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![Domain and boundaries of the heat-propagation problem defined on a squared plate with a hole [22].](publication/359276709/figure/fig1/AS:11431281096548144@1668177734434/Domain-and-boundaries-of-the-heat-propagation-problem-defined-on-a-squared-plate-with-a.png)
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We compare the Finite Element Method (FEM) simulation of a standard Partial Differential Equation thermal problem of a plate with a hole with a Neural Network (NN) simulation. The largest deviation from the true solution obtained from FEM (0.015 for a solution on the order of unity) is easily achieved with NN too without much tuning of the hyperpar...
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... As mentioned in Section 2.3, the existing methods for solving PDEs based on FEM require grid partitioning and solving nonlinear equations, which results in high computational costs and difficult technological breakthroughs. Actually, In addition to FEM, Frey et al (2022) and Sacchetti et al (2022) have also discussed the feasibility of using NN to solve PDE, and attempted to use NN to solve PDEs with different boundary conditions and types. Their conclusions indicate that using NN to solve PDEs can achieve more accurate results than FEM, especially when it comes to analyze data with time properties or high-dimensional data, but the network structure of NN is often relatively complex [43][44]. ...
... Actually, In addition to FEM, Frey et al (2022) and Sacchetti et al (2022) have also discussed the feasibility of using NN to solve PDE, and attempted to use NN to solve PDEs with different boundary conditions and types. Their conclusions indicate that using NN to solve PDEs can achieve more accurate results than FEM, especially when it comes to analyze data with time properties or high-dimensional data, but the network structure of NN is often relatively complex [43][44]. ...
The study of road traffic flow theory utilizes physics and applied mathematics to analyze relevant parameters and their relationships quanlitatively and quantitatively, in order to explore their dynamic changes. The fluid dynamics model used for traffic flow analysis is highly favored by scholars due to its solid mathematical foundation and good simulation results. However, existing models have two main shortcomings: firstly, existing research is mostly limited to non-viscoelastic fluid equation or incompressible non-Newtonian fluid equation, making it difficult to accurately describe the viscosity state and micro cluster properties of the actual traffic flow; secondly, the existing non-Newtonian fluid partial differential equations (PDEs) rely heavily on the finite element method (FEM) for solving, requiring higher computational cost, larger storage space, and more constraint conditions. Thus, in this paper, a traffic flow equation based on compressible non-Newtonian fluid has been constructed, and it has been solved by using physical-informed rational neural network (PIRNN) and noise heavy-ball acceleration gradient descent (NHAGD) to ensure learning and training speed and accuracy. Numerical results indicate that the proposed method can truly reflect the gradual change process in the viscosity of traffic flow, and has better solving performance than traditional FEM and physical-informed neural network (PINN) with activation functions; under the same conditions, the prediction error of the proposed method is also smaller than that of traditional traffic flow models.
Differential equations (DEs) are important mathematical models for describing natural phenomena and engineering problems. Finding analytical solutions for DEs has theoretical and practical benefits. However, traditional methods for finding analytical solutions only work for some special forms of DEs, such as separable variables or transformable to ordinary differential equations. For general nonlinear DEs, analytical solutions are often hard to obtain. The current popular method based on neural networks requires a lot of data to train the network and only gives approximate solutions with errors and instability. It is also a black-box model that is not interpretable. To obtain analytical solutions for DEs, this paper proposes a symbolic regression algorithm based on genetic programming with the simplification-pruning operator (SP-GPSR). This method introduces a new operator that can simplify the individual expressions in the population and randomly remove some structures in the formulas. Moreover, this method uses multiple fitness functions that consider the accuracy of the analytic solution satisfying the sampled data and the differential equations. In addition, this algorithm also uses a hybrid optimization technique to improve search efficiency and convergence speed. This paper conducts experiments on two typical classes of DEs. The results show that the proposed method can effectively find analytical solutions for DEs with high accuracy and simplicity.
Transport of volatile organic compounds (VOCs) through porous media with active surfaces takes place in many important applications, such as in cellulose-based materials for packaging. Generally, it is a complex process that combines diffusion with sorption at any time. To date, the data needed to use and validate the mathematical models proposed in literature to describe the mentioned processes are scarce and have not been systematically compiled. As an extension of the model of Ramarao et al. (Dry Technol 21(10):2007–2056, 2003) for the water vapor transport through paper, we propose to describe the transport of VOCs by a nonlinear Fisher–Kolmogorov–Petrovsky–Piskunov equation coupled to a partial differential equation (PDE) for the sorption process. The proposed PDE system contains specific material parameters such as diffusion coefficients and adsorption rates as multiplication factors. Although these parameters are essential for solving the PDEs at a given time scale, not all of the required parameters can be directly deduced from experiments, particularly diffusion coefficients and sorption constants. Therefore, we propose to use experimental concentration data, obtained for the migration of dimethyl sulfoxide (DMSO) through a stack of paper sheets, to infer the sorption constant. These concentrations are considered as the outcome of a model prediction and are inserted into an inverse boundary problem. We employ Physics-Informed Neural Networks (PINNs) to find the underlying sorption constant of DMSO on paper from this inverse problem. We illustrate how to practically combine PINN-based calculations with experimental data to obtain trustworthy transport-related material parameters. Finally we verify the obtained parameter by solving the forward migration problem via PINNs and finite element methods on the relevant time scale and show the satisfactory correspondence between the simulation and experimental results.
With the development of the times, intelligent space simulation technology has gradually emerged in the design of ceramic forms, and the development of modern ceramics has gradually transformed into artistic development. The development of ceramics must conform to the design trend, adjust the rhythm, and seek development opportunities. Organic design opens up a green channel for ceramics and integrates into the lives of today’s people. This paper mainly discusses the external form of ceramics and studies from two levels of curve form and bionic form. Through the intelligent space simulation technology, the shape design of ceramics is studied, and the optimal shape of ceramics is obtained, which enables people to have a better understanding of ceramic art.
