Analysis of the Generalization Error of deep learning based on Randomized Quasi-Monte Carlo for Solving Linear Kolmogorov PDEs

Deep learning algorithms have been widely used to solve linear Kolmogorov partial differential equations~(PDEs) in high dimensions, where the loss function is defined as a mathematical expectation. We propose to use the randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC) method for computing the loss function. In theory, we decompose the error from empirical risk minimization~(ERM) into the generalization error and the approximation error. Notably, the approximation error is independent of the sampling methods. We prove that the convergence order of the mean generalization error for the RQMC method is $O(n^{-1+\epsilon})$ for arbitrarily small $\epsilon>0$, while for the MC method it is $O(n^{-1/2+\epsilon})$ for arbitrarily small $\epsilon>0$. Consequently, we find that the overall error for the RQMC method is asymptotically smaller than that for the MC method as $n$ increases. Our numerical experiments show that the algorithm based on the RQMC method consistently achieves smaller relative $L^{2}$ error than that based on the MC method

A generalized switched-capacitor step-up multi-level inverter employing single DC source

In this paper, a new generalized step-up multilevel DC-AC converter is proposed, which is suitable for applications with low-voltage input sources such as photovoltaic power generation and electric vehicles. This inverter can achieve a high voltage gain by controlling the series-parallel conversion of DC power supply and capacitors. Only one DC voltage source and a few power devices are employed. The maximum output voltage and the number of output levels can be further increased through the switched-capacitor unit's extension and the submodule cascaded extension. Moreover, the capacitor voltages are self-balanced without complicated voltage control circuits. The complementary operating mechanism between each pair of switches simplifies the modulation algorithm. The inductive-load ability is fully taken into account in the proposed inverter. Additionally, a remarkable characteristic of the inverter is that the charging and discharging states among different capacitors are synchronous, which reduces the voltage ripple of the frontend capacitors. The circuit structure, the working principle, the modulation strategy, the capacitors and losses analysis are presented in detail. Afterwards, the advantages of the proposed inverter are analyzed by comparing with other recently proposed inverters. Finally, the steady-state and dynamic performance of the proposed inverter is verified and validated by simulation and experiment

TAB Series-Resonant DC-DC Converter and Multi-Phase-Shift Based Global Optimization Modulation

In this paper, a triple-active-bridge resonant dc-dc converter with the ability of topology-level power decoupling is proposed. The power coupling between the two dc ports is eliminated by adding a resonant capacitor to the common port. The operation principle and the steady-state power characteristics are analyzed. On this basis, a multi-phase-shift-based global optimization modulation is proposed to minimize the RMS values of the transformer currents in the entire power and voltage range, thus increasing the global efficiency. An experimental prototype is built to verify the correctness and availability of the proposed power decoupling topology and optimized modulation

Interfacial Behavior and Shear Strength of Al-25Si-4Cu-1Mg Joints by Transient Liquid Phase Bonding with Cu as Interlayer

Spray-formed hypereutectic Al-Si-Cu-Mg alloy is the candidate for automotive and aerospace industries due to its superior wear resistance, lower thermal expansion coefficient and density, and higher thermal conductivity. This paper aims to investigate the bonding properties of hypereutectic Al-25Si-4Cu-1Mg alloys using the transient liquid phase (TLP) method with Cu as an interlayer. To obtain the suitable bonding parameters, the interfacial microstructure and shear strength of Al-25Si-4Cu-1Mg joints were investigated with the effect of different bonding temperatures and holding times. The results showed that TLP bonding between Al-Si-Mg-Cu alloy was mainly realized by large amounts of Al2Cu intermetallic compounds (IMCs), primary Si and α-Al phases. With the brazing temperature increasing, the width of the brazing seam gradually increased, and the voids began to be produced. With the holding time increasing, θ-Al2Cu phases approached into the base metal and Si particles in the brazing seam were obviously coarsened. With the formation of θ-Al2Cu phases into the base metal, more Si particles were segregated at the interface between brazing seam and base metal, and the shear test confirmed that it was the weakest bonding location. Finally, the effect of bonding parameters on the joint strength indicated that the joint brazed at 540 °C for 7.5 min presented the best shear performance with the shear strength reaching 75 MPa because the size of Si particles in the brazing seam was closest to the size of Si particles in base metal under this parameter

