This research aims at developing RCPS （revised creative problem solving） teaching model, besides the authors designed the instructions of chemical reaction to promote eight grade students＇ scientific learning motiv...This research aims at developing RCPS （revised creative problem solving） teaching model, besides the authors designed the instructions of chemical reaction to promote eight grade students＇ scientific learning motivation and scientific concept learning. We adopted quasi-experiment study, the experimental group and controlled group all 28 students were chose, go on the parameter is analyzed together compared with textbook instructions, scale of scientific learning motivation and test of scientific conception learning were used for the two groups in prior test and post test, then they used statistical ANCOVA （analysis of covariance） to analyze the differences between the two teaching models. The result of this study finds that RCPS teaching model improved student＇s scientific learning motivation and learning scientific concept was superior to textbook instructions in controlled group, p = 0.001 （〈 0.01）, and all with high experimental treatment effects （〉 0.14）. The study also proposes that when RCPS teaching model was applied to scientific concept teaching, RCPS teaching model should be joined the conception introducing stage, and pay attention to students＇ scientific learning motivation.展开更多
Large language models(LLMs)have emerged as powerful tools for addressing a wide range of problems,including those in scientific computing,particularly in solving partial differential equations(PDEs).However,different ...Large language models(LLMs)have emerged as powerful tools for addressing a wide range of problems,including those in scientific computing,particularly in solving partial differential equations(PDEs).However,different models exhibit distinct strengths and preferences,resulting in varying levels of performance.In this paper,we compare the capabilities of the most advanced LLMs—DeepSeek,ChatGPT,and Claude—along with their reasoning-optimized versions in addressing computational challenges.Specifically,we evaluate their proficiency in solving traditional numerical problems in scientific computing as well as leveraging scientific machine learning techniques for PDE-based problems.We designed all our experiments so that a nontrivial decision is required,e.g,defining the proper space of input functions for neural operator learning.Our findings show that reasoning and hybrid-reasoning models consistently and significantly outperform non-reasoning ones in solving challenging problems,with ChatGPT o3-mini-high generally offering the fastest reasoning speed.展开更多
1.Introduction Since the publication of our original study comparing large language models(LLMs)in scientific computing and scientific machine learning tasks,Anthropic has released Claude 4.0[1],a major upgrade in its...1.Introduction Since the publication of our original study comparing large language models(LLMs)in scientific computing and scientific machine learning tasks,Anthropic has released Claude 4.0[1],a major upgrade in its Claude family of LLMs.Claude 4.0 is designed to introduce substantial improvements in reasoning,coding,and mathematical capabilities.展开更多
The data-driven machine learning paradigm typically requires high-quality,large-scale datasets for training neural networks,which are often unavailable in many scientific and engineering applications.Integrating physi...The data-driven machine learning paradigm typically requires high-quality,large-scale datasets for training neural networks,which are often unavailable in many scientific and engineering applications.Integrating physics equations into machine learning models,either fully or partially,can mitigate these data requirements and improve generalizability;however,such approaches frequently rely on differentiable programming frameworks.This ability poses significant challenges when legacy or commercial numerical solvers,which are often nondifferentiable and difficult to modify without introducing code changes,are integrated.This work addresses these challenges by leveraging the mini-batching iterative ensemble Kalman inversion(EKI)algorithm as a gradientfree training framework for hybrid neural models.The use of stochastic mini-batching significantly enhances the computational efficiency and convergence of EKI,making it well-suited for high-dimensional learning problems.The proposed method is demonstrated for modeling a fiber-reinforced composite plate,where heterogeneous local constitutive laws are parameterized by a trainable neural network embedded within the FEniCS finite element solver.Using the displacement field as indirect data,the hybrid neural FEM solver successfully predicts deformations by learning the local constitutive laws,even for unseen fiber volume fraction distributions and varying test loading conditions.These results demonstrate the effectiveness of iterative EKI in training hybrid neural models with non-differentiable components,paving the way for broader adoption of hybrid neural models in scientific and engineering applications.展开更多
