The paper is to introduce a computational methodology that is based on ordinary differential equations(ODE)solver for the structural systems adopted by a super tall building in its preliminary design stage so as to fa...The paper is to introduce a computational methodology that is based on ordinary differential equations(ODE)solver for the structural systems adopted by a super tall building in its preliminary design stage so as to facilitate the designers to adjust the dynamic properties of the adopted structural system.The construction of the study is composed by following aspects.The first aspect is the modelling of a structural system.As a typical example,a mega frame-core-tube structural system adopted by some famous super tall buildings such as Taipei 101 building,Shanghai World financial center,is employed to demonstrate the modelling of a computational model.The second aspect is the establishment of motion equations constituted by a group of ordinary differential equations for the analyses of free vibration and resonant response.The solutions of the motion equations(that constitutes the third aspect)resorted to ODE-solver technique.Finally,some valuable conclusions are summarized.展开更多
Diffusion probabilistic models(DPMs)have achieved impressive success in high-resolution image synthesis,especially in recent large-scale text-to-image generation applications.An essential technique for improving the s...Diffusion probabilistic models(DPMs)have achieved impressive success in high-resolution image synthesis,especially in recent large-scale text-to-image generation applications.An essential technique for improving the sample quality of DPMs is guided sampling,which usually needs a large guidance scale to obtain the best sample quality.The commonly-used fast sampler for guided sampling is denoising diffusion implicit models(DDIM),a first-order diffusion ordinary differential equation(ODE)solver that generally needs 100 to 250 steps for high-quality samples.Although recent works propose dedicated high-order solvers and achieve a further speedup for sampling without guidance,their effectiveness for guided sampling has not been well-tested before.In this work,we demonstrate that previous high-order fast samplers suffer from instability issues,and they even become slower than DDIM when the guidance scale grows larger.To further speed up guided sampling,we propose DPM-Solver++,a high-order solver for the guided sampling of DPMs.DPM-Solver++solves the diffusion ODE with the data prediction model and adopts thresholding methods to keep the solution matches training data distribution.We further propose a multistep variant of DPM-Solver++to address the instability issue by reducing the effective step size.Experiments show that DPM-Solver++can generate high-quality samples within only 15 to 20 steps for guided sampling by pixel-space and latent-space DPMs.展开更多
A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differe...A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differential equations(PDEs)combined with the automatic differentiation(AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation(i.e., the Laplace equation).展开更多
基金Acknowledgment The research work was financially supported both by the Natural Science Foundation of China(51178164)and the Priority Discipline Foundation of Henan Province(507909).
文摘The paper is to introduce a computational methodology that is based on ordinary differential equations(ODE)solver for the structural systems adopted by a super tall building in its preliminary design stage so as to facilitate the designers to adjust the dynamic properties of the adopted structural system.The construction of the study is composed by following aspects.The first aspect is the modelling of a structural system.As a typical example,a mega frame-core-tube structural system adopted by some famous super tall buildings such as Taipei 101 building,Shanghai World financial center,is employed to demonstrate the modelling of a computational model.The second aspect is the establishment of motion equations constituted by a group of ordinary differential equations for the analyses of free vibration and resonant response.The solutions of the motion equations(that constitutes the third aspect)resorted to ODE-solver technique.Finally,some valuable conclusions are summarized.
文摘Diffusion probabilistic models(DPMs)have achieved impressive success in high-resolution image synthesis,especially in recent large-scale text-to-image generation applications.An essential technique for improving the sample quality of DPMs is guided sampling,which usually needs a large guidance scale to obtain the best sample quality.The commonly-used fast sampler for guided sampling is denoising diffusion implicit models(DDIM),a first-order diffusion ordinary differential equation(ODE)solver that generally needs 100 to 250 steps for high-quality samples.Although recent works propose dedicated high-order solvers and achieve a further speedup for sampling without guidance,their effectiveness for guided sampling has not been well-tested before.In this work,we demonstrate that previous high-order fast samplers suffer from instability issues,and they even become slower than DDIM when the guidance scale grows larger.To further speed up guided sampling,we propose DPM-Solver++,a high-order solver for the guided sampling of DPMs.DPM-Solver++solves the diffusion ODE with the data prediction model and adopts thresholding methods to keep the solution matches training data distribution.We further propose a multistep variant of DPM-Solver++to address the instability issue by reducing the effective step size.Experiments show that DPM-Solver++can generate high-quality samples within only 15 to 20 steps for guided sampling by pixel-space and latent-space DPMs.
基金Project supported by the National Natural Science Foundation of China(No.11521091)
文摘A neural network(NN) is a powerful tool for approximating bounded continuous functions in machine learning. The NN provides a framework for numerically solving ordinary differential equations(ODEs) and partial differential equations(PDEs)combined with the automatic differentiation(AD) technique. In this work, we explore the use of NN for the function approximation and propose a universal solver for ODEs and PDEs. The solver is tested for initial value problems and boundary value problems of ODEs, and the results exhibit high accuracy for not only the unknown functions but also their derivatives. The same strategy can be used to construct a PDE solver based on collocation points instead of a mesh, which is tested with the Burgers equation and the heat equation(i.e., the Laplace equation).
