectiveness of our methods.展开更多
The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are establi...The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are established with n = 6m+l, 6m+5 and 6m+3, m is an odd positive integer and m is an even positive integer 9|6m + 3. It is seen that the number sets for constructing pandiagonal snowflake magic squares can be extended to the matrices with symmetric partial difference in each direction for orders 6m + 1 , 6m + 5; to the trisection matrices with symmetric partial difference in each direction for order 6m + 3.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.12071042)the Beijing Natural Science Foundation(Grant No.1202004)Promoting the Development of University Classification-Student Innovation and Entrepreneurship Training Programme(Grant No.5112410857)。
文摘In this paper,we propose a symmetric difference data enhancement physics-informed neural network(SDE-PINN)to study soliton solutions for discrete nonlinear lattice equations(NLEs).By considering known and unknown symmetric points,numerical simulations are conducted to one-soliton and two-soliton solutions of a discrete KdV equation,as well as a one-soliton solution of a discrete Toda lattice equation.Compared with the existing discrete deep learning approach,the numerical results reveal that within the specified spatiotemporal domain,the prediction accuracy by SDE-PINN is excellent regardless of the interior or extrapolation prediction,with a significant reduction in training time.The proposed data enhancement technique and symmetric structure development provides a new perspective for the deep learning approach to solve discrete NLEs.The newly proposed SDE-PINN can also be applied to solve continuous nonlinear equations and other discrete NLEs numerically.
基金supported by the National Key Scientific Instrument and Equipment Development Projects,China(Grant No.2013YQ470767)。
文摘Laser spot thermography is a novel technique for the detection of surface cracks with a laser to heat sample locally and with an IR camera to record the surface temperature distribution. Common methods to characterize cracks are only suitable for the situation that the laser scanning path is vertical to the crack. But due to the randomness of cracks,when the scanning path is parallel to the crack,surface cracks cannot be detected by these methods. To tackle this problem,a method is presented which is suitable for the situation that the scanning path is parallel to crack. The main idea is to evaluate the crack-caused asymmetries of the surface temperature distribution. The effect of temperature gradient and the maximum scanning interval are analyzed by a 2D simulation. A new crack imaging technique is presented that is based on delayed temperature difference at symmetric points to characterize the crack in the thermal image. Compared well with those obtained by the spatial first derivative method,experimental results are shown to efficiently prove this method.
基金The NNSF(10171026 and 60473114)of Chinathe Research Funds(2005TD03) for Young Innovation Group,Education Department of Anhui Province.
文摘For a univariate function given by its Taylor series expansion, a continued fraction expansion can be obtained with the Viscovatov＇s algorithm, as the limiting value of a Thiele interpolating continued fraction or by means of the determinantal formulas for inverse and reciprocal differences with coincident data points. In this paper, both Viscovatov-like algorithms and Taylor-like expansions are incorporated to yield bivariate blending continued expansions which are computed as the limiting value of bivariate blending rational interpolants, which are constructed based on symmetric blending differences. Numerical examples are given to show the effectiveness of our methods.
文摘The constructional methods of pandiagonal snowflake magic squares of orders 4m are established in paper [3]. In this paper, the constructional methods of pandiagonal snowflake magic squares of odd orders n are established with n = 6m+l, 6m+5 and 6m+3, m is an odd positive integer and m is an even positive integer 9|6m + 3. It is seen that the number sets for constructing pandiagonal snowflake magic squares can be extended to the matrices with symmetric partial difference in each direction for orders 6m + 1 , 6m + 5; to the trisection matrices with symmetric partial difference in each direction for order 6m + 3.
