Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs

While physics-informed neural networks (PINNs) have been proven effective for low-dimensional partial differential equations (PDEs), the computational cost remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling Monte Carlo methods for derivative approximation, eliminating the need for costly auto-differentiation. Despite its computational efficiency in high dimensions, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD). We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss and the PDE nonlinearity. We propose tailored bias correction techniques based on the order of PDE nonlinearity. The unbiased RS-PINN allows for a detailed examination of its pros and cons compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, HJB, viscous Burgers', Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.Comment: 21 pages, 5 figure

Global intron retention mediated gene regulation during CD4+ T cell activation.

T cell activation is a well-established model for studying cellular responses to exogenous stimulation. Using strand-specific RNA-seq, we observed that intron retention is prevalent in polyadenylated transcripts in resting CD4(+) T cells and is significantly reduced upon T cell activation. Several lines of evidence suggest that intron-retained transcripts are less stable than fully spliced transcripts. Strikingly, the decrease in intron retention (IR) levels correlate with the increase in steady-state mRNA levels. Further, the majority of the genes upregulated in activated T cells are accompanied by a significant reduction in IR. Of these 1583 genes, 185 genes are predominantly regulated at the IR level, and highly enriched in the proteasome pathway, which is essential for proper T cell proliferation and cytokine release. These observations were corroborated in both human and mouse CD4(+) T cells. Our study revealed a novel post-transcriptional regulatory mechanism that may potentially contribute to coordinated and/or quick cellular responses to extracellular stimuli such as an acute infection

Development of Smart Films of a Chitosan Base and <i>Robusta</i> Coffee Peel Extract for Monitoring the Fermentation Process of Pickles

Smart film is widely used in the field of food packaging. The smart film was prepared by adding anthocyanin-rich Robusta coffee peel (RCP) extract into a chitosan (CS)–glycerol (GL) matrix by a solution-casting method. By changing the content of RCP (0, 10%, 15% and 20%) in the CS–GL film, the related performance indicators of CS–GL–RCP films were studied. The results showed that the CS–GL–RCP films had excellent mechanical properties, and CS–GL–RCP15 film maintained the tensile strength (TS) of 16.69 MPa and an elongation-at-break (EAB) of 18.68% with RCP extract. CS–GL–RCP films had the best UV-vis light barrier property at 200–350 nm and the UV transmittance was close to 0. The microstructure observation results showed that CS–GL–RCP films had a dense and uniform cross section, which proved that the RCP extract had good compatibility with the polymer. In addition, the CS–GL–RCP15 film was pH-sensitive and could exhibit different color changes with different pH solutions. So, the CS–GL–RCP15 film was used to detect the fermentation process of pickles at 20 ± 1 °C for 15 days. The pickles were stored in a round pickle container after the boiling water had cooled. The color of the CS–GL–RCP15 film changed significantly, which was consistent with the change of pickles from fresh to mature. The color of the smart film changed significantly with the maturity of pickles, and the difference of ΔE of film increased to 8.89 (15 Days), which can be seen by the naked eye. Therefore, CS–GL–RCP films prepared in this study provided a new strategy for the development of smart packaging materials

Numerical Simulation of Random Cavitation Suppression Based on Variable NACA Airfoils

In order to suppress the cavitation of an airfoil under random operating conditions, a deformable covering was constructed in the cavitation prone area of the NACA0012 airfoil. By sensing the pressure difference between the inner and outer sides of the airfoil, the covering of the airfoil can be changed adaptively to meet the requirement of suppressing random cavitation of the airfoil. The simulation results show that the cavitation influence range of the airfoil with a shape memory alloy covering can be reduced by more than 70%, and the cavitation is well reduced and suppressed. Moreover, the backflow near the wall of the airfoil was reduced under random working conditions. When the maximum bulge deformation of the covering was between 3&ndash;6 mm, the airfoil produced a cavitation range only on the covering surface of the airfoil, and there was no cavitation erosion on other parts. This method with locally variable airfoil to suppress cavitation provides a good reference value for other hydraulic machinery to suppress cavitation

Numerical Simulation of Random Cavitation Suppression Based on Variable NACA Airfoils

