Establish a formal universality theorem for the Perceiver encoder architecture

Prove that the finite-depth Perceiver encoder, together with its self-attention, feedforward, permutation-invariant aggregation, and decoder components, uniformly approximates every continuous permutation-invariant sensor-set function on compact domains under the assumptions used for the Lagrangian attention sampler.

Background

The expressivity argument assumes an encoder-decoder universality property for continuous permutation-invariant functions of finite sensor sets. The paper gives supporting evidence by combining existing universality results for cross attention, self-attention with feedforward networks, and permutation-invariant set aggregation.

However, the cited results are not combined into a single theorem covering the exact finite-depth Perceiver architecture used in the sampler. The missing formal result is therefore an unresolved theoretical component of the universality justification.

References

A formal proof combining all three stages in a single theorem statement does not yet exist in the literature; we treat (EU) as an assumption, supported by the above components.

— Deep operator learning for efficient sampling from invariant measures of stochastic differential equations  (2609.11376 - Guo et al., 10 Sep 2026) in End of subsection “Universal approximation property of the Lagrangian attention neural operator,” Section 4.1