Title: Neural Functional Singular Value Decomposition for Irregularly Sampled Infinite Dimensional Data
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Functional Singular Value Decomposition, Scientific Machine Learning, Operator Learning

Abstract: This work studies an extension of the singular value decomposition to infinite dimensional spaces by considering neural networks as basis elements.
In contrast to the classical finite dimensional singular value decomposition, this approach is grid-less and can be used in cases where only irregularly sampled data is available and evaluation at arbitrary sample points is required.
To the best of our knowledge, we are the first work to propose a neural rank reduction method that is capable of handling irregularly sampled data.
Our approach is based on a regularized least squares formulation that fits the neural network to the given data while enforcing normalization for each function and orthogonality between pairs of functions.
Performing rank reduction for infinite dimensional operators is particularly interesting for scientific machine learning with focus on predicting the solution of partial differential equations given some boundary condition.
In this context the learned neural basis functions form a linear and finite dimensional approximation of the image of the solution operator.
We demonstrate the efficacy of our algorithm by first learning this approximation based on given irregularly sampled data.
In a second stage we train an artificial neural network as being a coefficient functional for the previously learned basis.