VC-PINN: Variable Coefficient Physical Information Neural Network For Forward And Inverse PDE Problems with Variable Coefficient

The paper proposes a deep learning method specifically dealing with the forward and inverse problem of variable coefficient partial differential equations -- Variable Coefficient Physical Information Neural Network (VC-PINN). The shortcut connections (ResNet structure) introduced into the network alleviates the "Vanishing gradient" and unifies the linear and nonlinear coefficients. The developed method was applied to four equations including the variable coefficient Sine-Gordon (vSG), the generalized variable coefficient Kadomtsev-Petviashvili equation (gvKP), the variable coefficient Korteweg-de Vries equation (vKdV), the variable coefficient Sawada-Kotera equation (vSK). Numerical results show that VC-PINN is successful in the case of high dimensionality, various variable coefficients (polynomials, trigonometric functions, fractions, oscillation attenuation coefficients), and the coexistence of multiple variable coefficients. We also conducted an in-depth analysis of VC-PINN in a combination of theory and numerical experiments, including four aspects, the necessity of ResNet, the relationship between the convexity of variable coefficients and learning, anti-noise analysis, the unity of forward and inverse problems/relationship with standard PINN

Holography and noncommutative Yang-Mills theory

Journal ArticleIn this Letter a recently proposed gravity dual of noncommutative Yang-Mills theory is derived from the relations between closed string moduli and open string moduli recently suggested by Seiberg and Witten. The only new input one needs is a simple form of the running string tension as a function of energy. This derivation provides convincing evidence that string theory integrates with the holographical principle and demonstrates a direct link between noncommutative Yang-Mills theory and holography

Linking Aboveground Traits to Root Traits and Local Environment: Implications of the Plant Economics Spectrum.

The plant economics spectrum proposes that ecological traits are functionally coordinated and adapt along environmental gradients. However, empirical evidence is mixed about whether aboveground and root traits are consistently linked and which environmental factors drive functional responses. Here we measure the strength of relationships between aboveground and root traits, and examine whether community-weighted mean trait values are adapted along gradients of light and soil fertility, based on the seedling censuses of 57 species in a subtropical forest. We found that aboveground traits were good predictors of root traits; specific leaf area, dry matter, nitrogen and phosphorus content were strongly correlated with root tissue density and specific root length. Traits showed patterns of adaptation along the gradients of soil fertility and light; species with fast resource-acquisitive strategies were more strongly associated with high soil phosphorus, potassium, openness, and with low nitrogen, organic matter conditions. This demonstrates the potential to estimate belowground traits from known aboveground traits in seedling communities, and suggests that soil fertility is one of the main factors driving functional responses. Our results extend our understanding of how ecological strategies shape potential responses of plant communities to environmental change