Adversarial Adaptive Sampling: Unify PINN and Optimal Transport for the Approximation of PDEs

Solving partial differential equations (PDEs) is a central task in scientific computing. Recently, neural network approximation of PDEs has received increasing attention due to its flexible meshless discretization and its potential for high-dimensional problems. One fundamental numerical difficulty is that random samples in the training set introduce statistical errors into the discretization of loss functional which may become the dominant error in the final approximation, and therefore overshadow the modeling capability of the neural network. In this work, we propose a new minmax formulation to optimize simultaneously the approximate solution, given by a neural network model, and the random samples in the training set, provided by a deep generative model. The key idea is to use a deep generative model to adjust random samples in the training set such that the residual induced by the approximate PDE solution can maintain a smooth profile when it is being minimized. Such an idea is achieved by implicitly embedding the Wasserstein distance between the residual-induced distribution and the uniform distribution into the loss, which is then minimized together with the residual. A nearly uniform residual profile means that its variance is small for any normalized weight function such that the Monte Carlo approximation error of the loss functional is reduced significantly for a certain sample size. The adversarial adaptive sampling (AAS) approach proposed in this work is the first attempt to formulate two essential components, minimizing the residual and seeking the optimal training set, into one minmax objective functional for the neural network approximation of PDEs

Has China gained significant influence over ASEAN? : from the golden decade to the diamond decade

published_or_final_versionInternational and Public AffairsMasterMaster of International and Public Affair

Early Cretaceous high-Ti and low-Ti mafic magmatism in Southeastern Tibet: Insights into magmatic evolution of the Comei Large Igneous Province

The Dala diabase intrusion, at the southeastern margin of the Yardoi gneiss dome, is located within the outcrop area of the ~ 132 Ma Comei Large Igneous Province (LIP), the result of initial activity of the Kerguelen plume. We present new zircon U-Pb geochronology results to show that the Dala diabase was emplaced at ~ 132 Ma and geochemical data (whole-rock element and Sr-Nd isotope ratios, zircon Hf isotopes and Fe-Ti oxide mineral chemistry) to confirm that the Dala diabase intrusion is part of the Comei LIP. The Dala diabase can be divided into a high-Mg/low-Ti series and a low-Mg/high-Ti series. The high-Mg/low-Ti series represents more primitive mafic magma compositions that we demonstrate are parental to the low-Mg/high-Ti series. Fractionation of olivine and clinopyroxene, followed by plagioclase within the low-Mg series, lead to systematic changes in concentrations of mantle compatible elements (Cr, Co, Ni, and V), REEs, HFSEs, and major elements such as Ti and P. Some Dala samples from the low-Mg/high-Ti series contain large ilmenite clusters and show extreme enrichment of Ti with elevated Ti/Y ratios, likely due to settling and accumulation of ilmenite during the magma chamber evolution. However, most samples from throughout the Comei LIP follow the Ti-evolution trend of the typical liquid line of descent (LLD) of primary OIB compositions, showing strong evidence of control of Ti contents by differentiation processes. In many other localities, however, primitive magmas are absent and observed Ti contents of evolved magmas cannot be quantitatively related to source processes. Careful examination of the petrogenetic relationship between co-existing low-Ti and high-Ti mafic rocks is essential to using observed rock chemistry to infer source composition, location, and degree of melting

A HISTORY OF THE STEREOLOGY IN CHINA

This review article introduces the formation and development of stereology in China under the background of the development of international stereology. In the early 1970s, some stereological monographs and collections were introduced into China, and Chinese scholars began to understand, study and promote stereology knowledge. Meanwhile, the widespread use of image analysis systems has contributed to the spread of stereology in China. On the other hand, academic exchanges and personnel training have played a catalytic role in the formation of stereology in China. According to China National Knowledge Infrastructure (CNKI) statistics, the number and impact of Chinese papers in stereology continues to grow during the past 30 years. After in-depth discussion, Chinese scholars have adopted a broader definition of stereology. With economic development and technological progress, China has great potential to develop, promote and apply the stereological methods and the related technologies

Basaltic and Solution Reference Materials for Iron, Copper and Zinc Isotope Measurements

Iron, Cu and Zn stable isotope systems are applied in constraining a variety of geochemical and environmental processes. Secondary reference materials have been developed by the Institute of Geology, Chinese Academy of Geological Sciences (CAGS), in collaboration with other participating laboratories, comprising three solutions (CAGS-Fe, CAGS-Cu and CAGS-Zn) and one basalt (CAGS-Basalt). These materials exhibit sufficient homogeneity and stability for application in Fe, Cu and Zn isotopic ratio determinations. Reference values were determined by inter-laboratory analytical comparisons involving up to eight participating laboratories employing MC-ICP-MS techniques, based on the unweighted means of submitted results. Isotopic compositions are reported in per mil notation, based on reference materials IRMM-014 for Fe, NIST SRM 976 for Cu and IRMM-3702 for Zn. Respective reference values of CAGS-Fe, CAGS-Cu and CAGS-Zn solutions are as follows: δ56Fe = 0.83 ± 0.06 and δ57Fe = 1.20 ± 0.12, δ65Cu = 0.57 ± 0.05, and δ66Zn = -0.79 ± 0.12 and δ68Zn = -1.65 ± 0.24, respectively. Those of CAGS-Basalt are δ56Fe = 0.15 ± 0.05, δ57Fe = 0.22 ± 0.05, δ65Cu = 0.12 ± 0.07, δ66Zn = 0.17 ± 0.11, and δ68Zn = 0.34 ± 0.21 (2s)