In these pedagogic notes I review the statistical mechanics approach to neural networks, focusing on the paradigmatic example of the perceptron architecture with binary an continuous weights, in the classification setting. I will review the Gardner's approach based on replica method and the derivation of the SAT/UNSAT transition in the storage setting. Then, I discuss some recent works that unveiled how the zero training error configurations are geometrically arranged, and how this arrangement changes as the size of the training set increases. I also illustrate how different regions of solution space can be explored analytically and how the landscape in the vicinity of a solution can be characterized. I give evidence how, in binary weight models, algorithmic hardness is a consequence of the disappearance of a clustered region of solutions that extends to very large distances. Finally, I demonstrate how the study of linear mode connectivity between solutions can give insights into the average shape of the solution manifold.
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Physics-informed neural networks (PINNs) are appealing data-driven tools for solving and inferring solutions to nonlinear partial differential equations (PDEs). Unlike traditional neural networks (NNs), which train only on solution data, a PINN incorporates a PDE's residual into its loss function and trains to minimize the said residual at a set of collocation points in the solution domain. This paper explores the use of the Schwarz alternating method as a means to couple PINNs with each other and with conventional numerical models (i.e., full order models, or FOMs, obtained via the finite element, finite difference or finite volume methods) following a decomposition of the physical domain. It is well-known that training a PINN can be difficult when the PDE solution has steep gradients. We investigate herein the use of domain decomposition and the Schwarz alternating method as a means to accelerate the PINN training phase. Within this context, we explore different approaches for imposing Dirichlet boundary conditions within each subdomain PINN: weakly through the loss and/or strongly through a solution transformation. As a numerical example, we consider the one-dimensional steady state advection-diffusion equation in the advection-dominated (high Peclet) regime. Our results suggest that the convergence of the Schwarz method is strongly linked to the choice of boundary condition implementation within the PINNs being coupled. Surprisingly, strong enforcement of the Schwarz boundary conditions does not always lead to a faster convergence of the method. While it is not clear from our preliminary study that the PINN-PINN coupling via the Schwarz alternating method accelerates PINN convergence in the advection-dominated regime, it reveals that PINN training can be improved substantially for Peclet numbers as high as 1e6 by performing a PINN-FOM coupling.
Understanding the ubiquitous phenomenon of neural synchronization across species and organizational levels is crucial for decoding brain function. Despite its prevalence, the specific functional role, origin, and dynamical implication of modular structures in correlation-based networks remains ambiguous. Using recurrent neural networks trained on systems neuroscience tasks, this study investigates these important characteristics of modularity in correlation networks. We demonstrate that modules are functionally coherent units that contribute to specialized information processing. We show that modules form spontaneously from asymmetries in the sign and weight of projections from the input layer to the recurrent layer. Moreover, we show that modules define connections with similar roles in governing system behavior and dynamics. Collectively, our findings clarify the function, formation, and operational significance of functional connectivity modules, offering insights into cortical function and laying the groundwork for further studies on brain function, development, and dynamics.
The paradigm of vertical federated learning (VFL), where institutions collaboratively train machine learning models via combining each other's local feature or label information, has achieved great success in applications to financial risk management (FRM). The surging developments of graph representation learning (GRL) have opened up new opportunities for FRM applications under FL via efficiently utilizing the graph-structured data generated from underlying transaction networks. Meanwhile, transaction information is often considered highly sensitive. To prevent data leakage during training, it is critical to develop FL protocols with formal privacy guarantees. In this paper, we present an end-to-end GRL framework in the VFL setting called VESPER, which is built upon a general privatization scheme termed perturbed message passing (PMP) that allows the privatization of many popular graph neural architectures.Based on PMP, we discuss the strengths and weaknesses of specific design choices of concrete graph neural architectures and provide solutions and improvements for both dense and sparse graphs. Extensive empirical evaluations over both public datasets and an industry dataset demonstrate that VESPER is capable of training high-performance GNN models over both sparse and dense graphs under reasonable privacy budgets.
We propose InCA, a lightweight method for transfer learning that cross-attends to any activation layer of a pre-trained model. During training, InCA uses a single forward pass to extract multiple activations, which are passed to external cross-attention adapters, trained anew and combined or selected for downstream tasks. We show that, even when selecting a single top-scoring adapter, InCA achieves performance comparable to full fine-tuning, at a cost comparable to fine-tuning just the last layer. For example, with a cross-attention probe 1.3% the size of a pre-trained ViT-L/16 model, we achieve performance within 0.2% of the full fine-tuning paragon at a computational training cost of 51% of the baseline, on average across 11 downstream classification. Unlike other forms of efficient adaptation, InCA does not require backpropagating through the pre-trained model, thus leaving its execution unaltered at both training and inference. The versatility of InCA is best illustrated in fine-grained tasks, which may require accessing information absent in the last layer but accessible in intermediate layer activations. Since the backbone is fixed, InCA allows parallel ensembling as well as parallel execution of multiple tasks. InCA achieves state-of-the-art performance in the ImageNet-to-Sketch multi-task benchmark.
The rising popularity of deep learning (DL) methods and techniques has invigorated interest in the topic of SE4DL, the application of software engineering (SE) practices on deep learning software. Despite the novel engineering challenges brought on by the data-driven and non-deterministic paradigm of DL software, little work has been invested into developing AI-targeted SE tools. On the other hand, tools tackling more general engineering issues in DL are actively used and referred to under the umbrella term of ``MLOps tools''. Furthermore, the available literature supports the utility of conventional SE tooling in DL software development. Building upon previous MSR research on tool usage in open-source software works, we identify conventional and MLOps tools adopted in popular applied DL projects that use Python as the main programming language. About 70% of the GitHub repositories mined contained at least one conventional SE tool. Software configuration management tools are the most adopted, while the opposite applies to maintenance tools. Substantially fewer MLOps tools were in use, with only 9 tools out of a sample of 80 used in at least one repository. The majority of them were open-source rather than proprietary. One of these tools, TensorBoard, was found to be adopted in about half of the repositories in our study. Consequently, the use of conventional SE tooling demonstrates its relevance to DL software. Further research is recommended on the adoption of MLOps tooling by open-source projects, focusing on the relevance of particular tool types, the development of required tools, as well as ways to promote the use of already available tools.
