The signature kernel [9] 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 dynamically truncated recursive local power series expansions. Building on the characterization of the signature kernel as the solution of a Goursat PDE [16], 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 local Goursat PDEs by recursively propagating boundary conditions along a directed graph via topological ordering, with dynamic truncation adaptively terminating each local power series expansion when coefficients fall below machine precision, striking an effective balance between computational cost and accuracy. This method achieves substantial performance improvements over state-of-the-art approaches for computing the signature kernel, providing (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 half a million points or more) on a single GPU. These advantages make our method particularly well-suited for rough-path-assisted machine learning, financial modeling, and signal processing applications that involve very long and highly volatile data. 1 Introduction Time series data is ubiquitous in contemporary data science and machine learning, appearing in diverse applications such as satellite communication, radio astronomy, healthcare monitoring, climate analysis, and language or video processing, among many others [20]. The sequential nature of this data presents unique challenges, as it is characterised by temporal dependencies and resulting structural patterns that must be captured efficiently to model and predict time-dependent systems and phenomena with accuracy.

Following the success of deep learning in computer vision, deep neural networks (DNNs) have now found their way to a wide range of imaging inverse problems [3, 19, 20]. In some applications, learning the distribution of images from data is the only option. In others, existing methods based on hand-crafted priors are well established. Magnetic resonance imaging (MRI) reconstruction, for which sparsity-based methods have been highly successful, is an example of the latter [17]. However, recent work suggests that image quality can be improved and computation times shortened significantly by the use of DNNs in MRI reconstruction [9]. At the same time, it is well known that DNNs trained for image classification admit socalled adversarial examples--images that have been altered in minor but very specific ways to change the label predicted by the network [4, 22]. In [2], it was discovered that DNNs used in inverse problems (MRI and computed tomography) exhibit similar behavior. Namely, the authors show that perturbing the measurements slightly can lead to undesirable artifacts in the image reconstructed by the network and that the same perturbations do not cause problems for state-of-the-art compressed sensing methods. On the other hand, [13] shows quantitatively that DNNs can be made robust, to a comparable level with total variation (TV) minimization, by injecting statistical noise to the measurement data during training.

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 possibly multi-scale solutions and are observed to significantly outperform baselines, paving the way for an alternative framework for robust and accurate operator learning. Our code is publicly available at https://github.com/bogdanraonic3/ConvolutionalNeuralOperator

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.

While deep neural networks have proven to be a powerful tool for many recognition and classification tasks, their stability properties are still not well understood. In the past, image classifiers have been shown to be vulnerable to so-called adversarial attacks, which are created by additively perturbing the correctly classified image. In this paper, we propose the ADef algorithm to construct a different kind of adversarial attack created by iteratively applying small deformations to the image, found through a gradient descent step. We demonstrate our results on MNIST with a convolutional neural network and on ImageNet with Inception-v3 and ResNet-101.