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Neural networks excel at discovering statistical patterns in high-dimensional data sets. In practice, higher-order cumulants, which quantify the non-Gaussian correlations between three or more variables, are particularly important for the performance of neural networks. But how efficient are neural networks at extracting features from higher-order cumulants? We study this question in the spiked cumulant model, where the statistician needs to recover a privileged direction or "spike" from the order-$p\ge 4$ cumulants of~$d$-dimensional inputs. We first characterise the fundamental statistical and computational limits of recovering the spike by analysing the number of samples~$n$ required to strongly distinguish between inputs from the spiked cumulant model and isotropic Gaussian inputs. We find that statistical distinguishability requires $n\gtrsim d$ samples, while distinguishing the two distributions in polynomial time requires $n \gtrsim d^2$ samples for a wide class of algorithms, i.e. those covered by the low-degree conjecture. These results suggest the existence of a wide statistical-to-computational gap in this problem. Numerical experiments show that neural networks learn to distinguish the two distributions with quadratic sample complexity, while "lazy" methods like random features are not better than random guessing in this regime. Our results show that neural networks extract information from higher-order correlations in the spiked cumulant model efficiently, and reveal a large gap in the amount of data required by neural networks and random features to learn from higher-order cumulants.

A posteriori reduced-order models, e.g. proper orthogonal decomposition, are essential to affordably tackle realistic parametric problems. They rely on a trustful training set, that is a family of full-order solutions (snapshots) representative of all possible outcomes of the parametric problem. Having such a rich collection of snapshots is not, in many cases, computationally viable. A strategy for data augmentation, designed for parametric laminar incompressible flows, is proposed to enrich poorly populated training sets. The goal is to include in the new, artificial snapshots emerging features, not present in the original basis, that do enhance the quality of the reduced-order solution. The methodologies devised are based on exploiting basic physical principles, such as mass and momentum conservation, to devise physically-relevant, artificial snapshots at a fraction of the cost of additional full-order solutions. Interestingly, the numerical results show that the ideas exploiting only mass conservation (i.e., incompressibility) are not producing significant added value with respect to the standard linear combinations of snapshots. Conversely, accounting for the linearized momentum balance via the Oseen equation does improve the quality of the resulting approximation and therefore is an effective data augmentation strategy in the framework of viscous incompressible laminar flows.

Microring resonators (MRRs) are promising devices for time-delay photonic reservoir computing, but the impact of the different physical effects taking place in the MRRs on the reservoir computing performance is yet to be fully understood. We numerically analyze the impact of linear losses as well as thermo-optic and free-carrier effects relaxation times on the prediction error of the time-series task NARMA-10. We demonstrate the existence of three regions, defined by the input power and the frequency detuning between the optical source and the microring resonance, that reveal the cavity transition from linear to nonlinear regimes. One of these regions offers very low error in time-series prediction under relatively low input power and number of nodes while the other regions either lack nonlinearity or become unstable. This study provides insight into the design of the MRR and the optimization of its physical properties for improving the prediction performance of time-delay reservoir computing.

We provide a non-unit disk framework to solve combinatorial optimization problems such as Maximum Cut (Max-Cut) and Maximum Independent Set (MIS) on a Rydberg quantum annealer. Our setup consists of a many-body interacting Rydberg system where locally controllable light shifts are applied to individual qubits in order to map the graph problem onto the Ising spin model. Exploiting the flexibility that optical tweezers offer in terms of spatial arrangement, our numerical simulations implement the local-detuning protocol while globally driving the Rydberg annealer to the desired many-body ground state, which is also the solution to the optimization problem. Using optimal control methods, these solutions are obtained for prototype graphs with varying sizes at time scales well within the system lifetime and with approximation ratios close to one. The non-blockade approach facilitates the encoding of graph problems with specific topologies that can be realized in two-dimensional Rydberg configurations and is applicable to both unweighted as well as weighted graphs. A comparative analysis with fast simulated annealing is provided which highlights the advantages of our scheme in terms of system size, hardness of the graph, and the number of iterations required to converge to the solution.

Forecast reconciliation is the post-forecasting process aimed to revise a set of incoherent base forecasts into coherent forecasts in line with given data structures. Most of the point and probabilistic regression-based forecast reconciliation results ground on the so called "structural representation" and on the related unconstrained generalized least squares reconciliation formula. However, the structural representation naturally applies to genuine hierarchical/grouped time series, where the top- and bottom-level variables are uniquely identified. When a general linearly constrained multiple time series is considered, the forecast reconciliation is naturally expressed according to a projection approach. While it is well known that the classic structural reconciliation formula is equivalent to its projection approach counterpart, so far it is not completely understood if and how a structural-like reconciliation formula may be derived for a general linearly constrained multiple time series. Such an expression would permit to extend reconciliation definitions, theorems and results in a straightforward manner. In this paper, we show that for general linearly constrained multiple time series it is possible to express the reconciliation formula according to a "structural-like" approach that keeps distinct free and constrained, instead of bottom and upper (aggregated), variables, establish the probabilistic forecast reconciliation framework, and apply these findings to obtain fully reconciled point and probabilistic forecasts for the aggregates of the Australian GDP from income and expenditure sides, and for the European Area GDP disaggregated by income, expenditure and output sides and by 19 countries.

