A Hyperbolic Neural Closure for M1 Radiation Transfer
Authors: Bongseok Kim, Jiahao Zhang, Johannes Krotz, Dinshaw Balsara, Ryan McClarren, Guang Lin
Organizations: School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA · Department of Mathematics, Purdue University, West Lafayette, IN, USA · Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, IN, USA · Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN, USA · Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA
In radiation transfer simulations, an M1 method achieves substantial computational savings by replacing the full angular transport equation with a low-order moment system. Because this reduced system is not closed, a closure model is required to represent the unknown higher-order moments using lower-order moments. While machine learning (ML)-based closures can improve accuracy beyond classical analytic closures, unconstrained learned closures may produce non-real characteristic speeds and consequently cause numerical solver breakdown. To guarantee real eigenvalues of the Jacobian associated with ML closures, we propose a hyperbolic neural closure for the M1 radiative transfer system. Rather than directly predicting closure terms, we parameterize the Jacobian through two neural networks: (i) a symmetric matrix network and (ii) a strictly convex entropy network whose Hessian defines a positive definite symmetrizer. These components are combined to yield a Jacobian that is similar to a symmetric matrix, thereby ensuring real eigenvalues. The closure is then reconstructed by numerical integration of the learned Jacobian field along a prescribed integration path. Numerical experiments show that the proposed closure not only achieves higher closure accuracy than classical analytic closures, but also improves solution accuracy and remains stable in discontinuous Galerkin simulations for radiative transfer problems.