In this paper,a deep collocation method(DCM)for thin plate bending problems is proposed.This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning.Besides,the proposed...In this paper,a deep collocation method(DCM)for thin plate bending problems is proposed.This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning.Besides,the proposed DCM is based on a feedforward deep neural network(DNN)and differs from most previous applications of deep learning for mechanical problems.First,batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries.A loss function is built with the aim that the governing partial differential equations(PDEs)of Kirchhoff plate bending problems,and the boundary/initial conditions are minimised at those collocation points.A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters.In Kirchhoff plate bending problems,the C^1 continuity requirement poses significant difficulties in traditional mesh-based methods.This can be solved by the proposed DCM,which uses a deep neural network to approximate the continuous transversal deflection,and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.展开更多
The integration of distributed generations(solar power,wind power),energy storage devices,and electric vehicles,causes unpredictable disturbances in power grids.It has become a top priority to coordinate the distribut...The integration of distributed generations(solar power,wind power),energy storage devices,and electric vehicles,causes unpredictable disturbances in power grids.It has become a top priority to coordinate the distributed generations,loads,and energy storages in order to better facilitate the utilization of new energy.Therefore,a novel algorithm based on deep reinforcement learning,namely the deep PDWoLF-PHC(policy dynamics based win or learn fast-policy hill climbing)network(DPDPN),is proposed to allocate power order among the various generators.The proposed algorithm combines the decision mechanism of reinforcement learning with the prediction mechanism of a deep neural network to obtain the optimal coordinated control for the source-grid-load.Consequently it solves the problem brought by stochastic disturbances and improves the utilization rate of new energy.Simulations are conducted with the case of the improved IEEE two-area and a case in the Guangdong power grid.Results show that the adaptability and control performance of the power system are improved using the proposed algorithm as compared with using other existing strategies.展开更多
The modeling of dynamic stall aerodynamics is essential to stall flutter, due to the flow separation in a large-amplitude pitching oscillation process. A newly neural network based Reduced Order Model(ROM) framework f...The modeling of dynamic stall aerodynamics is essential to stall flutter, due to the flow separation in a large-amplitude pitching oscillation process. A newly neural network based Reduced Order Model(ROM) framework for predicting the aerodynamic forces of an airfoil undergoing large-amplitude pitching oscillation at various velocities is presented in this work. First, the dynamic stall aerodynamics is calculated by solving RANS equations and the transitional SST-γ model. Afterwards, the stall flutter bifurcation behavior is calculated by the above CFD solver coupled with structural dynamic equation. The critical flutter speed and limit-cycle oscillation amplitudes are consistent with those obtained by experiments. A newly multi-layer Gated Recurrent Unit(GRU) neural network based ROM is constructed to accelerate the calculation of aerodynamic forces. The training and validation process are carried out upon the unsteady aerodynamic data obtained by the proposed CFD method. The well-trained ROM is then coupled with the structure equation at a specific velocity, the Limit-Cycle Oscillation(LCO) of stall flutter under this flow condition is predicted precisely and more quickly. In order to predict both the critical flutter velocity and LCO amplitudes after bifurcation at different velocities, a new ROM with GRU neural network considering the variation of flow velocities is developed. The stall flutter results predicted by ROM agree well with the CFD ones at different velocities. Finally, a brief sensitivity analysis of two structural parameters of ROM is carried out. It infers the potential of the presented modeling method to depict the nonlinearity of dynamic stall and stall flutter phenomenon.展开更多
At present, deep learning based methods are being employed to resolvethe computational challenges of high-dimensional partial differential equations(PDEs). But the computation of the high order derivatives of neural n...At present, deep learning based methods are being employed to resolvethe computational challenges of high-dimensional partial differential equations(PDEs). But the computation of the high order derivatives of neural networks iscostly, and high order derivatives lack robustness for training purposes. We proposea novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variablesto rewrite the PDEs into a system of low order differential equations as what is donein the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neuralnetwork. By taking the residual of the system as a loss function, we can optimizethe network parameters to approximate the solution. The whole process relies onlow order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularlywell-suited for high-dimensional PDEs with high order derivatives.展开更多
