Universal Approximation of Operators with Transformers and Neural Integral Operators
This provides foundational theoretical guarantees for using transformers in operator learning, which is incremental as it extends existing universal approximation results to more general settings.
The paper tackles the problem of approximating operators in Banach spaces, showing that transformers and neural integral operators can universally approximate integral operators between Hölder spaces and arbitrary operators between Banach spaces.
We study the universal approximation properties of transformers and neural integral operators for operators in Banach spaces. In particular, we show that the transformer architecture is a universal approximator of integral operators between Hölder spaces. Moreover, we show that a generalized version of neural integral operators, based on the Gavurin integral, are universal approximators of arbitrary operators between Banach spaces. Lastly, we show that a modified version of transformer, which uses Leray-Schauder mappings, is a universal approximator of operators between arbitrary Banach spaces.