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Utilizing physics-informed neural networks (PINN) to solve partial differential equations (PDEs) becomes a hot issue and also shows its great powers, but still suffers from the dilemmas of limited predicted accuracy in the sampling domain and poor prediction ability beyond the sampling domain which are usually mitigated by adding the physical properties of PDEs into the loss function or by employing smart techniques to change the form of loss function for special PDEs. In this paper, we design a symmetry-enhanced deep neural network (sDNN) which makes the architecture of neural networks invariant under the finite group through expanding the dimensions of weight matrixes and bias vectors in each hidden layers by the order of finite group if the group has matrix representations, otherwise extending the set of input data and the hidden layers except for the first hidden layer by the order of finite group. However, the total number of training parameters is only about one over the order of finite group of the original PINN size due to the symmetric architecture of sDNN. Furthermore, we give special forms of weight matrixes and bias vectors of sDNN, and rigorously prove that the architecture itself is invariant under the finite group and the sDNN has the universal approximation ability to learn the function keeping the finite group. Numerical results show that the sDNN has strong predicted abilities in and beyond the sampling domain and performs far better than the vanilla PINN with fewer training points and simpler architecture.

Domain decomposition provides an effective way to tackle the dilemma of physics-informed neural networks (PINN) which struggle to accurately and efficiently solve partial differential equations (PDEs) in the whole domain, but the lack of efficient tools for dealing with the interfaces between two adjacent sub-domains heavily hinders the training effects, even leads to the discontinuity of the learned solutions. In this paper, we propose a symmetry group based domain decomposition strategy to enhance the PINN for solving the forward and inverse problems of the PDEs possessing a Lie symmetry group. Specifically, for the forward problem, we first deploy the symmetry group to generate the dividing-lines having known solution information which can be adjusted flexibly and are used to divide the whole training domain into a finite number of non-overlapping sub-domains, then utilize the PINN and the symmetry-enhanced PINN methods to learn the solutions in each sub-domain and finally stitch them to the overall solution of PDEs. For the inverse problem, we first utilize the symmetry group acting on the data of the initial and boundary conditions to generate labeled data in the interior domain of PDEs and then find the undetermined parameters as well as the solution by only training the neural networks in a sub-domain. Consequently, the proposed method can predict high-accuracy solutions of PDEs which are failed by the vanilla PINN in the whole domain and the extended physics-informed neural network in the same sub-domains. Numerical results of the Korteweg-de Vries equation with a translation symmetry and the nonlinear viscous fluid equation with a scaling symmetry show that the accuracies of the learned solutions are improved largely.

As a typical application of deep learning, physics-informed neural network (PINN) {has been} successfully used to find numerical solutions of partial differential equations (PDEs), but how to improve the limited accuracy is still a great challenge for PINN. In this work, we introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries or non-classical symmetries of PDEs are embedded into the loss function in PINN, to improve the accuracy of PINN for solving the forward and inverse problems of PDEs. We test the effectiveness of SPINN for the forward problem via two groups of ten independent numerical experiments using different numbers of collocation points and neurons for the heat equation, Korteweg-de Vries (KdV) equation and potential Burgers {equations} respectively, and for the inverse problem by considering different layers and neurons as well as different training points for the Burgers equation in potential form. The numerical results show that SPINN performs better than PINN with fewer training points and simpler architecture of neural network. Furthermore, we discuss the computational overhead of SPINN in terms of the relative computational cost to PINN and show that the training time of SPINN has no obvious increases, even less than PINN for certain cases.

The DArk Matter Particle Explorer (DAMPE) is well suitable for searching for monochromatic and sharp $\gamma$-ray structures in the GeV$-$TeV range thanks to its unprecedented high energy resolution. In this work, we search for $\gamma$-ray line structures using five years of DAMPE data. To improve the sensitivity, we develop two types of dedicated data sets (including the BgoOnly data which is the first time to be used in the data analysis for the calorimeter-based gamma-ray observatories) and adopt the signal-to-noise ratio optimized regions of interest (ROIs) for different DM density profiles. No line signals or candidates are found between 10 and 300 GeV in the Galaxy. The constraints on the velocity-averaged cross section for $\chi\chi \to \gamma\gamma$ and the decay lifetime for $\chi \to \gamma\nu$, both at 95% confidence level, have been calculated and the systematic uncertainties have been taken into account. Comparing to the previous Fermi-LAT results, though DAMPE has an acceptance smaller by a factor of $\sim 10$, similar constraints on the DM parameters are achieved and below 100 GeV the lower limits on the decay lifetime are even stronger by a factor of a few. Our results demonstrate the potential of high-energy-resolution observations on dark matter detection., Comment: 14 pages, 8 figures. Update the content to keep up with the published version