The existing recognition algorithms of space-time block code(STBC)for multi-antenna(MA)orthogonal frequencydivision multiplexing(OFDM)systems use feature extraction and hypothesis testing to identify the signal types ...The existing recognition algorithms of space-time block code(STBC)for multi-antenna(MA)orthogonal frequencydivision multiplexing(OFDM)systems use feature extraction and hypothesis testing to identify the signal types in a complex communication environment.However,owing to the restrictions on the prior information and channel conditions,these existing algorithms cannot perform well under strong interference and noncooperative communication conditions.To overcome these defects,this study introduces deep learning into the STBCOFDM signal recognition field and proposes a recognition method based on the fourth-order lag moment spectrum(FOLMS)and attention-guided multi-scale dilated convolution network(AMDCNet).The fourth-order lag moment vectors of the received signals are calculated,and vectors are stitched to form two-dimensional FOLMS,which is used as the input of the deep learning-based model.Then,the multi-scale dilated convolution is used to extract the details of images at different scales,and a convolutional block attention module(CBAM)is introduced to construct the attention-guided multi-scale dilated convolution module(AMDCM)to make the network be more focused on the target area and obtian the multi-scale guided features.Finally,the concatenate fusion,residual block and fully-connected layers are applied to acquire the STBC-OFDM signal types.Simulation experiments show that the average recognition probability of the proposed method at−12 dB is higher than 98%.Compared with the existing algorithms,the recognition performance of the proposed method is significantly improved and has good adaptability to environments with strong disturbances.In addition,the proposed deep learning-based model can directly identify the pre-processed FOLMS samples without a priori information on channel and noise,which is more suitable for non-cooperative communication systems than the existing algorithms.展开更多
Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularl...Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularly deep learning(DL),applied and relevant to computational mechanics(solid,fluids,finite-element technology)are reviewed in detail.Both hybrid and pure machine learning(ML)methods are discussed.Hybrid methods combine traditional PDE discretizations with ML methods either(1)to help model complex nonlinear constitutive relations,(2)to nonlinearly reduce the model order for efficient simulation(turbulence),or(3)to accelerate the simulation by predicting certain components in the traditional integration methods.Here,methods(1)and(2)relied on Long-Short-Term Memory(LSTM)architecture,with method(3)relying on convolutional neural networks.Pure ML methods to solve(nonlinear)PDEs are represented by Physics-Informed Neural network(PINN)methods,which could be combined with attention mechanism to address discontinuous solutions.Both LSTM and attention architectures,together with modern and generalized classic optimizers to include stochasticity for DL networks,are extensively reviewed.Kernel machines,including Gaussian processes,are provided to sufficient depth for more advanced works such as shallow networks with infinite width.Not only addressing experts,readers are assumed familiar with computational mechanics,but not with DL,whose concepts and applications are built up from the basics,aiming at bringing first-time learners quickly to the forefront of research.History and limitations of AI are recounted and discussed,with particular attention at pointing out misstatements or misconceptions of the classics,even in well-known references.Positioning and pointing control of a large-deformable beam is given as an example.展开更多
We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equa...We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.展开更多
Based on the fifth-order Stokes regular wave theory, a simplified model for extreme-wave kinematics in deep sea was developed. In this model, from the wave records the average of two neighboring wave periods for the e...Based on the fifth-order Stokes regular wave theory, a simplified model for extreme-wave kinematics in deep sea was developed. In this model, from the wave records the average of two neighboring wave periods for the extreme crest or trough was defined as the period of the Stokes wave by the up and down zero-crossing methods. Then the input wave amplitude was deduced by substituting the wave period and extreme crest or trough into the expression for the fifth-order Stokes wave elevation. Thus the corresponding formula for the wave velocity can be used to describe kinematics beneath the extreme wave. By comparison with the published numerical models and experimental data, the proposed model is validated to be able to calculate the extreme wave velocity rather easily and accurately.展开更多