Interfacial Reaction and Mechanical Properties of Sn-Bi Solder joints

Sn-Bi solder with different Bi content can realize a low-to-medium-to-high soldering process. To obtain the effect of Bi content in Sn-Bi solder on the microstructure of solder, interfacial behaviors in solder joints with Cu and the joints strength, five Sn-Bi solders including Sn-5Bi and Sn-15Bi solid solution, Sn-30Bi and Sn-45Bi hypoeutectic and Sn-58Bi eutectic were selected in this work. The microstructure, interfacial reaction under soldering and subsequent aging and the shear properties of Sn-Bi solder joints were studied. Bi content in Sn-Bi solder had an obvious effect on the microstructure and the distribution of Bi phases. Solid solution Sn-Bi solder was composed of the β-Sn phases embedded with fine Bi particles, while hypoeutectic Sn-Bi solder was composed of the primary β-Sn phases and Sn-Bi eutectic structure from networked Sn and Bi phases, and eutectic Sn-Bi solder was mainly composed of a eutectic structure from short striped Sn and Bi phases. During soldering with Cu, the increase on Bi content in Sn-Bi solder slightly increased the interfacial Cu6Sn5 intermetallic compound (IMC)thickness, gradually flattened the IMC morphology, and promoted the accumulation of more Bi atoms to interfacial Cu6Sn5 IMC. During the subsequent aging, the growth rate of the IMC layer at the interface of Sn-Bi solder/Cu rapidly increased from solid solution Sn-Bi solder to hypoeutectic Sn-Bi solder, and then slightly decreased for Sn-58Bi solder joints. The accumulation of Bi atoms at the interface promoted the rapid growth of interfacial Cu6Sn5 IMC layer in hypoeutectic or eutectic Sn-Bi solder through blocking the formation of Cu6Sn5 in solder matrix and the transition from Cu6Sn5 to Cu3Sn. Ball shear tests on Sn-Bi as-soldered joints showed that the increase of Bi content in Sn-Bi deteriorated the shear strength of solder joints. The addition of Bi into Sn solder was also inclined to produce brittle morphology with interfacial fracture, which suggests that the addition of Bi increased the shear resistance strength of Sn-Bi solder

Implementation and Analysis of An Efficient Soft-Switching Battery Wireless Charger with Re-Configurable Rectifier

To significantly reduce switching loss of the high frequency inverter in battery wireless charger (BWC), improve the soft-switching stability and system reliability, a novel soft-switching BWC with low loss auxiliary network and reconfigurable rectifier is presented. The presented soft-switching BWC can implement the smooth switching between constant current mode and constant voltage mode without frequency switching and relays, which is beneficial to enhancing the system stability. With the help of a novel low energy consumption auxiliary circuit, the power switch tubes in the H-bridge inverter can implement zero-voltage soft switch- on and zero-current soft switch- off in the full range, completely eliminating switching loss of the inverter. The proposed scheme can enhance the system efficiency while refraining from the soft-switching range limitation and soft-switching failure caused by the system parameter perturbation. Finally, a 1-kW soft-switching BWC principle prototype is developed to validate the effectiveness and correctness of the proposed scheme

Microstructural Evolution and Tensile Properties of Sn-Ag-Cu Mixed with Sn-Pb Solder Alloys

The effect of incorporating eutectic Sn-Pb solder with Sn-3.0Ag-0.5Cu (SAC) Pb-free solder on the microstructure and tensile properties of the mixed alloys was investigated. Alloys containing 100, 75, 50, 25, 20, 15, 10, 5 and 0 wt% SAC, with the balance being Sn-37Pb eutectic solder alloy, were prepared and characterized. Optical and scanning electron microscopy were used to analyze the microstructures while “mini-tensile” test specimens were fabricated and tested to determine mechanical properties at the mm length scale, more closely matching that of the solder joints. Microstructural analysis indicated that a Pb-rich phase formed and was uniformly distributed at the boundary between the Sn-rich grains or between the Sn-rich and the intermetallic compounds in the solder. Tensile results showed that mixing of the alloys resulted in an increase in both the yield and the ultimate tensile strength compared to the original solders, with the 50% SAC-50% Sn-Pb mixture having the highest measured strength. Initial investigations indicate the formation and distribution of a Pb-rich phase in the mixed solder alloys as the source of the strengthening mechanism