Machine Learning Moment Closure for the Boltzmann Transport Equation: Enforcing Hyperbolicity

Bongseok Kim, Jiahao Zhang, Guang Lin

The Boltzmann transport equation describes particle transport across collisional regimes, but resolving its velocity dependence is expensive. Moment methods reduce this cost by evolving finitely many velocity moments, requiring a closure for unresolved higher-order moments. Data-driven closures can represent these moments accurately, yet without structural constraints they may produce complex characteristic speeds and unstable simulations. Learned Boltzmann closures with guaranteed hyperbolicity have so far been formulated in one spatial dimension; in multiple dimensions, hyperbolicity must hold in every direction simultaneously. We develop, to the best of our knowledge, the first data-driven closure for multidimensional Hermite moment systems of the linear Boltzmann equation that is hyperbolic by construction. The network learns a symmetric positive-definite symmetrizer shared by all spatial directions and restricted to a subspace that preserves the known lower-order transport equations; the unresolved rows follow from a symmetric completion. Every directional Jacobian thus has real eigenvalues and a complete set of eigenvectors for every network output and at any truncation…