The signature kernel is a recent state-of-the-art tool for analyzing high-dimensional sequential data, valued for its theoretical guarantees and strong empirical performance. In this paper, we present a novel method for efficiently computing the signature kernel of long, high-dimensional time series via adaptively truncated recursive local power series expansions. Building on the characterization of the signature kernel as the solution of a Goursat PDE, our approach employs tilewise Neumann‐series expansions to derive rapidly converging power series approximations of the signature kernel that are locally defined on subdomains and propagated iteratively across the entire domain of the Goursat solution by exploiting the geometry of the time series. Algorithmically, this involves solving a system of interdependent Goursat PDEs via adaptively truncated local power series expansions and recursive propagation of boundary conditions along a directed graph in a topological ordering. This method strikes an effective balance between computational cost and accuracy, achieving substantial performance improvements over state-of-the-art approaches for computing the signature kernel. It offers (a) adjustable and superior accuracy, even for time series with very high roughness; (b) drastically reduced memory requirements; and (c) scalability to efficiently handle very long time series (e.g., with up to one million data points or more) on a single GPU. As demonstrated in our benchmarks, these advantages make our method particularly well-suited for rough-path-assisted machine learning, financial modeling, and signal processing applications involving very long and highly volatile sequential data.

Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined on paper, neural operators necessitate discretization in the transition to computer implementations. This step can compromise their integrity, often causing them to deviate from the underlying operators. This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues. At its core is the concept of operator aliasing, which measures inconsistency between neural operators and their discrete representations. We explore this for widely-used operator learning techniques. Our findings detail how aliasing introduces errors when handling different discretizations and grids and loss of crucial continuous structures. More generally, this framework not only sheds light on existing challenges but, given its constructive and broad nature, also potentially offers tools for developing new neural operators.

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we present novel adaptations for convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is designed specifically to preserve its underlying continuous nature, even when implemented in a discretized form on a computer. We prove a universality theorem to show that CNOs can approximate operators arising in PDEs to desired accuracy. CNOs are tested on a novel suite of benchmarks, encompassing a diverse set of PDEs with multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning.

Abstract. As interest in deep neural networks (DNNs) for image reconstruction tasks grows, their reliability has been called into question [V. Antun, F. Renna, C. Poon, B. Adcock, and A. C. Hansen, Proc. Natl. Acad. Sci. USA, 117 (2020), pp. 30088–30095; N. M. Gottschling, V. Antun, B. Adcock, and A. C. Hansen, The Troublesome Kernel: Why Deep Learning for Inverse Problems Is Typically Unstable, preprint, arXiv:2001.01258 , 2020]. However, recent work has shown that, compared to total variation (TV) minimization, when appropriately regularized, DNNs show similar robustness to adversarial noise in terms of [Formula: see text]-reconstruction error [M. Genzel, J. Macdonald, and M. März, IEEE Trans. Pattern Anal., 45 (2022), pp. 1119–1134]. We consider a different notion of robustness, using the [Formula: see text]-norm, and argue that localized reconstruction artifacts are a more relevant defect than the [Formula: see text]-error. We create adversarial perturbations to undersampled magnetic resonance imaging measurements (in the frequency domain) which induce severe localized artifacts in the TV-regularized reconstruction. Notably, the same attack method is not as effective against DNN-based reconstruction. Finally, we show that this phenomenon is inherent to reconstruction methods for which exact recovery can be guaranteed, as with compressed sensing reconstructions with [Formula: see text]- or TV-minimization.