Back-streaming neutrons from the spallation target of the China Spallation Neutron Source (CSNS) that emit through the incoming proton channel were exploited to build a white neutron beam facility (the so-called Back-n white neutron source), which was completed in March 2018. The Back-n neutron beam is very intense, at approximately 2*10^7 n/cm^2/s at 55 m from the target, and has a nominal proton beam with a power of 100 kW in the CSNS-I phase and a kinetic energy of 1.6 GeV and a thick tungsten target in multiple slices with modest moderation from the cooling water through the slices. In addition, the excellent energy spectrum spanning from 0.5 eV to 200 MeV, and a good time resolution related to the time-of-flight measurements make it a typical white neutron source for nuclear data measurements; its overall performance is among that of the best white neutron sources in the world. Equipped with advanced spectrometers, detectors, and application utilities, the Back-n facility can serve wide applications, with a focus on neutron-induced cross-section measurements. This article presents an overview of the neutron beam characteristics, the experimental setups, and the ongoing applications at Back-n., Comment: 11 pages, 9 figures

Inductively coupled plasma mass (ICP-MS) spectroscopy is widely used for screening materials of low background detectors in dark matter and double beta decay searches due to its high sensitivity to trace $^{238}$U and $^{232}$Th. This work describes a novel co-precipitation approach to measure the amount of $^{238}$U in high-purity copper to sub-ppt level. Such an approach allows the pre-concentration of U and removal of the matrix, by selecting a proper precipitator to co-precipitate with $^{238}$U and using excess ammonia water to separate the uranium hydroxide from copper by forming water-soluble tetra-amminecopper (II). The isotope dilution method and standard addition method were both used to mitigate the matrix effect and cross-check each other. The latter was also used to measure the recovery efficiency of $^{238}$U by using $^{233}$U as the tracer. The method detection limit (MDL) reached $\sim$0.1 pg $^{238}$U /g Cu for both methods while the recovery efficiency of uranium robustly remains 65\%--85\%. Various sources of interference in the ICP-MS analysis were thoroughly evaluated, and the contamination from reagents were found to be the dominant factor that affected the MDL. Further purification will allow significant improvements in the MDL. This co-precipitate approach can be easily extended to measure $^{232}$Th by using $^{229}$Th as the tracer., Comment: 15 pages, 3 figures, 5 tables

In this paper, we concentrate on the Lie symmetry structure of a system of multi-dimensional time-fractional partial differential equations (PDEs). Specifically, we first give an explicit prolongation formula of the Lie infinitesimal generator involving time-fractional partial derivative in multi-dimensional case, and then show that the admitted infinitesimal generator has an elegant structure. Furthermore, we present two simple conditions to determine the infinitesimal generator where one is a system of linear time-fractional PDEs, the other is a system of integer-order PDEs and plays the dominant role in the whole procedure. We study three time-fractional PDEs to illustrate the efficiency of the results., Comment: Accepted

In \cite{hab-2016,hab-2017}, Habibullin \emph{et.al} proposed an approach to construct Lax pairs of a nonlinear integrable partial differential equation (PDE), where one is the linearized equation of the studied PDE and the other is the invariant manifold of the linearized equation. In this paper, we show that the invariant manifold is the characteristic of a generalized conditional symmetry of the system composed of the studied PDE and its linearized PDE. Then we give an upper order bound of the invariant manifold which provides a theoretical basis for a complete classification of such type of invariant manifold. Moreover, we suggest a modified method to construct Lax pair of the KdV equation which can not be obtained by the original method in \cite{hab-2016,hab-2017}.

The sparse difference resultant introduced in \citep{gao-2015} is a basic concept in difference elimination theory. In this paper, we show that the sparse difference resultant of a generic Laurent transformally essential system can be computed via the sparse resultant of a simple algebraic system arising from the difference system. Moreover, new order bounds of sparse difference resultant are found. Then we propose an efficient algorithm to compute sparse difference resultant which is the quotient of two determinants whose elements are the coefficients of the polynomials in the algebraic system. The complexity of the algorithm is analyzed and experimental results show the efficiency of the algorithm., Comment: 22 pages