ﻻ يوجد ملخص باللغة العربية
This is the third paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work cite{huang2021gradient}, we proposed an approach to learn the gradient of the unclosed high order moment, which performs much better than learning the moment itself and the conventional $P_N$ closure. However, while the ML moment closure has better accuracy, it is not able to guarantee hyperbolicity and has issues with long time stability. In our second paper cite{huang2021hyperbolic}, we identified a symmetrizer which leads to conditions that enforce that the gradient based ML closure is symmetrizable hyperbolic and stable over long time. The limitation of this approach is that in practice the highest moment can only be related to four, or fewer, lower moments. In this paper, we propose a new method to enforce the hyperbolicity of the ML closure model. Motivated by the observation that the coefficient matrix of the closure system is a lower Hessenberg matrix, we relate its eigenvalues to the roots of an associated polynomial. We design two new neural network architectures based on this relation. The ML closure model resulting from the first neural network is weakly hyperbolic and guarantees the physical characteristic speeds, i.e., the eigenvalues are bounded by the speed of light. The second model is strictly hyperbolic and does not guarantee the boundedness of the eigenvalues. Several benchmark tests including the Gaussian source problem and the two-material problem show the good accuracy, stability and generalizability of our hyperbolic ML closure model.
Physics-informed Machine Learning has recently become attractive for learning physical parameters and features from simulation and observation data. However, most existing methods do not ensure that the physics, such as balance laws (e.g., mass, mome
This article presents a general framework for recovering missing dynamical systems using available data and machine learning techniques. The proposed framework reformulates the prediction problem as a supervised learning problem to approximate a map
Neural networks (NNs) are often used as surrogates or emulators of partial differential equations (PDEs) that describe the dynamics of complex systems. A virtually negligible computational cost of such surrogates renders them an attractive tool for e
Linear kinetic transport equations play a critical role in optical tomography, radiative transfer and neutron transport. The fundamental difficulty hampering their efficient and accurate numerical resolution lies in the high dimensionality of the phy
As a nonlocal extension of continuum mechanics, peridynamics has been widely and effectively applied in different fields where discontinuities in the field variables arise from an initially continuous body. An important component of the constitutive