Rates and architectures for learning geometrically non-trivial operators

TL;DR

This paper extends the theory of data-efficient learning of operators to complex geometric transforms like Radon and geodesic ray transforms, showing they avoid the curse of dimensionality and can be learned with few samples using specialized architectures.

Contribution

It introduces a new theoretical framework for learning complex geometric operators, proving superalgebraic error decay and designing universal, stable architectures that encode geometric structure.

Findings

Operators do not suffer from curse of dimensionality.

Error decays faster than any fixed power of training samples.

Geometry-encoded architectures learn from few examples.

Abstract

Deep learning methods have proven capable of recovering operators between high-dimensional spaces, such as solution maps of PDEs and similar objects in mathematical physics, from very few training samples. This phenomenon of data-efficiency has been proven for certain classes of elliptic operators with simple geometry, i.e., operators that do not change the domain of the function or propagate singularities. However, scientific machine learning is commonly used for problems that do involve the propagation of singularities in a priori unknown ways, such as waves, advection, and fluid dynamics. In light of this, we expand the learning theory to include double fibration transforms--geometric integral operators that include generalized Radon and geodesic ray transforms. We prove that this class of operators does not suffer from the curse of dimensionality: the error decays superalgebraically, that is, faster than any fixed power of the reciprocal of the number of training samples. Furthermore, we investigate architectures that explicitly encode the geometry of these transforms, demonstrating that an architecture reminiscent of cross-attention based on levelset methods yields a parameterization that is universal, stable, and learns double fibration transforms from very few training examples. Our results contribute to a rapidly-growing line of theoretical work on learning operators for scientific machine learning.