For multi-scale problems, the conventional physics-informed neural networks (PINNs) face some challenges in obtaining available predictions. In this paper, based on PINNs, we propose a practical deep learning framework for multi-scale problems by reconstructing the loss function and associating it with special neural network architectures. New PINN methods derived from the improved PINN framework differ from the conventional PINN method mainly in two aspects. First, the new methods use a novel loss function by modifying the standard loss function through a (grouping) regularization strategy. The regularization strategy implements a different power operation on each loss term so that all loss terms composing the loss function are of approximately the same order of magnitude, which makes all loss terms be optimized synchronously during the optimization process. Second, for the multi-frequency or high-frequency problems, in addition to using the modified loss function, new methods upgrade the neural network architecture from the common fully-connected neural network to special network architectures such as the Fourier feature architecture, and the integrated architecture developed by us. The combination of the above two techniques leads to a significant improvement in the computational accuracy of multi-scale problems. Several challenging numerical examples demonstrate the effectiveness of the proposed methods. The proposed methods not only significantly outperform the conventional PINN method in terms of computational efficiency and computational accuracy, but also compare favorably with the state-of-the-art methods in the recent literature. The improved PINN framework facilitates better application of PINNs to multi-scale problems.
The joint analysis of multimodal neuroimaging data is critical in the field of brain research because it reveals complex interactive relationships between neurobiological structures and functions. In this study, we focus on investigating the effects of structural imaging (SI) features, including white matter micro-structure integrity (WMMI) and cortical thickness, on the whole brain functional connectome (FC) network. To achieve this goal, we propose a network-based vector-on-matrix regression model to characterize the FC-SI association patterns. We have developed a novel multi-level dense bipartite and clique subgraph extraction method to identify which subsets of spatially specific SI features intensively influence organized FC sub-networks. The proposed method can simultaneously identify highly correlated structural-connectomic association patterns and suppress false positive findings while handling millions of potential interactions. We apply our method to a multimodal neuroimaging dataset of 4,242 participants from the UK Biobank to evaluate the effects of whole-brain WMMI and cortical thickness on the resting-state FC. The results reveal that the WMMI on corticospinal tracts and inferior cerebellar peduncle significantly affect functional connections of sensorimotor, salience, and executive sub-networks with an average correlation of 0.81 (p<0.001).
We hypothesize that due to the greedy nature of learning in multi-modal deep neural networks, these models tend to rely on just one modality while under-fitting the other modalities. Such behavior is counter-intuitive and hurts the models' generalization, as we observe empirically. To estimate the model's dependence on each modality, we compute the gain on the accuracy when the model has access to it in addition to another modality. We refer to this gain as the conditional utilization rate. In the experiments, we consistently observe an imbalance in conditional utilization rates between modalities, across multiple tasks and architectures. Since conditional utilization rate cannot be computed efficiently during training, we introduce a proxy for it based on the pace at which the model learns from each modality, which we refer to as the conditional learning speed. We propose an algorithm to balance the conditional learning speeds between modalities during training and demonstrate that it indeed addresses the issue of greedy learning. The proposed algorithm improves the model's generalization on three datasets: Colored MNIST, Princeton ModelNet40, and NVIDIA Dynamic Hand Gesture.
Artificial neural networks thrive in solving the classification problem for a particular rigid task, acquiring knowledge through generalized learning behaviour from a distinct training phase. The resulting network resembles a static entity of knowledge, with endeavours to extend this knowledge without targeting the original task resulting in a catastrophic forgetting. Continual learning shifts this paradigm towards networks that can continually accumulate knowledge over different tasks without the need to retrain from scratch. We focus on task incremental classification, where tasks arrive sequentially and are delineated by clear boundaries. Our main contributions concern 1) a taxonomy and extensive overview of the state-of-the-art, 2) a novel framework to continually determine the stability-plasticity trade-off of the continual learner, 3) a comprehensive experimental comparison of 11 state-of-the-art continual learning methods and 4 baselines. We empirically scrutinize method strengths and weaknesses on three benchmarks, considering Tiny Imagenet and large-scale unbalanced iNaturalist and a sequence of recognition datasets. We study the influence of model capacity, weight decay and dropout regularization, and the order in which the tasks are presented, and qualitatively compare methods in terms of required memory, computation time, and storage.
Graph representation learning for hypergraphs can be used to extract patterns among higher-order interactions that are critically important in many real world problems. Current approaches designed for hypergraphs, however, are unable to handle different types of hypergraphs and are typically not generic for various learning tasks. Indeed, models that can predict variable-sized heterogeneous hyperedges have not been available. Here we develop a new self-attention based graph neural network called Hyper-SAGNN applicable to homogeneous and heterogeneous hypergraphs with variable hyperedge sizes. We perform extensive evaluations on multiple datasets, including four benchmark network datasets and two single-cell Hi-C datasets in genomics. We demonstrate that Hyper-SAGNN significantly outperforms the state-of-the-art methods on traditional tasks while also achieving great performance on a new task called outsider identification. Hyper-SAGNN will be useful for graph representation learning to uncover complex higher-order interactions in different applications.
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