In this paper,we propose a structure-preserving numerical scheme for the size-modified Poisson-Nernst-Planck-Cahn-Hilliard(SPNPCH)equations derived from the free energy including electrostatic energies,entropies,steri...In this paper,we propose a structure-preserving numerical scheme for the size-modified Poisson-Nernst-Planck-Cahn-Hilliard(SPNPCH)equations derived from the free energy including electrostatic energies,entropies,steric energies,and Cahn-Hilliard mixtures.Based on the Jordan-Kinderlehrer-Otto(JKO)framework and the Benamou-Brenier formula of quadratic Wasserstein distance,the SPNPCH equations are transformed into a constrained optimization problem.By exploiting the convexity of the objective function,we can prove the existence and uniqueness of the numerical solution to the optimization problem.Mass conservation and unconditional energy-dissipation are preserved automatically by this scheme.Furthermore,by making use of the singularity of the entropy term which keeps the concentration from approaching zero,we can ensure the positivity of concentration.To solve the optimization problem,we apply the quasi-Newton method,which can ensure the positivity of concentration in the iterative process.Numerical tests are performed to confirm the anticipated accuracy and the desired physical properties of the developed scheme.Finally,the proposed scheme can also be applied to study the influence of ionic sizes and gradient energy coefficients on ion distribution.展开更多
Deep learning techniques for solving elliptic interface problems have gained significant attentions.In this paper,we introduce a hybrid residual and weak form(HRW)loss aimed at mitigating the challenge of model traini...Deep learning techniques for solving elliptic interface problems have gained significant attentions.In this paper,we introduce a hybrid residual and weak form(HRW)loss aimed at mitigating the challenge of model training.HRW utilizes the functions residual loss and Ritz method in an adversary-system,which enhances the probability of jumping out of the local optimum even when the loss landscape comprises multiple soft constraints(regularization terms),thus improving model’s capability and robustness.For the problem with interface conditions,unlike existing methods that use the domain decomposition,we design a Pre-activated ResNet of ResNet(PRoR)network structure employing a single network to feed both coordinates and corresponding subdomain indicators,thus reduces the number of parameters.The effectiveness and improvements of the PRoR with HRW are verified on two-dimensional interface problems with regular or irregular interfaces.We then apply the PRoR with HRW to solve the size-modified Poisson-Boltzmann equation,an improved dielectric continuum model for predicting the electrostatic potentials in an ionic solvent by considering the steric effects.Our findings demonstrate that the PRoR with HRW accurately approximates solvation free-energies of three proteins with irregular interfaces,showing the competitive results compared to the ones obtained using the finite element method.展开更多
基金J.Ding was supported by the Natural Science Foundation of Jiangsu Province(Grant BK20210443)by the National Natural Science Foundation of China(Grant 12101264)by the Shuangchuang program of Jiangsu Province(Grant 1142024031211190)。
文摘In this paper,we propose a structure-preserving numerical scheme for the size-modified Poisson-Nernst-Planck-Cahn-Hilliard(SPNPCH)equations derived from the free energy including electrostatic energies,entropies,steric energies,and Cahn-Hilliard mixtures.Based on the Jordan-Kinderlehrer-Otto(JKO)framework and the Benamou-Brenier formula of quadratic Wasserstein distance,the SPNPCH equations are transformed into a constrained optimization problem.By exploiting the convexity of the objective function,we can prove the existence and uniqueness of the numerical solution to the optimization problem.Mass conservation and unconditional energy-dissipation are preserved automatically by this scheme.Furthermore,by making use of the singularity of the entropy term which keeps the concentration from approaching zero,we can ensure the positivity of concentration.To solve the optimization problem,we apply the quasi-Newton method,which can ensure the positivity of concentration in the iterative process.Numerical tests are performed to confirm the anticipated accuracy and the desired physical properties of the developed scheme.Finally,the proposed scheme can also be applied to study the influence of ionic sizes and gradient energy coefficients on ion distribution.
文摘Deep learning techniques for solving elliptic interface problems have gained significant attentions.In this paper,we introduce a hybrid residual and weak form(HRW)loss aimed at mitigating the challenge of model training.HRW utilizes the functions residual loss and Ritz method in an adversary-system,which enhances the probability of jumping out of the local optimum even when the loss landscape comprises multiple soft constraints(regularization terms),thus improving model’s capability and robustness.For the problem with interface conditions,unlike existing methods that use the domain decomposition,we design a Pre-activated ResNet of ResNet(PRoR)network structure employing a single network to feed both coordinates and corresponding subdomain indicators,thus reduces the number of parameters.The effectiveness and improvements of the PRoR with HRW are verified on two-dimensional interface problems with regular or irregular interfaces.We then apply the PRoR with HRW to solve the size-modified Poisson-Boltzmann equation,an improved dielectric continuum model for predicting the electrostatic potentials in an ionic solvent by considering the steric effects.Our findings demonstrate that the PRoR with HRW accurately approximates solvation free-energies of three proteins with irregular interfaces,showing the competitive results compared to the ones obtained using the finite element method.