This paper proposes a high order deep domain decomposition method(HOrderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HOrderDNN)with domain decomposition method(DD...This paper proposes a high order deep domain decomposition method(HOrderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HOrderDNN)with domain decomposition method(DDM).The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM,and apply HOrderDNNs to solve the high-frequency problem on each sub-domain.Besides,we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains,on the interface and the boundary during the optimization process.The performance of HOrderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface problems.The results indicate that:HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequency problems.In detail,HOrderDeepDDMs(p>1)could capture the high-frequency information very well.When compared to the deep domain decomposition method(DeepDDM),HOrderDeepDDMs(p>1)converge faster and achieve much smaller relative errors with the same number of trainable parameters.For example,when solving the high-frequency interface elliptic problems in Section 3.3.1,the minimum relative errors obtained by HOrderDeepDDMs(p=9)are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same,as shown in Fig.4.展开更多
FTT experiments with water as a hydrogen source and three types of possible carbon sources in the subsurface(diiron nonacarbonyl,siderite and formic acid,representing CO,CO_(2)and a simple organic acid,respectively)we...FTT experiments with water as a hydrogen source and three types of possible carbon sources in the subsurface(diiron nonacarbonyl,siderite and formic acid,representing CO,CO_(2)and a simple organic acid,respectively)were carried out in this study.Our experimental results showed that n-alkanes with the highest carbon number of C_(33)were produced when CO was used as a carbon source(series A);a variety of polycyclic aromatic hydrocarbons(PAHs)were detected in series B with CO_(2)as a carbon source;gaseous hydrocarbons were also detected with formic acid added(series C).The different products in the three series showed that there were different hydrocarbon generation mechanisms and reaction processes with different carbon sources.The generation of long-chain n-alkanes in series A provided experimental support for the formation of abiogenic petroleum underground,which was of significance to early membranes on the Earth.PAHs in series B provide experimental support for the possibility of an abiotic source of reduced carbon on other planets.The carbon isotopes of gaseous hydrocarbons produced by CO exhibited a partial reversed order(δ^(13)C_(1)<δ^(13)C_(2)>δ^(13)C_(3)>δ^(13)C_(4)>δ^(13)C_(5)),while the gaseous hydrocarbons produced by CO_(2)and HCOOH showed a positive order(δ^(13)C_(1)<δ^(13)C_(2)<δ^(13)C_(3)<δ^(13)C_(4)<δ^(13)C_(5)).Based on these,the alkylene mechanism and the carbonyl insertion mechanism were used to reasonably explain these characteristics.展开更多
This paper proposes a high order deep neural network(HOrderDNN)for solving high frequency partial differential equations(PDEs),which incorporates the idea of“high order”from finite element methods(FEMs)into commonly...This paper proposes a high order deep neural network(HOrderDNN)for solving high frequency partial differential equations(PDEs),which incorporates the idea of“high order”from finite element methods(FEMs)into commonly-used deep neural networks(DNNs)to obtain greater approximation ability.The main idea of HOrderDNN is introducing a nonlinear transformation layer between the input layer and the first hidden layer to form a high order polynomial space with the degree not exceeding p,followed by a normal DNN.The order p can be guided by the regularity of solutions of PDEs.The performance of HOrderDNNis evaluated on high frequency function fitting problems and high frequency Poisson and Helmholtz equations.The results demonstrate that:HOrderDNNs(p>1)can efficiently capture the high frequency information in target functions;and when compared to physics-informed neural network(PINN),HOrderDNNs(p>1)converge faster and achieve much smaller relative errors with same number of trainable parameters.In particular,when solving the high frequency Helmholtz equation in Section 3.5,the relative error of PINN stays around 1 with its depth and width increase,while the relative error can be reduced to around 0.02 as p increases(see Table 5).展开更多
文摘In this paper,a deep collocation method(DCM)for thin plate bending problems is proposed.This method takes advantage of computational graphs and backpropagation algorithms involved in deep learning.Besides,the proposed DCM is based on a feedforward deep neural network(DNN)and differs from most previous applications of deep learning for mechanical problems.First,batches of randomly distributed collocation points are initially generated inside the domain and along the boundaries.A loss function is built with the aim that the governing partial differential equations(PDEs)of Kirchhoff plate bending problems,and the boundary/initial conditions are minimised at those collocation points.A combination of optimizers is adopted in the backpropagation process to minimize the loss function so as to obtain the optimal hyperparameters.In Kirchhoff plate bending problems,the C^1 continuity requirement poses significant difficulties in traditional mesh-based methods.This can be solved by the proposed DCM,which uses a deep neural network to approximate the continuous transversal deflection,and is proved to be suitable to the bending analysis of Kirchhoff plate of various geometries.
