Leray-Schauder Mappings for Operator Learning

We present an algorithm for learning operators between Banach spaces, based on the use of Leray-Schauder mappings to learn a finite-dimensional approximation of compact subspaces. We show that the resulting method is a universal approximator of (possibly nonlinear) operators. We demonstrate the efficiency of the approach on two benchmark datasets showing it achieves results comparable to state of the art models.

💡 Research Summary

The paper introduces a novel framework for learning operators between Banach spaces that leverages Leray‑Schauder mappings to obtain finite‑dimensional approximations of compact subsets of the domain and codomain. Traditional operator learning approaches such as DeepONet, Fourier Neural Operators (FNO), and recent transformer‑based models first discretize the input and output functions on a fixed grid, then train neural networks on the resulting high‑dimensional vectors. This discretization creates several drawbacks: the learned operator is tied to the training grid, up‑sampling or changing resolution at inference requires additional interpolation, and tokenization can induce step‑function artifacts that degrade generalization.

The authors propose to replace the discretization‑dependent projection with a Leray‑Schauder map (P_n). Given a compact set (K\subset X) and an (\varepsilon)-net ({x_i}{i=1}^n), the map assigns non‑negative weights (\mu_i(x)) based on the distance (|x-x_i|) and normalizes them, thereby defining a nonlinear projection onto the span (E_n=\text{span}{x_i}). The key theoretical contribution is Theorem 2.1, which shows that for any continuous (possibly nonlinear) operator (T:X\to Y) and any compact (K\subset X), there exist finite‑dimensional subspaces (E_n\subset X), (E_m\subset Y), an isomorphism (\varphi_k) between each subspace and (\mathbb{R}^k), and a neural network (f{n,m}) such that \