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Spatial networks are networks whose graph topology is constrained by their embedded spatial space. Understanding the coupled spatial-graph properties is crucial for extracting powerful representations from spatial networks. Therefore,…

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…

Image set-based visual classification methods have achieved remarkable performance, via characterising the image set in terms of a non-singular covariance matrix on a symmetric positive definite (SPD) manifold. To adapt to complicated…

Utilizing physics-informed neural networks (PINN) to solve partial differential equations (PDEs) becomes a hot issue and also shows its great powers, but still suffers from the dilemmas of limited predicted accuracy in the sampling domain…

Monocular depth estimation (MDE) has witnessed remarkable progress driven by Convolutional Neural Networks and transformer-based architectures. However, these approaches typically treat the problem as a generic image-to-image regression on…

Incorporating prior knowledge of physics laws and structural properties of dynamical systems into the design of deep learning architectures has proven to be a powerful technique for improving their computational efficiency and…

Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data…

We view disentanglement learning as discovering an underlying structure that equivariantly reflects the factorized variations shown in data. Traditionally, such a structure is fixed to be a vector space with data variations represented by…

Self-supervised learning of deep neural networks has become a prevalent paradigm for learning representations that transfer to a variety of downstream tasks. Similar to proposed models of the ventral stream of biological vision, it is…

Dynamic relational data arise in many machine learning applications, yet their evolving structure poses challenges for learning representations that remain consistent and interpretable over time. A common approach is to learn time varying…

We investigate recurrent neural networks with asymmetric interactions and demonstrate that the inclusion of self-couplings or sparse excitatory inter-module connections leads to the emergence of a densely connected manifold of dynamically…

This paper introduces Lie groups in degenerate geometric (Clifford) algebras that preserve four fundamental subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations. We prove that…

Predicting the evolution of systems that exhibit spatio-temporal dynamics in response to external stimuli is a key enabling technology fostering scientific innovation. Traditional equations-based approaches leverage first principles to…

The coincidence between polynomial neural networks and matrix Lie maps is discussed in the article. The matrix form of Lie transform is an approximation of the general solution of the nonlinear system of ordinary differential equations. It…

Non-Euclidean constraints are inherent in many kinds of data in computer vision and machine learning, typically as a result of specific invariance requirements that need to be respected during high-level inference. Often, these geometric…

We examine the algebraic and geometric properties of a uni-directional GRU and word embeddings trained end-to-end on a text classification task. A hyperparameter search over word embedding dimension, GRU hidden dimension, and a linear…

Learning continuous-time dynamics on complex networks is crucial for understanding, predicting and controlling complex systems in science and engineering. However, this task is very challenging due to the combinatorial complexities in the…

Neural ordinary differential equations (NODE) have garnered significant attention for their design of continuous-depth neural networks and the ability to learn data/feature dynamics. However, for high-dimensional systems, estimating…

Many relevant applications of group theoretical methods to physical problems are related, in some manner, to classification schemes by means of symmetry groups. In these schemes, irreducible representations of a Lie group have to be…

Neural networks are a central technique in machine learning. Recent years have seen a wave of interest in applying neural networks to physical systems for which the governing dynamics are known and expressed through differential equations.…