Extreme Learning Machine-Assisted Solution of Biharmonic Equations via Its Coupled Schemes

Obtaining the solutions of partial differential equations based on various machine learning methods has drawn more and more attention in the fields of scientific computation and engineering applications. In this work, we first propose a coupled Extreme Learning Machine (called CELM) method incorporated with the physical laws to solve a class of fourth-order biharmonic equations by reformulating it into two well-posed Poisson problems. In addition, some activation functions including tangent, gauss, sine, and trigonometric (sin+cos) functions are introduced to assess our CELM method. Notably, the sine and trigonometric functions demonstrate a remarkable ability to effectively minimize the approximation error of the CELM model. In the end, several numerical experiments are performed to study the initializing approaches for both the weights and biases of the hidden units in our CELM model and explore the required number of hidden units. Numerical results show the proposed CELM algorithm is high-precision and efficient to address the biharmonic equation in both regular and irregular domains

The Overseeing Mother: Revisiting the Frontal-Pose Lady in the Wu Family Shrines in Second Century China

Located in present-day Jiaxiang in Shandong province, the Wu family shrines built during the second century in the Eastern Han dynasty (25–220) were among the best-known works in Chinese art history. Although for centuries scholars have exhaustively studied the pictorial programs, the frontal-pose female image situated on the second floor of the central pavilion carved at the rear wall of the shrines has remained a question. Beginning with the woman’s eyes, this article demonstrates that the image is more than a generic portrait (“hard motif ”), but rather represents “feminine overseeing from above” (“soft motif ”). This synthetic motif combines three different earlier motifs – the frontal-pose hostess enjoying entertainment, the elevated spectator, and the Queen Mother of the West. By creatively fusing the three motifs into one unity, the Jiaxiang artists lent to the frontal-pose lady a unique power: she not only dominated the center of the composition, but also, like a divine being, commanded a unified view of the surroundings on the lofty building, hence echoing the political reality of the empress mother’s “overseeing the court” in the second century during Eastern Han dynasty

Physical informed neural networks with soft and hard boundary constraints for solving advection-diffusion equations using Fourier expansions

Deep learning methods have gained considerable interest in the numerical solution of various partial differential equations (PDEs). One particular focus is on physics-informed neural networks (PINNs), which integrate physical principles into neural networks. This transforms the process of solving PDEs into optimization problems for neural networks. In order to address a collection of advection-diffusion equations (ADE) in a range of difficult circumstances, this paper proposes a novel network structure. This architecture integrates the solver, which is a multi-scale deep neural network (MscaleDNN) utilized in the PINN method, with a hard constraint technique known as HCPINN. This method introduces a revised formulation of the desired solution for advection-diffusion equations (ADE) by utilizing a loss function that incorporates the residuals of the governing equation and penalizes any deviations from the specified boundary and initial constraints. By surpassing the boundary constraints automatically, this method improves the accuracy and efficiency of the PINN technique. To address the ``spectral bias'' phenomenon in neural networks, a subnetwork structure of MscaleDNN and a Fourier-induced activation function are incorporated into the HCPINN, resulting in a hybrid approach called SFHCPINN. The effectiveness of SFHCPINN is demonstrated through various numerical experiments involving advection-diffusion equations (ADE) in different dimensions. The numerical results indicate that SFHCPINN outperforms both standard PINN and its subnetwork version with Fourier feature embedding. It achieves remarkable accuracy and efficiency while effectively handling complex boundary conditions and high-frequency scenarios in ADE.Comment: 2

Solving a class of multi-scale elliptic PDEs by means of Fourier-based mixed physics informed neural networks

