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How to simulate shock waves and other discontinuities is a long history topic. As a new and booming method, the physics-informed neural network (PINN) is still weak in calculating shock waves than traditional shock-capturing methods. In this paper, we propose a ‘retreat in order to advance’ way to improve the shock capturing ability of PINN by using a weighted equations (WE) method with PINN. The primary strategy of the method is to weaken the expression of the network in high compressible regions by adding a local positive and compression-dependent weight into governing equations at each interior point. With this strategy, the network will focus on training smooth parts of the solutions. Then automatically affected by the compressible property near shock waves, a sharp discontinuity appears with the wrong inside-shock-points ‘compressed’ into well-trained smooth regions just like passive particles. In this paper, we study one-dimensional and two-dimensional Euler equations. And illustrated by the comparisons with high-order classical WENO-Z method in numerical examples, the proposed method can significantly improve the discontinuity computing ability

An ellipsoid, the simplest non-spherical shape, has been extensively used as models for elongated building blocks for a wide spectrum of molecular, colloidal and granular systems. Yet the densest packing of congruent hard ellipsoids, which is intimately related to the high-density phase of many condensed matter systems, is still an open problem. We discover a novel dense crystalline packing of ellipsoids containing 24 particles with a quasi-square-triangular (SQ-TR) tiling arrangement, whose packing density {\phi} exceeds that of the SM2 crystal for aspect ratios {\alpha} in (1.365, 1.5625), attaining a maximal {\phi} about 0.75806 at {\alpha} = 93/64. We show that SQ-TR phase is thermodynamically stable at high densities over the aforementioned {\alpha} range and report a novel phase diagram for self-dual ellipsoids. The discovery of SQ-TR crystal suggests novel organizing principles for non-spherical particles and self-assembly of colloidal systems., Comment: 16 pages, 10 figures