In this paper,physics-informed liquid networks(PILNs)are proposed based on liquid time-constant networks(LTC)for solving nonlinear partial differential equations(PDEs).In this approach,the network state is controlled ...In this paper,physics-informed liquid networks(PILNs)are proposed based on liquid time-constant networks(LTC)for solving nonlinear partial differential equations(PDEs).In this approach,the network state is controlled via ordinary differential equations(ODEs).The significant advantage is that neurons controlled by ODEs are more expressive compared to simple activation functions.In addition,the PILNs use difference schemes instead of automatic differentiation to construct the residuals of PDEs,which avoid information loss in the neighborhood of sampling points.As this method draws on both the traveling wave method and physics-informed neural networks(PINNs),it has a better physical interpretation.Finally,the KdV equation and the nonlinear Schr¨odinger equation are solved to test the generalization ability of the PILNs.To the best of the authors’knowledge,this is the first deep learning method that uses ODEs to simulate the numerical solutions of PDEs.展开更多
Under the frame of the （2＋1）-dimensional zero curvature equation and Tu model, （2＋1）-dimensional Tu hierarchy is obtained. Again by employing a subalgebra of the loop algebra ↑-A2 the integrable coupling system...Under the frame of the （2＋1）-dimensional zero curvature equation and Tu model, （2＋1）-dimensional Tu hierarchy is obtained. Again by employing a subalgebra of the loop algebra ↑-A2 the integrable coupling system of the above hierarchy is presented. Finally, A multi-component integrable hierarchy is obtained by employing a multi-component loop algebra ↑-GM.展开更多
This paper considers the Cohen-Grossberg BAM neural networks(CG-BAMNNs) on time scale, which can unify and generalize the continuous and discrete systems. First, the criteria for the existence and uniqueness of the eq...This paper considers the Cohen-Grossberg BAM neural networks(CG-BAMNNs) on time scale, which can unify and generalize the continuous and discrete systems. First, the criteria for the existence and uniqueness of the equilibrium of CG-BAMNNs are derived on time scale. Then based on that, the authors give the criteria for the stability and estimation of equilibrium of the CG-BAMNNs on time scale. The method proposed in this paper unifies and generalizes the continuous and discrete CGBAMNNs systems, and is applicable to some other neural network systems on time scale with practical meaning. The effectiveness of the proposed criteria for delayed CG-BAMNNs is demonstrated by numerical simulation.展开更多
基金supported by the National Natural Science Foundation of China under Grant Nos.11975143 and 12105161.
文摘In this paper,physics-informed liquid networks(PILNs)are proposed based on liquid time-constant networks(LTC)for solving nonlinear partial differential equations(PDEs).In this approach,the network state is controlled via ordinary differential equations(ODEs).The significant advantage is that neurons controlled by ODEs are more expressive compared to simple activation functions.In addition,the PILNs use difference schemes instead of automatic differentiation to construct the residuals of PDEs,which avoid information loss in the neighborhood of sampling points.As this method draws on both the traveling wave method and physics-informed neural networks(PINNs),it has a better physical interpretation.Finally,the KdV equation and the nonlinear Schr¨odinger equation are solved to test the generalization ability of the PILNs.To the best of the authors’knowledge,this is the first deep learning method that uses ODEs to simulate the numerical solutions of PDEs.
文摘Under the frame of the （2＋1）-dimensional zero curvature equation and Tu model, （2＋1）-dimensional Tu hierarchy is obtained. Again by employing a subalgebra of the loop algebra ↑-A2 the integrable coupling system of the above hierarchy is presented. Finally, A multi-component integrable hierarchy is obtained by employing a multi-component loop algebra ↑-GM.
基金supported by the National Natural Science Foundation of China under Grant Nos.12105161,11975143the Natural Science Foundation of Shandong Province under Grant No.ZR2019QD018。
文摘This paper considers the Cohen-Grossberg BAM neural networks(CG-BAMNNs) on time scale, which can unify and generalize the continuous and discrete systems. First, the criteria for the existence and uniqueness of the equilibrium of CG-BAMNNs are derived on time scale. Then based on that, the authors give the criteria for the stability and estimation of equilibrium of the CG-BAMNNs on time scale. The method proposed in this paper unifies and generalizes the continuous and discrete CGBAMNNs systems, and is applicable to some other neural network systems on time scale with practical meaning. The effectiveness of the proposed criteria for delayed CG-BAMNNs is demonstrated by numerical simulation.
