非局部Allen-Cahn的数值模拟在实际应用中得到了广泛应用。然而,开发一种既高效又符合物理定律的数值方法仍是一个重大挑战。近年来,利用神经网络求解偏微分方程显示出了巨大的潜力。受这些研究的启发,我们提出在Evolutionary Deep Neur...非局部Allen-Cahn的数值模拟在实际应用中得到了广泛应用。然而,开发一种既高效又符合物理定律的数值方法仍是一个重大挑战。近年来,利用神经网络求解偏微分方程显示出了巨大的潜力。受这些研究的启发,我们提出在Evolutionary Deep Neural Network (EDNN)中引入一个辅助变量,以保持偏微分方程的基本物理特性。该方法确保离散数值格式具有无条件能量耗散特性,从而将问题框定为一个最小化任务。我们对非局部Allen-Cahn方程进行了数值模拟,验证了我们修正过的EDNN的准确性和效率。The numerical simulation of nonlocal Allen-Cahn equations has been widely applied in practical applications. However, developing an efficient numerical method that adheres to physical laws remains a significant challenge. Recently, the use of neural networks to solve partial differential equations has demonstrated great potential. Inspired by these studies, we propose incorporating an auxiliary variable into the Evolutionary Deep Neural Network (EDNN) framework to preserve the fundamental physical properties of partial differential equations. This approach ensures that the discrete numerical scheme possesses an unconditionally energy dissipation property, thereby framing the problem as a minimization task. To validate the accuracy and efficiency of our modified EDNN, we conducted numerical simulations of the nonlocal Allen-Cahn equation.展开更多
我们主要关注如下非局部Choquard方程解的存在性:−Δu=(∫Ω| u |2μ∗| x−y |μdy)| u |2μ∗−2u+λ(∫Ω| u |q| x−y |μdy)| u |q−2u+βulogu2in Ω这里Ω是ℝN中一个具有光滑边界的有界区域,λ,β>0为实参数,2q2μ∗,2μ∗=2N−μN−2(...我们主要关注如下非局部Choquard方程解的存在性:−Δu=(∫Ω| u |2μ∗| x−y |μdy)| u |2μ∗−2u+λ(∫Ω| u |q| x−y |μdy)| u |q−2u+βulogu2in Ω这里Ω是ℝN中一个具有光滑边界的有界区域,λ,β>0为实参数,2q2μ∗,2μ∗=2N−μN−2(N≥5)是Hardy-Littlewood-Sobolev不等式意义下的上临界指标。We are interested in the existence of the following nonlocal Choquard equation: −Δu=(∫Ω| u |2μ∗| x−y |μdy)| u |2μ∗−2u+λ(∫Ω| u |q| x−y |μdy)| u |q−2u+βulogu2in Ωwhere Ωis a bounded domain of ℝNwith smooth boundary, λ,β>0are real parameters, 2q2μ∗, 2μ∗=2N−μN−2(N≥5)is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.展开更多
文摘非局部Allen-Cahn的数值模拟在实际应用中得到了广泛应用。然而,开发一种既高效又符合物理定律的数值方法仍是一个重大挑战。近年来,利用神经网络求解偏微分方程显示出了巨大的潜力。受这些研究的启发,我们提出在Evolutionary Deep Neural Network (EDNN)中引入一个辅助变量,以保持偏微分方程的基本物理特性。该方法确保离散数值格式具有无条件能量耗散特性,从而将问题框定为一个最小化任务。我们对非局部Allen-Cahn方程进行了数值模拟,验证了我们修正过的EDNN的准确性和效率。The numerical simulation of nonlocal Allen-Cahn equations has been widely applied in practical applications. However, developing an efficient numerical method that adheres to physical laws remains a significant challenge. Recently, the use of neural networks to solve partial differential equations has demonstrated great potential. Inspired by these studies, we propose incorporating an auxiliary variable into the Evolutionary Deep Neural Network (EDNN) framework to preserve the fundamental physical properties of partial differential equations. This approach ensures that the discrete numerical scheme possesses an unconditionally energy dissipation property, thereby framing the problem as a minimization task. To validate the accuracy and efficiency of our modified EDNN, we conducted numerical simulations of the nonlocal Allen-Cahn equation.
文摘我们主要关注如下非局部Choquard方程解的存在性:−Δu=(∫Ω| u |2μ∗| x−y |μdy)| u |2μ∗−2u+λ(∫Ω| u |q| x−y |μdy)| u |q−2u+βulogu2in Ω这里Ω是ℝN中一个具有光滑边界的有界区域,λ,β>0为实参数,2q2μ∗,2μ∗=2N−μN−2(N≥5)是Hardy-Littlewood-Sobolev不等式意义下的上临界指标。We are interested in the existence of the following nonlocal Choquard equation: −Δu=(∫Ω| u |2μ∗| x−y |μdy)| u |2μ∗−2u+λ(∫Ω| u |q| x−y |μdy)| u |q−2u+βulogu2in Ωwhere Ωis a bounded domain of ℝNwith smooth boundary, λ,β>0are real parameters, 2q2μ∗, 2μ∗=2N−μN−2(N≥5)is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.
