In this paper, we define lower dimensional volumes of spin manifolds with boundary. We compute the lower dimensional volume Vol (1,3) 6 for 6-dimensional spin manifolds with boundary and derive the gravity on boundary from the noncommutative residue associated with Dirac operators. For 6-dimensional manifolds with boundary, we also get a Kastler-Kalau-Walze type theorem for a general fourth order operator. C 2015 AIP Publishing LLC. [http://dx.

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Summary While deep learning has achieved great success in solving partial differential equations (PDEs) that accurately describe engineering systems, it remains a big challenge to obtain efficient and accurate solutions for complex problems instead of traditional numerical simulation. In the field of reservoir engineering, the current mainstream machine learning methods have been successfully applied. However, these popular methods cannot directly solve the problem of 2D two-phase oil/water PDEs well, which is the core of reservoir numerical simulation. Fourier neural operator (FNO) is a recently proposed high-efficiency PDE solution architecture that overcomes the shortcomings of the above popular methods, which can handle this type of PDE problem well in our work. In this paper, a deep-learning-based model is developed to solve three categories of problems controlled by the subsurface 2D oil/water two-phase flow PDE based on the FNO. For this complex engineering equation, we consider many factors, select characteristic variables, increase the dimension channel, expand the network structure, and realize the solution of the engineering problem. The first category is to predict the distribution of saturation and pressure fields by PDE parameters. The second category is the prediction of time series. The third category is for the inverse problem. It has achieved good results on both forward and inverse problems. The network uses fast Fourier transform (FFT) to extract PDE information in Fourier space to approximate differential operators, making the network faster and with greater physics significance. The model is mesh-independent and has good generalization, which also shows superresolution. Compared to the original FNO, we improve the network structure, add physical constraints to deal with boundary conditions (BCs), and use a shape matrix to control irregular boundaries. Also, we have improved the FFT module to make the transformation smoother. Compared with advanced deep learning-based solvers at different resolutions, the results show that this model overcomes some shortcomings of popular algorithms such as physics-informed neural networks (PINNs) and fully convolutional network (FCN) and has stronger accuracy and applicability. Our work has great potential in the replacement of traditional numerical methods with neural networks for reservoir numerical simulation.

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