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Recent advancements in the realm of deep learning, particularly in the development of large language models (LLMs), have demonstrated AI's ability to tackle complex mathematical problems or solving programming challenges. However, the capability to solve well-defined problems based on extensive training data differs significantly from the nuanced process of making scientific discoveries. Trained on almost all human knowledge available, today's sophisticated LLMs basically learn to predict sequences of tokens. They generate mathematical derivations and write code in a similar way as writing an essay, and do not have the ability to pioneer scientific discoveries in the manner a human scientist would do. In this study we delve into the potential of using deep learning to rediscover a fundamental mathematical concept: integrals. By defining integrals as area under the curve, we illustrate how AI can deduce the integral of a given function, exemplified by inferring $\int_{0}^{x} t^2 dt = \frac{x^3}{3}$ and $\int_{0}^{x} ae^{bt} dt = \frac{a}{b} e^{bx} - \frac{a}{b}$. Our experiments show that deep learning models can approach the task of inferring integrals either through a sequence-to-sequence model, akin to language translation, or by uncovering the rudimentary principles of integration, such as $\int_{0}^{x} t^n dt = \frac{x^{n+1}}{n+1}$.

This manuscript investigates the information-theoretic limits of integrated sensing and communications (ISAC), aiming for simultaneous reliable communication and precise channel state estimation. We model such a system with a state-dependent discrete memoryless channel (SD-DMC) with present or absent channel feedback and generalized side information at the transmitter and the receiver, where the joint task of message decoding and state estimation is performed at the receiver. The relationship between the achievable communication rate and estimation error, the capacity-distortion (C-D) trade-off, is characterized across different causality levels of the side information. This framework is shown to be capable of modeling various practical scenarios by assigning the side information with different meanings, including monostatic and bistatic radar systems. The analysis is then extended to the two-user degraded broadcast channel, and we derive an achievable C-D region that is tight under certain conditions. To solve the optimization problem arising in the computation of C-D functions/regions, we propose a proximal block coordinate descent (BCD) method, prove its convergence to a stationary point, and derive a stopping criterion. Finally, several representative examples are studied to demonstrate the versatility of our framework and the effectiveness of the proposed algorithm.

We reformulate models in epidemiology and population dynamics in terms of probability distributions. This allows us to construct the Fisher information, which we interpret as the metric of a one-dimensional differentiable manifold. For systems that can be effectively described by a single degree of freedom, we show that their time evolution is fully captured by this metric. In this way, we discover universal features across seemingly very different models. This further motivates a reorganisation of the dynamics around zeroes of the Fisher metric, corresponding to extrema of the probability distribution. Concretely, we propose a simple form of the metric for which we can analytically solve the dynamics of the system that well approximates the time evolution of various established models in epidemiology and population dynamics, thus providing a unifying framework.

We investigate both the theoretical and algorithmic aspects of likelihood-based methods for recovering a complex-valued signal from multiple sets of measurements, referred to as looks, affected by speckle (multiplicative) noise. Our theoretical contributions include establishing the first existing theoretical upper bound on the Mean Squared Error (MSE) of the maximum likelihood estimator under the deep image prior hypothesis. Our theoretical results capture the dependence of MSE upon the number of parameters in the deep image prior, the number of looks, the signal dimension, and the number of measurements per look. On the algorithmic side, we introduce the concept of bagged Deep Image Priors (Bagged-DIP) and integrate them with projected gradient descent. Furthermore, we show how employing Newton-Schulz algorithm for calculating matrix inverses within the iterations of PGD reduces the computational complexity of the algorithm. We will show that this method achieves the state-of-the-art performance.

With the incorporation of the UNet architecture, diffusion probabilistic models have become a dominant force in image generation tasks. One key design in UNet is the skip connections between the encoder and decoder blocks. Although skip connections have been shown to improve training stability and model performance, we reveal that such shortcuts can be a limiting factor for the complexity of the transformation. As the sampling steps decrease, the generation process and the role of the UNet get closer to the push-forward transformations from Gaussian distribution to the target, posing a challenge for the network's complexity. To address this challenge, we propose Skip-Tuning, a simple yet surprisingly effective training-free tuning method on the skip connections. Our method can achieve 100% FID improvement for pretrained EDM on ImageNet 64 with only 19 NFEs (1.75), breaking the limit of ODE samplers regardless of sampling steps. Surprisingly, the improvement persists when we increase the number of sampling steps and can even surpass the best result from EDM-2 (1.58) with only 39 NFEs (1.57). Comprehensive exploratory experiments are conducted to shed light on the surprising effectiveness. We observe that while Skip-Tuning increases the score-matching losses in the pixel space, the losses in the feature space are reduced, particularly at intermediate noise levels, which coincide with the most effective range accounting for image quality improvement.

