Data-Driven Modeling and Forecasting of Chaotic Dynamics on Inertial Manifolds Constructed as Spectral Submanifolds

We present a data-driven and interpretable approach for reducing the dimensionality of chaotic systems using spectral submanifolds (SSMs). Emanating from fixed points or periodic orbits, these SSMs are low-dimensional inertial manifolds containing the chaotic attractor of the underlying high-dimensional system. The reduced dynamics on the SSMs turn out to predict chaotic dynamics accurately over a few Lyapunov times and also reproduce long-term statistical features, such as the largest Lyapunov exponents and probability distributions, of the chaotic attractor. We illustrate this methodology on numerical data sets including a delay-embedded Lorenz attractor, a nine-dimensional Lorenz model, and a Duffing oscillator chain. We also demonstrate the predictive power of our approach by constructing an SSM-reduced model from unforced trajectories of a buckling beam, and then predicting its periodically forced chaotic response without using data from the forced beam.Comment: Submitted to Chao

Nonlinear Model Reduction to Fractional and Mixed-Mode Spectral Submanifolds

A primary spectral submanifold (SSM) is the unique smoothest nonlinear continuation of a nonresonant spectral subspace $E$ of a dynamical system linearized at a fixed point. Passing from the full nonlinear dynamics to the flow on an attracting primary SSM provides a mathematically precise reduction of the full system dynamics to a very low-dimensional, smooth model in polynomial form. A limitation of this model reduction approach has been, however, that the spectral subspace yielding the SSM must be spanned by eigenvectors of the same stability type. A further limitation has been that in some problems, the nonlinear behavior of interest may be far away from the smoothest nonlinear continuation of the invariant subspace $E$. Here we remove both of these limitations by constructing a significantly extended class of SSMs that also contains invariant manifolds with mixed internal stability types and of lower smoothness class arising from fractional powers in their parametrization. We show on examples how fractional and mixed-mode SSMs extend the power of data-driven SSM reduction to transitions in shear flows, dynamic buckling of beams and periodically forced nonlinear oscillatory systems. More generally, our results reveal the general function library that should be used beyond integer-powered polynomials in fitting nonlinear reduced-order models to data.Comment: To appear in Chao

Accuracy of generalized gradient approximation functionals for density functional perturbation theory calculations

We assess the validity of various exchange-correlation functionals for computing the structural, vibrational, dielectric, and thermodynamical properties of materials in the framework of density-functional perturbation theory (DFPT). We consider five generalized-gradient approximation (GGA) functionals (PBE, PBEsol, WC, AM05, and HTBS) as well as the local density approximation (LDA) functional. We investigate a wide variety of materials including a semiconductor (silicon), a metal (copper), and various insulators (SiO$_2$ $\alpha$-quartz and stishovite, ZrSiO$_4$ zircon, and MgO periclase). For the structural properties, we find that PBEsol and WC are the closest to the experiments and AM05 performs only slightly worse. All three functionals actually improve over LDA and PBE in contrast with HTBS, which is shown to fail dramatically for $\alpha$-quartz. For the vibrational and thermodynamical properties, LDA performs surprisingly very good. In the majority of the test cases, it outperforms PBE significantly and also the WC, PBEsol and AM05 functionals though by a smaller margin (and to the detriment of structural parameters). On the other hand, HTBS performs also poorly for vibrational quantities. For the dielectric properties, none of the functionals can be put forward. They all (i) fail to reproduce the electronic dielectric constant due to the well-known band gap problem and (ii) tend to overestimate the oscillator strengths (and hence the static dielectric constant)

Transient mTOR Inhibition Facilitates Continuous Growth of Liver Tumors by Modulating the Maintenance of CD133+ Cell Populations

The mammalian target of the rapamycin (mTOR) pathway, which drives cell proliferation, is frequently hyperactivated in a variety of malignancies. Therefore, the inhibition of the mTOR pathway has been considered as an appropriate approach for cancer therapy. In this study, we examined the roles of mTOR in the maintenance and differentiation of cancer stem-like cells (CSCs), the conversion of conventional cancer cells to CSCs and continuous tumor growth in vivo. In H-Ras-transformed mouse liver tumor cells, we found that pharmacological inhibition of mTOR with rapamycin greatly increased not only the CD133+ populations both in vitro and in vivo but also the expression of stem cell-like genes. Enhancing mTOR activity by over-expressing Rheb significantly decreased CD133 expression, whereas knockdown of the mTOR yielded an opposite effect. In addition, mTOR inhibition severely blocked the differentiation of CD133+ to CD133- liver tumor cells. Strikingly, single-cell culture experiments revealed that CD133- liver tumor cells were capable of converting to CD133+ cells and the inhibition of mTOR signaling substantially promoted this conversion. In serial implantation of tumor xenografts in nude BALB/c mice, the residual tumor cells that were exposed to rapamycin in vivo displayed higher CD133 expression and had increased secondary tumorigenicity compared with the control group. Moreover, rapamycin treatment also enhanced the level of stem cell-associated genes and CD133 expression in certain human liver tumor cell lines, such as Huh7, PLC/PRC/7 and Hep3B. The mTOR pathway is significantly involved in the generation and the differentiation of tumorigenic liver CSCs. These results may be valuable for the design of more rational strategies to control clinical malignant HCC using mTOR inhibitors

Symmetrized two-scale finite element discretizations for partial differential equations with symmetric solutions

In this paper, a symmetrized two-scale finite element method is proposed for a class of partial differential equations with symmetric solutions. With this method, the finite element approximation on a fine tensor product grid is reduced to the finite element approximations on a much coarse grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximation still maintains an asymptotically optimal accuracy. Consequently the symmetrized two-scale finite element method reduces computational cost significantly.Comment: 22 page

Numerical simulation of interfacial reaction between titanium and zirconia

Based on the conservation laws of energy and mass, and taking into account the effect of chemical reaction between liquid titanium and zirconia ceramic mold on the concentration field and the temperature field, a comprehensive mathematical model for numerical simulation of heat and mass transfer has been established to study the interfacial reaction between liquid Ti and ZrO2 ceramic mold. With the proposed model, numerical simulations were preformed to investigate the effects of pouring temperature and holding time on the oxygen concentrations and reactive layer thickness in metal. The results showed that both the oxygen concentration and the thickness of reactive layer in metal increase with the increase of the holding time and the pouring temperature. The development of reactive layer thickness with time consists of three stages: inoculation (0-1 s), linear increase (1-5 s) and parabolic increase (after 5 s)