Implicit graph neural networks (IGNNs) – that solve a fixed-point equilibrium equation using Picard iteration for representation learning – have shown remarkable performance in learning long-range 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 Anderson-accelerated 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.

In this paper, we explore applications of deep learning in statistical physics. We choose the Boltzmann equation as a typical example, where neural networks serve as a closure to its moment system. We present two types of neural networks to embed the convexity of entropy and to preserve the minimum entropy principle and intrinsic mathematical structures of the moment system of the Boltzmann equation. We derive an error bound for the generalization gap of convex neural networks which are trained in Sobolev norm and use the results to construct data sampling methods for neural network training. Numerical experiments demonstrate that the neural entropy closure is significantly faster than classical optimizers while maintaining sufficient accuracy.