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Many material response functions depend strongly on microstructure, such as inhomogeneities in phase or orientation. Homogenization presents the task of predicting the mean response of a sample of the microstructure to external loading for…
We present a new approach for predictive modeling and its uncertainty quantification for mechanical systems, where coarse-grained models such as constitutive relations are derived directly from observation data. We explore the use of a…
We study the relationship between the physics of topology and zero modes in frustrated systems and metama- terials. Zero modes that exist in topological matters are distinct from the ones arising from symmetry breaking. Incidentally, a…
Multi-layer feedforward networks have been used to approximate a wide range of nonlinear functions. An important and fundamental problem is to understand the learnability of a network model through its statistical risk, or the expected…
We present a graph-based deep learning framework for predicting the magnetic properties of quasi-one-dimensional Ising spin systems. The lattice geometry is encoded as a graph and processed by a graph neural network (GNN) followed by fully…
Despite their increasing popularity and success in a variety of supervised learning problems, deep neural networks are extremely hard to interpret and debug: Given and already trained Deep Neural Net, and a set of test inputs, how can we…
Although overparameterized models have achieved remarkable practical success, their theoretical properties, particularly their generalization behavior, remain incompletely understood. The well known double descents phenomenon suggests that…
Deep learning models have been shown to be vulnerable to adversarial attacks. This perception led to analyzing deep learning models not only from the perspective of their performance measures but also their robustness to certain types of…
Geometrically frustrated assemblies where building blocks misfit have been shown to generate intriguing phenomena from self-limited growth, fiber formation, to structural complexity. We introduce a graph theory formulation of geometrically…
Neural networks are powerful function estimators, leading to their status as a paradigm of choice for modeling structured data. However, unlike other structured representations that emphasize the modularity of the problem -- e.g., factor…
Deep learning has arguably achieved tremendous success in recent years. In simple words, deep learning uses the composition of many nonlinear functions to model the complex dependency between input features and labels. While neural networks…
Learning monotonic models with respect to a subset of the inputs is a desirable feature to effectively address the fairness, interpretability, and generalization issues in practice. Existing methods for learning monotonic neural networks…
When training deep neural networks with gradient descent, sharpness often increases -- a phenomenon known as progressive sharpening -- before saturating at the edge of stability. Although commonly observed in practice, the underlying…
Recently, graph-based models designed for downstream tasks have significantly advanced research on graph neural networks (GNNs). GNN baselines based on neural message-passing mechanisms such as GCN and GAT perform worse as the network…
While many approaches to make neural networks more fathomable have been proposed, they are restricted to interrogating the network with input data. Measures for characterizing and monitoring structural properties, however, have not been…
We study the properties of alignment, a form of implicit regularization, in linear neural networks under gradient descent. We define alignment for fully connected networks with multidimensional outputs and show that it is a natural…
Previous work has shown that DNNs with large depth $L$ and $L_{2}$-regularization are biased towards learning low-dimensional representations of the inputs, which can be interpreted as minimizing a notion of rank $R^{(0)}(f)$ of the learned…
Characterizing the in uence of network properties on the global emerging behavior of interacting elements constitutes a central question in many areas, from physical to social sciences. In this article we study a primary model of disordered…
Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation…
We perform an empirical study of the behaviour of deep networks when fully linearizing some of its feature channels through a sparsity prior on the overall number of nonlinear units in the network. In experiments on image classification and…