As a crossover frontier of physics and mechanics, quantum computing is showing its great potential in computational mechanics. However, quantum hardware noise remains a critical barrier to achieving accurate simulation results due to the limitation of the current hardware level. In this paper, we integrate error-mitigated quantum computing in data-driven computational mechanics, where the zero-noise extrapolation (ZNE) technique is employed to improve the accuracy of quantum computing. Numerical examples including multiscale simulation of a composite L-shaped beam are conducted with the quantum computer simulator Qpanda, and the results validate the effectiveness of the proposed method. We believe this work presents a promising step towards using the power of quantum computing in computational mechanics.

Deep neural networks (DNNs) often fail silently with over-confident predictions on out-of-distribution (OOD) samples, posing risks in real-world deployments. Existing techniques predominantly emphasize either the feature representation space or the gradient norms computed with respect to DNN parameters, yet they overlook the intricate gradient distribution and the topology of classification regions. To address this gap, we introduce GRadient-aware Out-Of-Distribution detection in interpolated manifolds (GROOD), a novel framework that relies on the discriminative power of gradient space to distinguish between in-distribution (ID) and OOD samples. To build this space, GROOD relies on class prototypes together with a prototype that specifically captures OOD characteristics. Uniquely, our approach incorporates a targeted mix-up operation at an early intermediate layer of the DNN to refine the separation of gradient spaces between ID and OOD samples. We quantify OOD detection efficacy using the distance to the nearest neighbor gradients derived from the training set, yielding a robust OOD score. Experimental evaluations substantiate that the introduction of targeted input mix-upamplifies the separation between ID and OOD in the gradient space, yielding impressive results across diverse datasets. Notably, when benchmarked against ImageNet-1k, GROOD surpasses the established robustness of state-of-the-art baselines. Through this work, we establish the utility of leveraging gradient spaces and class prototypes for enhanced OOD detection for DNN in image classification.

Partitioned neural network functions are used to approximate the solution of partial differential equations. The problem domain is partitioned into non-overlapping subdomains and the partitioned neural network functions are defined on the given non-overlapping subdomains. Each neural network function then approximates the solution in each subdomain. To obtain the convergent neural network solution, certain continuity conditions on the partitioned neural network functions across the subdomain interface need to be included in the loss function, that is used to train the parameters in the neural network functions. In our work, by introducing suitable interface values, the loss function is reformulated into a sum of localized loss functions and each localized loss function is used to train the corresponding local neural network parameters. In addition, to accelerate the neural network solution convergence, the localized loss function is enriched with an augmented Lagrangian term, where the interface condition and the boundary condition are enforced as constraints on the local solutions by using Lagrange multipliers. The local neural network parameters and Lagrange multipliers are then found by optimizing the localized loss function. To take the advantage of the localized loss function for the parallel computation, an iterative algorithm is also proposed. For the proposed algorithms, their training performance and convergence are numerically studied for various test examples.

Conventional computing paradigm struggles to fulfill the rapidly growing demands from emerging applications, especially those for machine intelligence, because much of the power and energy is consumed by constant data transfers between logic and memory modules. A new paradigm, called "computational random-access memory (CRAM)" has emerged to address this fundamental limitation. CRAM performs logic operations directly using the memory cells themselves, without having the data ever leave the memory. The energy and performance benefits of CRAM for both conventional and emerging applications have been well established by prior numerical studies. However, there lacks an experimental demonstration and study of CRAM to evaluate its computation accuracy, which is a realistic and application-critical metrics for its technological feasibility and competitiveness. In this work, a CRAM array based on magnetic tunnel junctions (MTJs) is experimentally demonstrated. First, basic memory operations as well as 2-, 3-, and 5-input logic operations are studied. Then, a 1-bit full adder with two different designs is demonstrated. Based on the experimental results, a suite of modeling has been developed to characterize the accuracy of CRAM computation. Further analysis of scalar addition, multiplication, and matrix multiplication shows promising results. These results are then applied to a complete application: a neural network based handwritten digit classifier, as an example to show the connection between the application performance and further MTJ development. The classifier achieved almost-perfect classification accuracy, with reasonable projections of future MTJ development. With the confirmation of MTJ-based CRAM's accuracy, there is a strong case that this technology will have a significant impact on power- and energy-demanding applications of machine intelligence.

Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.