Para-CFlows: $C^k$-universal diffeomorphism approximators as superior neural surrogates

Invertible neural networks based on Coupling Flows (CFlows) have various applications such as image synthesis and data compression. The approximation universality for CFlows is of paramount importance to ensure the model expressiveness. In this paper, we prove that CFlows}can approximate any diffeomorphism in $C^k$-norm if its layers can approximate certain single-coordinate transforms. Specifically, we derive that a composition of affine coupling layers and invertible linear transforms achieves this universality. Furthermore, in parametric cases where the diffeomorphism depends on some extra parameters, we prove the corresponding approximation theorems for parametric coupling flows named Para-CFlows. In practice, we apply Para-CFlows as a neural surrogate model in contextual Bayesian optimization tasks, to demonstrate its superiority over other neural surrogate models in terms of optimization performance and gradient approximations.