In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural netwo...In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion.This map takes input of the element geometry and the PDE’s parameters on that element,and gives output of two operators:(1)the in2out operator for inter-element communication,and(2)the in2sol operator(Green’s function)for element-wise solution recovery.A significant advantage of this approach is that,once trained,this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining.Also,the training is significantly simpler since it is done on the element level instead on the entire domain.We call this approach element learning.This method is closely related to hybridizable discontinuous Galerkin(HDG)methods in the sense that the local solvers of HDG are replaced by machine learning approaches.Numerical tests are presented for an example PDE,the radiative transfer or radiation transport equation,in a variety of scenarios with idealized or realistic cloud fields,with smooth or sharp gradient in the cloud boundary transition.Under a fixed accuracy level of 10^(−3) in the relative L^(2) error,and polynomial degree p=6 in each element,we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method.展开更多
Partial Differential Equation(PDE)is among the most fundamental tools employed to model dynamic systems.Existing PDE modeling methods are typically derived from established knowledge and known phenomena,which are time...Partial Differential Equation(PDE)is among the most fundamental tools employed to model dynamic systems.Existing PDE modeling methods are typically derived from established knowledge and known phenomena,which are time-consuming and labor-intensive.Recently,discovering governing PDEs from collected actual data via Physics Informed Neural Networks(PINNs)provides a more efficient way to analyze fresh dynamic systems and establish PEDmodels.This study proposes Sequentially Threshold Least Squares-Lasso(STLasso),a module constructed by incorporating Lasso regression into the Sequentially Threshold Least Squares(STLS)algorithm,which can complete sparse regression of PDE coefficients with the constraints of l0 norm.It further introduces PINN-STLasso,a physics informed neural network combined with Lasso sparse regression,able to find underlying PDEs from data with reduced data requirements and better interpretability.In addition,this research conducts experiments on canonical inverse PDE problems and compares the results to several recent methods.The results demonstrated that the proposed PINN-STLasso outperforms other methods,achieving lower error rates even with less data.展开更多
Wave propagation problems are typically formulated as partial differential equations(PDEs)on unbounded domains to be solved.The classical approach to solving such problems involves truncating them to problems on bound...Wave propagation problems are typically formulated as partial differential equations(PDEs)on unbounded domains to be solved.The classical approach to solving such problems involves truncating them to problems on bounded domains by designing the artificial boundary conditions or perfectly matched layers,which typically require significant effort,and the presence of nonlinearity in the equation makes such designs even more challenging.Emerging deep learning-based methods for solving PDEs,with the physics-informed neural networks(PINNs)method as a representative,still face significant challenges when directly used to solve PDEs on unbounded domains.Calculations performed in a bounded domain of interest without imposing boundary constraints can lead to a lack of unique solutions thus causing the failure of PINNs.In light of this,this paper proposes a novel and effective data generationbased operator learning method for solving PDEs on unbounded domains.The key idea behind this method is to generate high-quality training data.Specifically,we construct a family of approximate analytical solutions to the target PDE based on its initial condition and source term.Then,using these constructed data comprising exact solutions,initial conditions,and source terms,we train an operator learning model called MIONet,which is capable of handling multiple inputs,to learn the mapping from the initial condition and source term to the PDE solution on a bounded domain of interest.Finally,we utilize the generalization ability of this model to predict the solution of the target PDE.The effectiveness of this method is exemplified by solving the wave equation and the Schr¨odinger equation defined on unbounded domains.More importantly,the proposed method can deal with nonlinear problems,which has been demonstrated by solving Burgers’equation and Korteweg-de Vries(KdV)equation.The code is available at https://github.com/ZJLAB-AMMI/DGOL.展开更多
Traditionally,classical numerical schemes have been employed to solve partial differential equations(PDEs)using computational methods.Recently,neural network-based methods have emerged.Despite these advancements,neura...Traditionally,classical numerical schemes have been employed to solve partial differential equations(PDEs)using computational methods.Recently,neural network-based methods have emerged.Despite these advancements,neural networkbased methods,such as physics-informed neural networks(PINNs)and neural operators,exhibit deficiencies in robustness and generalization.To address these issues,numerous studies have integrated classical numerical frameworks with machine learning techniques,incorporating neural networks into parts of traditional numerical methods.In this study,we focus on hyperbolic conservation laws by replacing traditional numerical fluxes with neural operators.To this end,we developed loss functions inspired by established numerical schemes related to conservation laws and approximated numerical fluxes using Fourier neural operators(FNOs).Our experiments demonstrated that our approach combines the strengths of both traditional numerical schemes and FNOs,outperforming standard FNO methods in several respects.For instance,we demonstrate that our method is robust,has resolution invariance,and is feasible as a data-driven method.In particular,our method can make continuous predictions over time and exhibits superior generalization capabilities with out-of-distribution(OOD)samples,which are challenges that existing neural operator methods encounter.展开更多