Deep neural networks have received widespread attention due to their simplicity and flexibility in the fields of engineering and scientific calculation. In this work, we probe into solving a class of elliptic Partial Differential Equations (PDEs) with multiple scales by means of Fourier-based mixed physics informed neural networks(dubbed FMPINN), the solver of FMPINN is configured as a multi-scale deep neural networks. Unlike the classical PINN method, a dual (flux) variable about the rough coefficient of PDEs is introduced to avoid the ill-condition of neural tangent kernel matrix that resulted from the oscillating coefficient of multi-scale PDEs. Therefore, apart from the physical conservation laws, the discrepancy between the auxiliary variables and the gradients of multi-scale coefficients is incorporated into the cost function, then obtaining a satisfactory solution of PDEs by minimizing the defined loss through some optimization methods. Additionally, a trigonometric activation function is introduced for FMPINN, which is suited for representing the derivatives of complex target functions. Handling the input data by Fourier feature mapping, it will effectively improve the capacity of deep neural networks to solve high-frequency problems. Finally, by introducing several numerical examples of multi-scale problems in various dimensional Euclidean spaces, we validate the efficiency and robustness of the proposed FMPINN algorithm in both low-frequency and high-frequency oscillation cases

High-performance Fe–Si soft magnetic composites with controllable silicate/nano-Fe composite coating

The design of magnetic insulation coating structure has always been a challenge for high-performance soft magnetic composites (SMCs). In this work, we prepared Fe–Si SMCs with silicate/nano-Fe composite coating successfully by in-situ oxidation method combined with spark plasma sintering (SPS). The formation mechanism of the composite coating and its effect on the electro-magnetic properties of Fe–Si SMCs were investigated. The results showed that a uniform Fe2O3 coating can be obtained by reactions between Fe and H2O/O2 during in-situ oxidation process, and became thicker with the increased oxidation time. After sintering, the oxide coating was transformed into a composite coating composed of Fe2SiO4 with excellent insulation and nano-Fe with high ferromagnetism, which resulted from the interfacial reaction between Fe2O3 coating and Fe–Si core. The increased oxidation time led to the gradually thicker composite coating, and resulted in a linear decrease in saturation magnetization, indicating good controllability of the coating. However, excessive oxidation time led to the increased eddy current loss as well as the core loss due to the weakened resistivity. Thus, the Fe–Si SMCs exhibited high saturation magnetic induction (1.66T) and very low core loss (643.9 kW/m3 at 0.1 T/50 kHz) especially when the oxidation time was 1 h

Expression and Bioinformatic Analysis of Ornithine Aminotransferase in Non-small Cell Lung Cancer

Background and objective It has been proven that ornithine aminotransferase (OAT) might play an important role in the oncogenesis and progression of numerous malignant tumors. The aim of this study is to detect the mRNA and protein expression of OAT in non-small cell lung cancer (NSCLC), as well as to analyze the bioinformatic features and binary interactions. Methods OAT mRNA expression was detected in A549 and 16HBE cell lines by reverse transcription-polymerase chain reaction. OAT protein expression was determined in 55 cases of NSCLC and 17 cases of adjacent non-tumor lung tissues by immunohistochemical staining. The bioinformatic features and binary interactions of OAT were analyzed. Gene ontology annotation and signal pathway analysis were performed. Results OAT mRNA expression in A549 cells was 2.85-fold lower than that in 16HBE cells. OAT protein expression was significantly higher in NSCLC tissues than that in adjacent non-tumor lung tissues. A significant difference of OAT protein expression was existed between squamous cell lung cancer and adenocarcinoma (P<0.05), but was not correlated with the gender, age, lymph node metastasis, tumor size, and TNM stages. Bioinformatic analysis suggested that OAT was a highly homologous and stable protein located in the mitochondria. An aminotran-3 domain and several sites of phosphorylation, which may function in signal transduction, gene transcription, and molecular transit, were found. In the 54 selected binary interactions of OAT, TNF and TRAF6 play roles in the NF-κB pathway. Conclusion OAT may play an important role in the oncogenesis and progression of NSCLC. Thus, OAT may be a novel biomarker for the diagnosis of NSCLC or a new target for its treatment