Reasoning, a crucial ability for complex problem-solving, plays a pivotal role in various real-world settings such as negotiation, medical diagnosis, and criminal investigation. It serves as a fundamental methodology in the field of Artificial General Intelligence (AGI). With the ongoing development of foundation models, e.g., Large Language Models (LLMs), there is a growing interest in exploring their abilities in reasoning tasks. In this paper, we introduce seminal foundation models proposed or adaptable for reasoning, highlighting the latest advancements in various reasoning tasks, methods, and benchmarks. We then delve into the potential future directions behind the emergence of reasoning abilities within foundation models. We also discuss the relevance of multimodal learning, autonomous agents, and super alignment in the context of reasoning. By discussing these future research directions, we hope to inspire researchers in their exploration of this field, stimulate further advancements in reasoning with foundation models, and contribute to the development of AGI.

Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.

Deep neural networks have revolutionized many machine learning tasks in power systems, ranging from pattern recognition to signal processing. The data in these tasks is typically represented in Euclidean domains. Nevertheless, there is an increasing number of applications in power systems, where data are collected from non-Euclidean domains and represented as the graph-structured data with high dimensional features and interdependency among nodes. The complexity of graph-structured data has brought significant challenges to the existing deep neural networks defined in Euclidean domains. Recently, many studies on extending deep neural networks for graph-structured data in power systems have emerged. In this paper, a comprehensive overview of graph neural networks (GNNs) in power systems is proposed. Specifically, several classical paradigms of GNNs structures (e.g., graph convolutional networks, graph recurrent neural networks, graph attention networks, graph generative networks, spatial-temporal graph convolutional networks, and hybrid forms of GNNs) are summarized, and key applications in power systems such as fault diagnosis, power prediction, power flow calculation, and data generation are reviewed in detail. Furthermore, main issues and some research trends about the applications of GNNs in power systems are discussed.

Recent developments in image classification and natural language processing, coupled with the rapid growth in social media usage, have enabled fundamental advances in detecting breaking events around the world in real-time. Emergency response is one such area that stands to gain from these advances. By processing billions of texts and images a minute, events can be automatically detected to enable emergency response workers to better assess rapidly evolving situations and deploy resources accordingly. To date, most event detection techniques in this area have focused on image-only or text-only approaches, limiting detection performance and impacting the quality of information delivered to crisis response teams. In this paper, we present a new multimodal fusion method that leverages both images and texts as input. In particular, we introduce a cross-attention module that can filter uninformative and misleading components from weak modalities on a sample by sample basis. In addition, we employ a multimodal graph-based approach to stochastically transition between embeddings of different multimodal pairs during training to better regularize the learning process as well as dealing with limited training data by constructing new matched pairs from different samples. We show that our method outperforms the unimodal approaches and strong multimodal baselines by a large margin on three crisis-related tasks.

While it is nearly effortless for humans to quickly assess the perceptual similarity between two images, the underlying processes are thought to be quite complex. Despite this, the most widely used perceptual metrics today, such as PSNR and SSIM, are simple, shallow functions, and fail to account for many nuances of human perception. Recently, the deep learning community has found that features of the VGG network trained on the ImageNet classification task has been remarkably useful as a training loss for image synthesis. But how perceptual are these so-called "perceptual losses"? What elements are critical for their success? To answer these questions, we introduce a new Full Reference Image Quality Assessment (FR-IQA) dataset of perceptual human judgments, orders of magnitude larger than previous datasets. We systematically evaluate deep features across different architectures and tasks and compare them with classic metrics. We find that deep features outperform all previous metrics by huge margins. More surprisingly, this result is not restricted to ImageNet-trained VGG features, but holds across different deep architectures and levels of supervision (supervised, self-supervised, or even unsupervised). Our results suggest that perceptual similarity is an emergent property shared across deep visual representations.