基金This work was supported in part by the National Natural Science Foundation of China under Grant No.51707102.
文摘The integration of distributed generations(solar power,wind power),energy storage devices,and electric vehicles,causes unpredictable disturbances in power grids.It has become a top priority to coordinate the distributed generations,loads,and energy storages in order to better facilitate the utilization of new energy.Therefore,a novel algorithm based on deep reinforcement learning,namely the deep PDWoLF-PHC(policy dynamics based win or learn fast-policy hill climbing)network(DPDPN),is proposed to allocate power order among the various generators.The proposed algorithm combines the decision mechanism of reinforcement learning with the prediction mechanism of a deep neural network to obtain the optimal coordinated control for the source-grid-load.Consequently it solves the problem brought by stochastic disturbances and improves the utilization rate of new energy.Simulations are conducted with the case of the improved IEEE two-area and a case in the Guangdong power grid.Results show that the adaptability and control performance of the power system are improved using the proposed algorithm as compared with using other existing strategies.
基金supported by the National Natural Science Foundation of China(No.11672018).
文摘The modeling of dynamic stall aerodynamics is essential to stall flutter, due to the flow separation in a large-amplitude pitching oscillation process. A newly neural network based Reduced Order Model(ROM) framework for predicting the aerodynamic forces of an airfoil undergoing large-amplitude pitching oscillation at various velocities is presented in this work. First, the dynamic stall aerodynamics is calculated by solving RANS equations and the transitional SST-γ model. Afterwards, the stall flutter bifurcation behavior is calculated by the above CFD solver coupled with structural dynamic equation. The critical flutter speed and limit-cycle oscillation amplitudes are consistent with those obtained by experiments. A newly multi-layer Gated Recurrent Unit(GRU) neural network based ROM is constructed to accelerate the calculation of aerodynamic forces. The training and validation process are carried out upon the unsteady aerodynamic data obtained by the proposed CFD method. The well-trained ROM is then coupled with the structure equation at a specific velocity, the Limit-Cycle Oscillation(LCO) of stall flutter under this flow condition is predicted precisely and more quickly. In order to predict both the critical flutter velocity and LCO amplitudes after bifurcation at different velocities, a new ROM with GRU neural network considering the variation of flow velocities is developed. The stall flutter results predicted by ROM agree well with the CFD ones at different velocities. Finally, a brief sensitivity analysis of two structural parameters of ROM is carried out. It infers the potential of the presented modeling method to depict the nonlinearity of dynamic stall and stall flutter phenomenon.
基金supported by the National Natural Science Foundation of China/Hong Kong RRC Joint Research Scheme(NSFC/RGC 11961160718)the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design(No.2019B030301001)+1 种基金supported by the National Science Foundation of China(NSFC-11871264)the Guangdong Basic and Applied Basic Research Foundation(2018A0303130123).
文摘At present, deep learning based methods are being employed to resolvethe computational challenges of high-dimensional partial differential equations(PDEs). But the computation of the high order derivatives of neural networks iscostly, and high order derivatives lack robustness for training purposes. We proposea novel approach to solving PDEs with high order derivatives by simultaneously approximating the function value and derivatives. We introduce intermediate variablesto rewrite the PDEs into a system of low order differential equations as what is donein the local discontinuous Galerkin method. The intermediate variables and the solutions to the PDEs are simultaneously approximated by a multi-output deep neuralnetwork. By taking the residual of the system as a loss function, we can optimizethe network parameters to approximate the solution. The whole process relies onlow order derivatives. Numerous numerical examples are carried out to demonstrate that our local deep learning is efficient, robust, flexible, and is particularlywell-suited for high-dimensional PDEs with high order derivatives.
基金supported by the National Natural Science Foundation of China(91538201)the Taishan Scholar Foundation of China(ts201511020).
