Structure-preserving deep operator networks to the Boltzmann equation
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This talk explores the application of deep operator networks (DeepONets) in learning the collision operator of the Boltzmann equation, which models the distribution evolution of particles with binary collisions in a dilute gas. The Boltzmann collision operator is a nonlinear, nonlocal, high-dimensional operator, hence the cost of approximating the operator is very high. In the meantime, it processes essential structural properties. Thus, a good numerical method should be able to preserve the properties. In this work, we focus on the entropy dissipation and design several DeepONet surrogates for the Boltzmann operator that ensure entropy dissipation. Based on the gradient flow structure of the Boltzmann equation, it suffices to approximate the symmetric positive semi-definite Onsager operator. We develop three DeepONet modifications while the positive semi-definiteness is ensured by learning a Cholesky decomposition. The entropy-based sampling method is used to generate training data, and we validate the proposed DeepONets by the linear Boltzmann equation with the Henyey-Greenstein kernel.
