🤖 AI Summary
This work addresses the learning of nonlinear operators between Banach spaces. Methodologically, it introduces the first general approximation framework for operator learning grounded in Leray–Schauder mapping theory—incorporating compact operator theory and finite-dimensional projections—to construct provably convergent approximations within compact subspaces, without assuming any specific parametric form. The framework guarantees uniform approximation for a broad class of nonlinear operators, underpinned by rigorous functional-analytic guarantees. Evaluated on two standard PDE-solving benchmarks, the proposed approach achieves accuracy comparable to state-of-the-art models, confirming both its theoretical soundness and empirical effectiveness. The core contribution is the establishment of the first fixed-point-theoretic paradigm for operator learning, endowed with functional-analytic verifiability: it provides principled, non-empirical justification for approximation capability, bridging deep operator learning with classical nonlinear functional analysis.
📝 Abstract
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.