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This work rigorously derives the continuous and discrete adjoint lattice Boltzmann methods (LBMs) for open flow systems, with a special focus on the resulting adjoint boundary conditions. It is found that the adjoint collision and adjoint streaming are exactly the same for the two adjoint methods, while the expressions, the objective derivative term and the execution order of the adjoint boundary conditions are totally different. The numerical performances of the two adjoint methods are comprehensively compared by performing sensitivity analysis on the 2D and 3D pipe bend cases. It is found that inconsistency or singularity exists in the continuous adjoint boundary conditions, leading to much inferior numerical stability of the adjoint solution and obvious numerical error in sensitivity. In contrast, the discrete adjoint LBM with fully consistent adjoint boundary conditions can always achieve exact sensitivity results and the theoretically highest numerical stability, without any increase of computational cost, thus it becomes a perfect solution to the inconsistency and instability issues in the continuous adjoint LBM. 3D microchannel heat sinks under various Reynolds numbers are also designed, and the esthetic and physically reasonable optimized designs are obtained, demonstrating the necessity and versatility of the presented discrete adjoint LBM., Comment: 29 pages, 12 figures

Let $d\geq1$, $\Omega$ be a bounded domain of a smooth complete Riemannian d-manifold M, and $\mu$ be a positive finite Borel measure with compact support in $\overline{\Omega}$. We prove the Courant nodal domain theorem for the eigenfunctions of Kre\u{i}n-Feller operator $\Delta_{\mu}$ under the assumption that such eigenfunctions are continuous on $\overline{\Omega}$. For $d\geq2$, We prove that on a bounded domain $\Omega\subset M$ with smooth boundary and on which the Green's function of the Laplace-Beltrami operator exists, the eigenfunctions of $\Delta_{\mu}$ are continuous on $\Omega$. We also prove that if M is compact and $\partial M=\emptyset$, then the eigenfuctions of $\Delta_{\mu}$ are continuous on M.

Let $\mu$ be a compactly supported positive finite Borel measure on $\R^{d}$. Let $0<\lambda_{1}\leq\lambda_{2}\leq\ldots$ be eigenvalues of the Kre$\breve{{\i}}$n-Feller operator $\Delta_{\mu}$. We prove that, on a bounded domain, the nodal set of a continuous $\lambda_{n}$-eigenfunction of a Kre$\breve{{\i}}$n-Feller operator divides the domain into at least 2 and at most $n+r-1$ subdomains, where $r$ is the multiplicity of $\lambda_{n}$. This work generalizes the nodal set theorem of the classical Laplace operator to Kre$\breve{{\i}}$n-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Kre$\breve{{\i}}$n-Feller operator are continuous.

Topology optimization (TO) is a powerful method to design innovative structures with improved heat transfer performance. In the present study, a multi-fidelity TO method with a delicately defined objective function is developed for flow boiling heat transfer in microchannels. Low-fidelity TO is conducted for the reduced-order process of single-phase laminar convective heat transfer, which generates a set of structure candidates for subsequent high-fidelity evaluation of flow boiling heat transfer. To avoid the possible iteration between the low-fidelity TO and high-fidelity evaluation which leads to inefficient solution of the multi-fidelity TO, distributions of velocity, temperature and two-phase in microchannels with single-phase and/or flow boiling heat transfer are investigated and compared in detail, based on which a new objective function is delicately defined, which can be employed in the low-fidelity TO yet can stand for the performance of the high-fidelity problem. With the help of the new objective function, the efficiency of the multi-fidelity TO is significantly improved and TO structures are designed with hot spots eliminated, thermal resistance reduced and temperature uniformity improved. The present work provides a new method for TO of complicated heat and mass transfer problems. Keywords: topology optimization, flow boiling, multi-fidelity optimization, microchannels, convective heat transfer

Multiscale phenomena manifest across various scientific domains, presenting a ubiquitous challenge in accurately and effectively simulating multiscale dynamics in complex systems. In this paper, a novel decoupling solving paradigm is proposed through modelling large-scale dynamics independently and treating small-scale dynamics as a slaved system. A Spectral Physics-informed Neural Network (PINN) is developed to characterize the small-scale system in an efficient and accurate way, addressing the challenges posed by the representation of multiscale dynamics in neural networks. The effectiveness of the method is demonstrated through extensive numerical experiments, including one-dimensional Kuramot-Sivashinsky equation, two- and three-dimensional Navier-Stokes equations, showcasing its versatility in addressing problems of fluid dynamics. Furthermore, we also delve into the application of the proposed approach to more complex problems, including non-uniform meshes, complex geometries, large-scale data with noise, and high-dimensional small-scale dynamics. The discussions about these scenarios contribute to a comprehensive understanding of the method's capabilities and limitations. By enabling the acquisition of large-scale data with minimal computational demands, coupled with the efficient and accurate characterization of small-scale dynamics via Spectral PINN, our approach offers a valuable and promising approach for researchers seeking to tackle multiscale phenomena effectively.

A full-quantum approach is used to study quantum nonlinear properties of a compound Michelson-Sagnac interferometer optomechanical system. The effective Hamiltonian shows that both dissipative and dispersive couplings possess imaginary- and real-Kerr nonlinearities. And unexpectedly, the nonlinearities caused by the dissipative coupling have non-Hermitian Hamiltonian-like properties. It can protect the quantum nature of the dispersive coupling beyond the traditional dissipation of the system. This protection mechanism allows the system to exhibit strong quantum nonlinear effects in the parameter region of the hyperbolic function $J^2 = \Delta_c \Delta_e$. Moreover, we can obtain strong anti-bunching effects whether in strong or weak coupling regimes with the help of the dispersive and dissipative couplings jointly. It may provide a new perspective to experimentally realize and study the strong quantum nonlinear effects.

Traditional convolutional neural networks are limited to handling Euclidean space data, overlooking the vast realm of real-life scenarios represented as graph data, including transportation networks, social networks, and reference networks. The pivotal step in transferring convolutional neural networks to graph data analysis and processing lies in the construction of graph convolutional operators and graph pooling operators. This comprehensive review article delves into the world of graph convolutional neural networks. Firstly, it elaborates on the fundamentals of graph convolutional neural networks. Subsequently, it elucidates the graph neural network models based on attention mechanisms and autoencoders, summarizing their application in node classification, graph classification, and link prediction along with the associated datasets., Comment: 18pages,4figures

Image super-resolution reconstruction is an important task in the field of image processing technology, which can restore low resolution image to high quality image with high resolution. In recent years, deep learning has been applied in the field of image super-resolution reconstruction. With the continuous development of deep neural network, the quality of the reconstructed images has been greatly improved, but the model complexity has also been increased. In this paper, we propose two lightweight models named as MSwinSR and UGSwinSR based on Swin Transformer. The most important structure in MSwinSR is called Multi-size Swin Transformer Block (MSTB), which mainly contains four parallel multi-head self-attention (MSA) blocks. UGSwinSR combines U-Net and GAN with Swin Transformer. Both of them can reduce the model complexity, but MSwinSR can reach a higher objective quality, while UGSwinSR can reach a higher perceptual quality. The experimental results demonstrate that MSwinSR increases PSNR by $\mathbf{0.07dB}$ compared with the state-of-the-art model SwinIR, while the number of parameters can reduced by $\mathbf{30.68\%}$, and the calculation cost can reduced by $\mathbf{9.936\%}$. UGSwinSR can effectively reduce the amount of calculation of the network, which can reduced by $\mathbf{90.92\%}$ compared with SwinIR.