Consider the following system of coupled Korteweg-de Vries equations, <img src="Edit_81ea1215-e696-403f-9d6c-1449e107359f.bmp" alt="" /><span style="white-space:nowrap;">where...Consider the following system of coupled Korteweg-de Vries equations, <img src="Edit_81ea1215-e696-403f-9d6c-1449e107359f.bmp" alt="" /><span style="white-space:nowrap;">where<em> u</em>, <em>v </em><span style="white-space:nowrap;">⊆</span> <em>W</em><sup>2,2</sup>, 2≤<em>N</em>≤7 and <em>λ</em><sub><em>i</em></sub>,<em>β</em> > 0, <em>β</em> </span>denotes a real coupling parameter. Firstly, we prove the existence of the solutions of a coupled system of Korteweg-de Vries equations using variation approach and minimization techniques on Nehari manifold. Then, we show the multiplicity of the equations by a bifurcation theory which is rare for studying higher order equations.展开更多
In this work,we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck(TFP)equations.It is well known that solutions of such equations are probability densit...In this work,we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck(TFP)equations.It is well known that solutions of such equations are probability density functions,and thus our approach relies on modelling the target solutions with the temporal normalizing flows.The temporal normalizing flow is then trained based on the TFP loss function,without requiring any labeled data.Being a machine learning scheme,the proposed approach is mesh-free and can be easily applied to high dimensional problems.We present a variety of test problems to show the effectiveness of the learning approach.展开更多
文摘Consider the following system of coupled Korteweg-de Vries equations, <img src="Edit_81ea1215-e696-403f-9d6c-1449e107359f.bmp" alt="" /><span style="white-space:nowrap;">where<em> u</em>, <em>v </em><span style="white-space:nowrap;">⊆</span> <em>W</em><sup>2,2</sup>, 2≤<em>N</em>≤7 and <em>λ</em><sub><em>i</em></sub>,<em>β</em> > 0, <em>β</em> </span>denotes a real coupling parameter. Firstly, we prove the existence of the solutions of a coupled system of Korteweg-de Vries equations using variation approach and minimization techniques on Nehari manifold. Then, we show the multiplicity of the equations by a bifurcation theory which is rare for studying higher order equations.
基金supported by the NSF of China(under grant numbers 12288201 and 11731006)the National Key R&D Program of China(2020YFA0712000)the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA25010404).
文摘In this work,we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck(TFP)equations.It is well known that solutions of such equations are probability density functions,and thus our approach relies on modelling the target solutions with the temporal normalizing flows.The temporal normalizing flow is then trained based on the TFP loss function,without requiring any labeled data.Being a machine learning scheme,the proposed approach is mesh-free and can be easily applied to high dimensional problems.We present a variety of test problems to show the effectiveness of the learning approach.
