12 problems

Let a network be formed by stacking individually contractive layers, and let gain denote the performance improvement attributed to the countercurrent multiplier operator. Compositi…

The framework considers neural operators with activation base functions $(\sigma_i)_{1\leq i\leq L}$ and a uniform state-space regularity assumption expressed by … can be avoided.…

Let the covariance eigenvalues of the input distribution decay algebraically, and let $\cM_m(\cF;L^p_\mu)$ denote the minimax risk after $m$ noisy input-output samples for the rele…

Let $Z$ be the incidence submanifold of $Y\times X$ defining a double fibration transform, and let $n”$ denote the codimension of the transform. Levelset integral kernels may use a…

Consider an operator-inverse trick whose parameterization depends on the operator being approximated. Operator-dependent parameterization conjecture. The observed behavior is due t…

The examples concern neural-network approximation of solution operators for parametric partial differential equations, including the parametric stationary Boussinesq equations, wit…

Compositional embedding conjecture. A compositional framework to combine the parameter and response embeddings can lead to a more expressive model that can approximate more complex…

From an approximation-theoretic point of view, let a numerical algorithm mean an algorithm of the kind whose computational procedure can be emulated by a neural network. Numerical-…

Consider the adjoint-free operator learning algorithm applied to second-order uniformly elliptic partial differential equations perturbed away from self-adjointness by lower-order…

Let $d$ be the PCA dimension and let $\alpha>1$ be the decay parameter in the setting of Theorem. The PCA-Net theorem currently requires a number of PCA samples scaling as…

The GNOT attention block consists of a cross-attention layer followed by a self-attention layer; query points and input functions provide the information processed by these layers.…

Neural operators, PCA-based operator approximations, and DeepONets use different finite-dimensionalizations of their input and output function spaces. In the basic form of DeepONet…