Fractional partial differential equations(FPDEs)can effectively represent anomalous transport and nonlocal interactions.However,inherent uncertainties arise naturally in real applications due to random forcing or unkn...Fractional partial differential equations(FPDEs)can effectively represent anomalous transport and nonlocal interactions.However,inherent uncertainties arise naturally in real applications due to random forcing or unknown material properties.Mathematical models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations(SFPDEs).There are many challenges in solving SFPDEs numerically,especially for long-time integration since such problems are high-dimensional and nonlocal.Here,we combine the bi-orthogonal(BO)method for representing stochastic processes with physicsinformed neural networks(PINNs)for solving partial differential equations to formulate the bi-orthogonal PINN method(BO-fPINN)for solving time-dependent SFPDEs.Specifically,we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs,and include the BO constraints in the loss function following a weak formulation.Since automatic differentiation is not currently applicable to fractional derivatives,we employ discretization on a grid to compute the fractional derivatives of the neural network output.The weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue crossings.Moreover,the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward problems.We demonstrate the effectiveness of the BO-fPINN method for different benchmark problems.Specifically,we first consider an SFPDE with eigenvalue crossing and obtain good results while the original BO method fails.We then solve several forward and inverse problems governed by SFPDEs,including problems with noisy initial conditions.We study the effect of the fractional order as well as the number of the BO modes on the accuracy of the BO-fPINN method.The results demonstrate the flexibility and efficiency of the proposed method,especially for inverse problems.We also present a simple example of transfer learning(for the fractional order)that can help in accelerating the training of BO-fPINN for SFPDEs.Taken together,the simulation results show that the BO-fPINN method can be employed to effectively solve time-dependent SFPDEs and may provide a reliable computational strategy for real applications exhibiting anomalous transport.展开更多
Developing a well-predictive machine learning model that also offers improved interpretability is a key challenge to widen the application of artificial intelligence in various application domains. In this work, we pr...Developing a well-predictive machine learning model that also offers improved interpretability is a key challenge to widen the application of artificial intelligence in various application domains. In this work, we present a Data Information integrated Neural Network (DINN) algorithm that incorporates the correlation information present in the dataset for the model development. The predictive performance of DINN is also compared with a standard artificial neural network (ANN) model. The DINN algorithm is applied on two case studies of energy systems namely energy efficiency cooling (ENC) & energy efficiency heating (ENH) of the buildings, and power generation from a 365 MW capacity industrial gas turbine. For ENC, DINN presents lower mean RMSE for testing datasets (RMSE_test = 1.23 %) in comparison with the ANN model (RMSE_test = 1.41 %). Similarly, DINN models have presented better predictive performance to model the output variables of the two case studies. The input perturbation analysis following the Gaussian distribution for noise generation reveals the order of significance of the variables, as made by DINN, can be better explained by the domain knowledge of the power generation operation of the gas turbine. This research work demonstrates the potential advantage to integrate the information present in the data for the well-predictive model development complemented with improved interpretation performance thereby opening avenues for industry-wide inclusion and other potential applications of machine learning.展开更多
文摘This research aims at developing RCPS （revised creative problem solving） teaching model, besides the authors designed the instructions of chemical reaction to promote eight grade students＇ scientific learning motivation and scientific concept learning. We adopted quasi-experiment study, the experimental group and controlled group all 28 students were chose, go on the parameter is analyzed together compared with textbook instructions, scale of scientific learning motivation and test of scientific conception learning were used for the two groups in prior test and post test, then they used statistical ANCOVA （analysis of covariance） to analyze the differences between the two teaching models. The result of this study finds that RCPS teaching model improved student＇s scientific learning motivation and learning scientific concept was superior to textbook instructions in controlled group, p = 0.001 （〈 0.01）, and all with high experimental treatment effects （〉 0.14）. The study also proposes that when RCPS teaching model was applied to scientific concept teaching, RCPS teaching model should be joined the conception introducing stage, and pay attention to students＇ scientific learning motivation.
基金supported by the ONR Vannevar Bush Faculty Fellowship(Grant No.N00014-22-1-2795).
文摘Large language models(LLMs)have emerged as powerful tools for addressing a wide range of problems,including those in scientific computing,particularly in solving partial differential equations(PDEs).However,different models exhibit distinct strengths and preferences,resulting in varying levels of performance.In this paper,we compare the capabilities of the most advanced LLMs—DeepSeek,ChatGPT,and Claude—along with their reasoning-optimized versions in addressing computational challenges.Specifically,we evaluate their proficiency in solving traditional numerical problems in scientific computing as well as leveraging scientific machine learning techniques for PDE-based problems.We designed all our experiments so that a nontrivial decision is required,e.g,defining the proper space of input functions for neural operator learning.Our findings show that reasoning and hybrid-reasoning models consistently and significantly outperform non-reasoning ones in solving challenging problems,with ChatGPT o3-mini-high generally offering the fastest reasoning speed.
文摘1.Introduction Since the publication of our original study comparing large language models(LLMs)in scientific computing and scientific machine learning tasks,Anthropic has released Claude 4.0[1],a major upgrade in its Claude family of LLMs.Claude 4.0 is designed to introduce substantial improvements in reasoning,coding,and mathematical capabilities.
基金supported by the Air Force Office of Scientific Research(AFOSR),United States of America(Grant No.FA9550-22-1-0065).
文摘The data-driven machine learning paradigm typically requires high-quality,large-scale datasets for training neural networks,which are often unavailable in many scientific and engineering applications.Integrating physics equations into machine learning models,either fully or partially,can mitigate these data requirements and improve generalizability;however,such approaches frequently rely on differentiable programming frameworks.This ability poses significant challenges when legacy or commercial numerical solvers,which are often nondifferentiable and difficult to modify without introducing code changes,are integrated.This work addresses these challenges by leveraging the mini-batching iterative ensemble Kalman inversion(EKI)algorithm as a gradientfree training framework for hybrid neural models.The use of stochastic mini-batching significantly enhances the computational efficiency and convergence of EKI,making it well-suited for high-dimensional learning problems.The proposed method is demonstrated for modeling a fiber-reinforced composite plate,where heterogeneous local constitutive laws are parameterized by a trainable neural network embedded within the FEniCS finite element solver.Using the displacement field as indirect data,the hybrid neural FEM solver successfully predicts deformations by learning the local constitutive laws,even for unseen fiber volume fraction distributions and varying test loading conditions.These results demonstrate the effectiveness of iterative EKI in training hybrid neural models with non-differentiable components,paving the way for broader adoption of hybrid neural models in scientific and engineering applications.
基金partially supported by the NSF(Grant No.DMS 2324368)by the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin-Madison with funding from the Wisconsin Alumni Research Foundation.
