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The standard Universal Approximation Theorem for operator neural networks(NNs) holds for arbitrary width and bounded depth. Here, we prove that operatorNNs of bounded width and arbitrary depth are universal approximators forcontinuous nonlinear operators. In our main result, we prove that fornon-polynomial activation functions that are continuously differentiable at apoint with a nonzero derivative, one can construct an operator NN of widthfive, whose inputs are real numbers with finite decimal representations, thatis arbitrarily close to any given continuous nonlinear operator. We derive ananalogous result for non-affine polynomial activation functions. We also showthat depth has theoretical advantages by constructing operator ReLU NNs ofdepth $2k^3+8$ and constant width that cannot be well-approximated by anyoperator ReLU NN of depth $k$, unless its width is exponential in $k$.

Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks