The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corr...
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The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel deep learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.
Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation o...
Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation of empirical data. One advantage of the neural network methods for PDEs lies in its automatic differentiation (AD), which necessitates only the sample points themselves, unlike traditional finite difference (FD) approximations that require nearby local points to compute derivatives. In this paper, we quantitatively demonstrate the advantage of AD in training neural networks. The concept of truncated entropy is introduced to characterize the training property. Specifically, through comprehensive experimental and theoretical analyses conducted on random feature models and two-layer neural networks, we discover that the defined truncated entropy serves as a reliable metric for quantifying the residual loss of random feature models and the training speed of neural networks for both AD and FD methods. Our experimental and theoretical analyses demonstrate that, from a training perspective, AD outperforms FD in solving PDEs.
In this paper, we investigate the training process of generative networks that use a type of probability density distance named particle-based distance as the objective function, e.g. MMD GAN, Cramér GAN, EIEG GA...
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Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machinelearning. Training neural networks for sharp interface problems is particularly challenging due to c...
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The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corr...
详细信息
Neural network-based approaches have recently shown significant promise in solving partial differential equations (PDEs) in science and engineering, especially in scenarios featuring complex domains or incorporation o...
Recently, multimodal electroencephalogram (EEG) learning has shown great promise in disease detection. At the same time, ensuring privacy in clinical studies has become increasingly crucial due to legal and ethical co...
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Recently, multimodal electroencephalogram (EEG) learning has shown great promise in disease detection. At the same time, ensuring privacy in clinical studies has become increasingly crucial due to legal and ethical co...
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Recently, multimodal electroencephalogram (EEG) learning has shown great promise in disease detection. At the same time, ensuring privacy in clinical studies has become increasingly crucial due to legal and ethical co...
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The task of lane detection involves identifying the boundaries of driving areas in real-time. Recognizing lanes with variable and complex geometric structures remains a challenge. In this paper, we explore a novel and...
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