文摘In this paper,we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations(PDEs).The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion.This map takes input of the element geometry and the PDE’s parameters on that element,and gives output of two operators:(1)the in2out operator for inter-element communication,and(2)the in2sol operator(Green’s function)for element-wise solution recovery.A significant advantage of this approach is that,once trained,this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining.Also,the training is significantly simpler since it is done on the element level instead on the entire domain.We call this approach element learning.This method is closely related to hybridizable discontinuous Galerkin(HDG)methods in the sense that the local solvers of HDG are replaced by machine learning approaches.Numerical tests are presented for an example PDE,the radiative transfer or radiation transport equation,in a variety of scenarios with idealized or realistic cloud fields,with smooth or sharp gradient in the cloud boundary transition.Under a fixed accuracy level of 10^(−3) in the relative L^(2) error,and polynomial degree p=6 in each element,we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method.
文摘Partial Differential Equation(PDE)is among the most fundamental tools employed to model dynamic systems.Existing PDE modeling methods are typically derived from established knowledge and known phenomena,which are time-consuming and labor-intensive.Recently,discovering governing PDEs from collected actual data via Physics Informed Neural Networks(PINNs)provides a more efficient way to analyze fresh dynamic systems and establish PEDmodels.This study proposes Sequentially Threshold Least Squares-Lasso(STLasso),a module constructed by incorporating Lasso regression into the Sequentially Threshold Least Squares(STLS)algorithm,which can complete sparse regression of PDE coefficients with the constraints of l0 norm.It further introduces PINN-STLasso,a physics informed neural network combined with Lasso sparse regression,able to find underlying PDEs from data with reduced data requirements and better interpretability.In addition,this research conducts experiments on canonical inverse PDE problems and compares the results to several recent methods.The results demonstrated that the proposed PINN-STLasso outperforms other methods,achieving lower error rates even with less data.
基金supported by the China Postdoctoral Science Foundation(No.2023M743266)Zhejiang Provincial Postdoctoral Research Project Merit-based Funding(No.ZJ2023067)+1 种基金Exploratory Research Project(No.2022RC0AN02)Research Initiation Project(No.K2022RC0PI01)of Zhejiang Lab.
文摘Wave propagation problems are typically formulated as partial differential equations(PDEs)on unbounded domains to be solved.The classical approach to solving such problems involves truncating them to problems on bounded domains by designing the artificial boundary conditions or perfectly matched layers,which typically require significant effort,and the presence of nonlinearity in the equation makes such designs even more challenging.Emerging deep learning-based methods for solving PDEs,with the physics-informed neural networks(PINNs)method as a representative,still face significant challenges when directly used to solve PDEs on unbounded domains.Calculations performed in a bounded domain of interest without imposing boundary constraints can lead to a lack of unique solutions thus causing the failure of PINNs.In light of this,this paper proposes a novel and effective data generationbased operator learning method for solving PDEs on unbounded domains.The key idea behind this method is to generate high-quality training data.Specifically,we construct a family of approximate analytical solutions to the target PDE based on its initial condition and source term.Then,using these constructed data comprising exact solutions,initial conditions,and source terms,we train an operator learning model called MIONet,which is capable of handling multiple inputs,to learn the mapping from the initial condition and source term to the PDE solution on a bounded domain of interest.Finally,we utilize the generalization ability of this model to predict the solution of the target PDE.The effectiveness of this method is exemplified by solving the wave equation and the Schr¨odinger equation defined on unbounded domains.More importantly,the proposed method can deal with nonlinear problems,which has been demonstrated by solving Burgers’equation and Korteweg-de Vries(KdV)equation.The code is available at https://github.com/ZJLAB-AMMI/DGOL.
文摘Traditionally,classical numerical schemes have been employed to solve partial differential equations(PDEs)using computational methods.Recently,neural network-based methods have emerged.Despite these advancements,neural networkbased methods,such as physics-informed neural networks(PINNs)and neural operators,exhibit deficiencies in robustness and generalization.To address these issues,numerous studies have integrated classical numerical frameworks with machine learning techniques,incorporating neural networks into parts of traditional numerical methods.In this study,we focus on hyperbolic conservation laws by replacing traditional numerical fluxes with neural operators.To this end,we developed loss functions inspired by established numerical schemes related to conservation laws and approximated numerical fluxes using Fourier neural operators(FNOs).Our experiments demonstrated that our approach combines the strengths of both traditional numerical schemes and FNOs,outperforming standard FNO methods in several respects.For instance,we demonstrate that our method is robust,has resolution invariance,and is feasible as a data-driven method.In particular,our method can make continuous predictions over time and exhibits superior generalization capabilities with out-of-distribution(OOD)samples,which are challenges that existing neural operator methods encounter.
