Physics-informed neural networks(PINNs)have become an attractive machine learning framework for obtaining solutions to partial differential equations(PDEs).PINNs embed initial,boundary,and PDE constraints into the los...Physics-informed neural networks(PINNs)have become an attractive machine learning framework for obtaining solutions to partial differential equations(PDEs).PINNs embed initial,boundary,and PDE constraints into the loss function.The performance of PINNs is generally affected by both training and sampling.Specifically,training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs,and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished.However,a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category,namely,time-dependent PDEs,where temporal information plays a key role in the algorithms used.There is one method,called Causal PINN,that considers temporal causality at the training level but not special temporal utilization at the sampling level.Incorporating temporal knowledge into sampling remains to be studied.To fill this gap,we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality.By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain,we provide a practical solution by incorporating temporal information into sampling.Numerical experiments of several nonlinear time-dependent PDEs,including the Cahn–Hilliard,Korteweg–de Vries,Allen–Cahn and wave equations,show that our proposed sampling method can improve the performance.We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods,especially when points are limited.展开更多
近年来,在线学习由于其巨大的实际应用价值,已经得到人们广泛的研究.然而,在许多开放环境应用场景下,当前时刻数据可能会增加新的特征,而下一时刻只有部分原有特征得以继承.例如,在环境监测中,新的传感器部署会产生数据新特征;下一时刻...近年来,在线学习由于其巨大的实际应用价值,已经得到人们广泛的研究.然而,在许多开放环境应用场景下,当前时刻数据可能会增加新的特征,而下一时刻只有部分原有特征得以继承.例如,在环境监测中,新的传感器部署会产生数据新特征;下一时刻部分旧的传感器失效,部分原有特征被保留.这样的数据被称为特征继承性增减的流式数据.传统的在线学习算法大多建立在数据特征空间稳定不变的基础之上,无法直接处理此种情形.针对上述问题,提出了一种面向特征继承性增减的在线分类算法(online classification algorithm with feature inheritably increasing and decreasing,OFID)及其2种变体.当新特征出现时,通过结合在线被动主动方法与结构风险最小化原则分别更新原始特征与新增特征上的分类器;当旧特征消失时,对数据流使用Frequent-Directions算法进行补全,使得旧分类器得以继续更新迭代.从理论上证明了OFID系列算法的损失上界,同时通过大量的实验验证了所提算法的有效性.展开更多
基金Project supported by the Key National Natural Science Foundation of China(Grant No.62136005)the National Natural Science Foundation of China(Grant Nos.61922087,61906201,and 62006238)。
文摘Physics-informed neural networks(PINNs)have become an attractive machine learning framework for obtaining solutions to partial differential equations(PDEs).PINNs embed initial,boundary,and PDE constraints into the loss function.The performance of PINNs is generally affected by both training and sampling.Specifically,training methods focus on how to overcome the training difficulties caused by the special PDE residual loss of PINNs,and sampling methods are concerned with the location and distribution of the sampling points upon which evaluations of PDE residual loss are accomplished.However,a common problem among these original PINNs is that they omit special temporal information utilization during the training or sampling stages when dealing with an important PDE category,namely,time-dependent PDEs,where temporal information plays a key role in the algorithms used.There is one method,called Causal PINN,that considers temporal causality at the training level but not special temporal utilization at the sampling level.Incorporating temporal knowledge into sampling remains to be studied.To fill this gap,we propose a novel temporal causality-based adaptive sampling method that dynamically determines the sampling ratio according to both PDE residual and temporal causality.By designing a sampling ratio determined by both residual loss and temporal causality to control the number and location of sampled points in each temporal sub-domain,we provide a practical solution by incorporating temporal information into sampling.Numerical experiments of several nonlinear time-dependent PDEs,including the Cahn–Hilliard,Korteweg–de Vries,Allen–Cahn and wave equations,show that our proposed sampling method can improve the performance.We demonstrate that using such a relatively simple sampling method can improve prediction performance by up to two orders of magnitude compared with the results from other methods,especially when points are limited.
文摘近年来,在线学习由于其巨大的实际应用价值,已经得到人们广泛的研究.然而,在许多开放环境应用场景下,当前时刻数据可能会增加新的特征,而下一时刻只有部分原有特征得以继承.例如,在环境监测中,新的传感器部署会产生数据新特征;下一时刻部分旧的传感器失效,部分原有特征被保留.这样的数据被称为特征继承性增减的流式数据.传统的在线学习算法大多建立在数据特征空间稳定不变的基础之上,无法直接处理此种情形.针对上述问题,提出了一种面向特征继承性增减的在线分类算法(online classification algorithm with feature inheritably increasing and decreasing,OFID)及其2种变体.当新特征出现时,通过结合在线被动主动方法与结构风险最小化原则分别更新原始特征与新增特征上的分类器;当旧特征消失时,对数据流使用Frequent-Directions算法进行补全,使得旧分类器得以继续更新迭代.从理论上证明了OFID系列算法的损失上界,同时通过大量的实验验证了所提算法的有效性.