In order to suppress the cavitation of an airfoil under random operating conditions, a deformable covering was constructed in the cavitation prone area of the NACA0012 airfoil. By sensing the pressure difference between the inner and outer sides of the airfoil, the covering of the airfoil can be changed adaptively to meet the requirement of suppressing random cavitation of the airfoil. The simulation results show that the cavitation influence range of the airfoil with a shape memory alloy covering can be reduced by more than 70%, and the cavitation is well reduced and suppressed. Moreover, the backflow near the wall of the airfoil was reduced under random working conditions. When the maximum bulge deformation of the covering was between 3–6 mm, the airfoil produced a cavitation range only on the covering surface of the airfoil, and there was no cavitation erosion on other parts. This method with locally variable airfoil to suppress cavitation provides a good reference value for other hydraulic machinery to suppress cavitation

Research on Flow Field Characteristics in Water Jet Nozzle and Surface Damage Caused by Target Impact

As a new processing method, water jet processing technology has risen rapidly due to its wide range of applications, no pollution, and zero discharge. In this paper, the flow characteristics and failure characteristics of ultra-high-pressure gas-liquid jet in the range of 300 MPa are analyzed by numerical calculation. The research conclusion shows that the jet atomization diffusion is caused by the friction between the liquid medium and the surrounding gas, the mixed flow of broken water droplets and cavitation. The jet diffusion process is essentially the energy exchange process between the jet in the core area and the turbulent flow in the atomization area. The distribution of the turbulent kinetic energy in the atomization area can determine the degree of jet diffusion and the rate of energy decay. The water jet impacted the surface of the target to form a crater-like annular erosion pit. With the increase of the impact pressure, the deformation showed an overall increasing trend, and the increasing trend increased significantly. The central depression of the erosion area is caused by the damage of the material by the stagnation pressure in the core area. The flow characteristics of gas-liquid flow in the process of formation and diffusion in the high-pressure water jet nozzle are explored from the microscopic point of view, and it also provides a theoretical basis for equipment optimization in engineering

Research on Flow Field Characteristics in Water Jet Nozzle and Surface Damage Caused by Target Impact

As a new processing method, water jet processing technology has risen rapidly due to its wide range of applications, no pollution, and zero discharge. In this paper, the flow characteristics and failure characteristics of ultra-high-pressure gas-liquid jet in the range of 300 MPa are analyzed by numerical calculation. The research conclusion shows that the jet atomization diffusion is caused by the friction between the liquid medium and the surrounding gas, the mixed flow of broken water droplets and cavitation. The jet diffusion process is essentially the energy exchange process between the jet in the core area and the turbulent flow in the atomization area. The distribution of the turbulent kinetic energy in the atomization area can determine the degree of jet diffusion and the rate of energy decay. The water jet impacted the surface of the target to form a crater-like annular erosion pit. With the increase of the impact pressure, the deformation showed an overall increasing trend, and the increasing trend increased significantly. The central depression of the erosion area is caused by the damage of the material by the stagnation pressure in the core area. The flow characteristics of gas-liquid flow in the process of formation and diffusion in the high-pressure water jet nozzle are explored from the microscopic point of view, and it also provides a theoretical basis for equipment optimization in engineering

Key Factors, Planning Strategy and Policy for Low-Carbon Transport Development in Developing Cities of China

Exploring key impact factors and their effects on urban residents&rsquo; transport carbon dioxide (CO2) emissions is significant for effective low-carbon transport planning. Researchers face the model uncertainty problem to seek a rational and better explanatory model and the key variables in the model set containing various factors after they are arranged and combined. This paper uses the Bayesian Model Averaging method to solve the above problem, explore the key variables, and determine their relative significance and averaging effects. Beijing, Xi&rsquo;an, and Wuhan are selected as three case cities for their representation of developing Chinese cities. We found that the initial key factor increasing transport emissions is the high dependence on cars, and the second is the geographical location factor that much more suburban residents suffer longer commuting. Developing satellite city rank first for reducing transport emissions due to more local trips with an average short distance, the second is the metro accessibility, and the third is polycentric form. Key planning strategies and policies are proposed: (i) combining policies of car restriction based on vehicle plate number, encouraging clean fuel cars, a carbon tax on oil uses, and rewarding public transit passengers; (ii) fostering subcenters&rsquo; strong industries to develop self-contained polycentric structures and satellite cities, and forming employment and life circle within 5 km radius; and (iii) integrating bus and rail transit services in the peripheral areas and suburbs and increasing the integration level of muti-modes transferring in transport hubs. The findings will offer empirical evidence and reference value in developing cities globally