文摘The existing recognition algorithms of space-time block code(STBC)for multi-antenna(MA)orthogonal frequencydivision multiplexing(OFDM)systems use feature extraction and hypothesis testing to identify the signal types in a complex communication environment.However,owing to the restrictions on the prior information and channel conditions,these existing algorithms cannot perform well under strong interference and noncooperative communication conditions.To overcome these defects,this study introduces deep learning into the STBCOFDM signal recognition field and proposes a recognition method based on the fourth-order lag moment spectrum(FOLMS)and attention-guided multi-scale dilated convolution network(AMDCNet).The fourth-order lag moment vectors of the received signals are calculated,and vectors are stitched to form two-dimensional FOLMS,which is used as the input of the deep learning-based model.Then,the multi-scale dilated convolution is used to extract the details of images at different scales,and a convolutional block attention module(CBAM)is introduced to construct the attention-guided multi-scale dilated convolution module(AMDCM)to make the network be more focused on the target area and obtian the multi-scale guided features.Finally,the concatenate fusion,residual block and fully-connected layers are applied to acquire the STBC-OFDM signal types.Simulation experiments show that the average recognition probability of the proposed method at−12 dB is higher than 98%.Compared with the existing algorithms,the recognition performance of the proposed method is significantly improved and has good adaptability to environments with strong disturbances.In addition,the proposed deep learning-based model can directly identify the pre-processed FOLMS samples without a priori information on channel and noise,which is more suitable for non-cooperative communication systems than the existing algorithms.
文摘Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularly deep learning(DL),applied and relevant to computational mechanics(solid,fluids,finite-element technology)are reviewed in detail.Both hybrid and pure machine learning(ML)methods are discussed.Hybrid methods combine traditional PDE discretizations with ML methods either(1)to help model complex nonlinear constitutive relations,(2)to nonlinearly reduce the model order for efficient simulation(turbulence),or(3)to accelerate the simulation by predicting certain components in the traditional integration methods.Here,methods(1)and(2)relied on Long-Short-Term Memory(LSTM)architecture,with method(3)relying on convolutional neural networks.Pure ML methods to solve(nonlinear)PDEs are represented by Physics-Informed Neural network(PINN)methods,which could be combined with attention mechanism to address discontinuous solutions.Both LSTM and attention architectures,together with modern and generalized classic optimizers to include stochasticity for DL networks,are extensively reviewed.Kernel machines,including Gaussian processes,are provided to sufficient depth for more advanced works such as shallow networks with infinite width.Not only addressing experts,readers are assumed familiar with computational mechanics,but not with DL,whose concepts and applications are built up from the basics,aiming at bringing first-time learners quickly to the forefront of research.History and limitations of AI are recounted and discussed,with particular attention at pointing out misstatements or misconceptions of the classics,even in well-known references.Positioning and pointing control of a large-deformable beam is given as an example.
基金supported by National Science Foundation of China(52171251)Liao Ning Revitalization Talents Program(XLYC1907014)+2 种基金the Fundamental Research Funds for the Central Universities(DUT21ZD205)Ministry of Industry and Information Technology(2019-357)the Project of State Key Laboratory of Satellite Ocean Environment Dynamics,Second Institute of Oceanography,MNR(QNHX2112)。
文摘We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.
基金Supported by the NSFC (under Grant Nos.5070900 and 10772040)the National High Tech Research and Development Program of China (2006AA09A109-3)
文摘Based on the fifth-order Stokes regular wave theory, a simplified model for extreme-wave kinematics in deep sea was developed. In this model, from the wave records the average of two neighboring wave periods for the extreme crest or trough was defined as the period of the Stokes wave by the up and down zero-crossing methods. Then the input wave amplitude was deduced by substituting the wave period and extreme crest or trough into the expression for the fifth-order Stokes wave elevation. Thus the corresponding formula for the wave velocity can be used to describe kinematics beneath the extreme wave. By comparison with the published numerical models and experimental data, the proposed model is validated to be able to calculate the extreme wave velocity rather easily and accurately.
