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In this paper we show global well-posedness near vacuum for the binary–ternary Boltzmann equation. The binary–ternary Boltzmann equation provides a correction term to the classical Boltzmann equation, taking into account both binary and ternary interactions of particles, and may serve as a more accurate description model for denser gases in non-equilibrium. Well-posedness of the classical Boltzmann equation and, independently, the purely ternary Boltzmann equation follow as special cases. To prove global well-posedness, we use a Kaniel–Shinbrot iteration and related work to approximate the solution of the non-linear equation by monotone sequences of super- solutions and subsolutions. This analysis required establishing new convolution-type estimates to control the contribution of the ternary collisional operator to the model. We show that the ternary operator allows consideration of softer potentials than the one binary operator, and consequently our solution to the ternary correction of the Boltzmann equation preserves all the properties of the binary interactions solution. These results are novel for collisional operators of monoatomic gases with either hard or soft potentials that model both binary and ternary interactions.
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Implicit Graph Neural Networks: A Monotone Operator Viewpoint
Implicit graph neural networks (IGNNs) – that solve a fixed-point equilibrium equation using Picard iteration for representation learning – have shown remarkable performance in learning longrange dependencies (LRD) in the underlying graphs. However, IGNNs suffer from several issues, including 1) their expressivity is limited by their parameterizations for the well-posedness guarantee, 2) IGNNs are unstable in learning LRD, and 3) IGNNs become computationally inefficient when learning LRD. In this paper, we provide a new well-posedness characterization for IGNNs leveraging monotone operator theory, resulting in a much more expressive parameterization than the existing one. We also propose an orthogonal parameterization for IGNN based on Cayley transform to stabilize learning LRD. Furthermore, we leverage Andersonaccelerated operator splitting schemes to efficiently solve for the fixed point of the equilibrium equation of IGNN with monotone or orthogonal parameterization. We verify the computational efficiency and accuracy of the new models over existing IGNNs on various graph learning tasks at both graph and node levels. Code is available at https://github.com/ Utah-Math-Data-Science/MIGNN
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- Award ID(s):
- 1952339
- PAR ID:
- 10529108
- Publisher / Repository:
- Proceedings of Machine Learning Research
- Date Published:
- Volume:
- 202
- Page Range / eLocation ID:
- 1521-1548
- Format(s):
- Medium: X
- Location:
- https://proceedings.mlr.press/v202/baker23a.html
- Sponsoring Org:
- National Science Foundation
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