基金supported by the NSF of China(92270115,12071301)and the Shanghai Municipal Science and Technology Commission(20JC1412500)Fanhai Zeng is supported by the National Key R&D Program of China(2021YFA1000202,2021YFA1000200)+2 种基金the NSF of China(12171283,12120101001)the startup fund from Shandong University(11140082063130)the Science Foundation Program for Distinguished Young Scholars of Shandong(Overseas)(2022HWYQ-045).
文摘Fractional partial differential equations(FPDEs)can effectively represent anomalous transport and nonlocal interactions.However,inherent uncertainties arise naturally in real applications due to random forcing or unknown material properties.Mathematical models considering nonlocal interactions with uncertainty quantification can be formulated as stochastic fractional partial differential equations(SFPDEs).There are many challenges in solving SFPDEs numerically,especially for long-time integration since such problems are high-dimensional and nonlocal.Here,we combine the bi-orthogonal(BO)method for representing stochastic processes with physicsinformed neural networks(PINNs)for solving partial differential equations to formulate the bi-orthogonal PINN method(BO-fPINN)for solving time-dependent SFPDEs.Specifically,we introduce a deep neural network for the stochastic solution of the time-dependent SFPDEs,and include the BO constraints in the loss function following a weak formulation.Since automatic differentiation is not currently applicable to fractional derivatives,we employ discretization on a grid to compute the fractional derivatives of the neural network output.The weak formulation loss function of the BO-fPINN method can overcome some drawbacks of the BO methods and thus can be used to solve SFPDEs with eigenvalue crossings.Moreover,the BO-fPINN method can be used for inverse SFPDEs with the same framework and same computational complexity as for forward problems.We demonstrate the effectiveness of the BO-fPINN method for different benchmark problems.Specifically,we first consider an SFPDE with eigenvalue crossing and obtain good results while the original BO method fails.We then solve several forward and inverse problems governed by SFPDEs,including problems with noisy initial conditions.We study the effect of the fractional order as well as the number of the BO modes on the accuracy of the BO-fPINN method.The results demonstrate the flexibility and efficiency of the proposed method,especially for inverse problems.We also present a simple example of transfer learning(for the fractional order)that can help in accelerating the training of BO-fPINN for SFPDEs.Taken together,the simulation results show that the BO-fPINN method can be employed to effectively solve time-dependent SFPDEs and may provide a reliable computational strategy for real applications exhibiting anomalous transport.
文摘Developing a well-predictive machine learning model that also offers improved interpretability is a key challenge to widen the application of artificial intelligence in various application domains. In this work, we present a Data Information integrated Neural Network (DINN) algorithm that incorporates the correlation information present in the dataset for the model development. The predictive performance of DINN is also compared with a standard artificial neural network (ANN) model. The DINN algorithm is applied on two case studies of energy systems namely energy efficiency cooling (ENC) & energy efficiency heating (ENH) of the buildings, and power generation from a 365 MW capacity industrial gas turbine. For ENC, DINN presents lower mean RMSE for testing datasets (RMSE_test = 1.23 %) in comparison with the ANN model (RMSE_test = 1.41 %). Similarly, DINN models have presented better predictive performance to model the output variables of the two case studies. The input perturbation analysis following the Gaussian distribution for noise generation reveals the order of significance of the variables, as made by DINN, can be better explained by the domain knowledge of the power generation operation of the gas turbine. This research work demonstrates the potential advantage to integrate the information present in the data for the well-predictive model development complemented with improved interpretation performance thereby opening avenues for industry-wide inclusion and other potential applications of machine learning.