基金supported partly by National Key R&D Program of China(grants Nos.2019YFA0709600 and 2019YFA0709602)National Natural Science Foundation of China(grants Nos.11831016 and 12101609)the Innovation Foundation of Qian Xuesen Laboratory of Space Technology。
文摘This paper proposes a high order deep domain decomposition method(HOrderDeepDDM)for solving high-frequency interface problems,which combines high order deep neural network(HOrderDNN)with domain decomposition method(DDM).The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM,and apply HOrderDNNs to solve the high-frequency problem on each sub-domain.Besides,we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains,on the interface and the boundary during the optimization process.The performance of HOrderDeepDDM is evaluated on high-frequency elliptic and Helmholtz interface problems.The results indicate that:HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequency problems.In detail,HOrderDeepDDMs(p>1)could capture the high-frequency information very well.When compared to the deep domain decomposition method(DeepDDM),HOrderDeepDDMs(p>1)converge faster and achieve much smaller relative errors with the same number of trainable parameters.For example,when solving the high-frequency interface elliptic problems in Section 3.3.1,the minimum relative errors obtained by HOrderDeepDDMs(p=9)are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same,as shown in Fig.4.
基金funded by a grant from the National Key R&D Program of China(Grant No.2017YFC0603102)partially supported by the Strategic Priority Research Program of the Chinese Academy of Sciences(Grant No.XDA14010102)a Chinese NSF grant(Grant No.41973069)。
文摘FTT experiments with water as a hydrogen source and three types of possible carbon sources in the subsurface(diiron nonacarbonyl,siderite and formic acid,representing CO,CO_(2)and a simple organic acid,respectively)were carried out in this study.Our experimental results showed that n-alkanes with the highest carbon number of C_(33)were produced when CO was used as a carbon source(series A);a variety of polycyclic aromatic hydrocarbons(PAHs)were detected in series B with CO_(2)as a carbon source;gaseous hydrocarbons were also detected with formic acid added(series C).The different products in the three series showed that there were different hydrocarbon generation mechanisms and reaction processes with different carbon sources.The generation of long-chain n-alkanes in series A provided experimental support for the formation of abiogenic petroleum underground,which was of significance to early membranes on the Earth.PAHs in series B provide experimental support for the possibility of an abiotic source of reduced carbon on other planets.The carbon isotopes of gaseous hydrocarbons produced by CO exhibited a partial reversed order(δ^(13)C_(1)<δ^(13)C_(2)>δ^(13)C_(3)>δ^(13)C_(4)>δ^(13)C_(5)),while the gaseous hydrocarbons produced by CO_(2)and HCOOH showed a positive order(δ^(13)C_(1)<δ^(13)C_(2)<δ^(13)C_(3)<δ^(13)C_(4)<δ^(13)C_(5)).Based on these,the alkylene mechanism and the carbonyl insertion mechanism were used to reasonably explain these characteristics.
基金supported partly by National Key R&D Program of China with grants 2019YFA0709600,2019YFA0709602National Natural Science Foundation of China with grants 11831016,12101609the Innovation Foundation of Qian Xuesen Laboratory of Space Technology.
文摘This paper proposes a high order deep neural network(HOrderDNN)for solving high frequency partial differential equations(PDEs),which incorporates the idea of“high order”from finite element methods(FEMs)into commonly-used deep neural networks(DNNs)to obtain greater approximation ability.The main idea of HOrderDNN is introducing a nonlinear transformation layer between the input layer and the first hidden layer to form a high order polynomial space with the degree not exceeding p,followed by a normal DNN.The order p can be guided by the regularity of solutions of PDEs.The performance of HOrderDNNis evaluated on high frequency function fitting problems and high frequency Poisson and Helmholtz equations.The results demonstrate that:HOrderDNNs(p>1)can efficiently capture the high frequency information in target functions;and when compared to physics-informed neural network(PINN),HOrderDNNs(p>1)converge faster and achieve much smaller relative errors with same number of trainable parameters.In particular,when solving the high frequency Helmholtz equation in Section 3.5,the relative error of PINN stays around 1 with its depth and width increase,while the relative error can be reduced to around 0.02 as p increases(see Table 5).
