An adaptive optimal selection approach of the Mixture-of-Experts model embedded with PINNs for one-dimensional hyperbolic conservation laws

In this paper, we propose a method of the mixture-of-experts (MoE) model embedded with physics-informed neural networks (PINNs) for the hyperbolic conservation laws. The issue on solving hyperbolic conservation laws with PINNs is still challenging since the solutions of conservation laws may contain discontinuities. PINNs, as functional approximators, nearly fail in such cases, and numerical solutions for its variants may suffer from various problems. Some specially designed variants of PINNs can be well applied to specific hyperbolic equations, but these models usually pay less attention to the generalization capability, and improvement can be made in computing efficiency. In view of this, we propose the adaptive algorithm that embeds PINNs with different strategies into the MoE model, which the algorithm selects “experts of PINNs” through a gating network, choosing the optimal strategy that every “expert” shows its expertise for different structures of the solution. We prove that the generalization error of the proposed model is not higher than that of any single expert, and the bounds for generalization error are also obtained. The numerical experiment results demonstrate the validity of our model and confirm the algorithm's generalization capability that it is fully adaptable for different equations.