Related papers: Pioneer: Physics-informed Riemannian Graph ODE for…
We propose a geometrical approach to the investigation of Hamiltonian systems on (Pseudo) Riemannian manifolds. A new geometrical criterion of instability and chaos is proposed. This approach is more generic than well known reduction to the…
This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimizing a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given…
The order/dimension of models derived on the basis of data is commonly restricted by the number of observations, or in the context of monitored systems, sensing nodes. This is particularly true for structural systems (e.g., civil or…
Oscillator Ising machines (OIMs) represent an exemplar case of using physics-inspired non-linear dynamical systems to solve computationally challenging combinatorial optimization problems (COPs). The computational performance of such…
The data-driven discovery of interpretable models approximating the underlying dynamics of a physical system has gained attraction in the past decade. Current approaches employ pre-specified functional forms or basis functions and often…
Neural networks with physics based inductive biases such as Lagrangian neural networks (LNN), and Hamiltonian neural networks (HNN) learn the dynamics of physical systems by encoding strong inductive biases. Alternatively, Neural ODEs with…
We study the Riemannian geometry of the Deep Linear Network (DLN) as a foundation for a thermodynamic description of the learning process. The main tools are the use of group actions to analyze overparametrization and the use of Riemannian…
Steering a language model - intervening on its internal activations to change downstream behaviour - has recently expanded beyond linear interpolation to nonlinear methods such as angular and kernelized steering, which define intervention…
Incorporating the Hamiltonian structure of physical dynamics into deep learning models provides a powerful way to improve the interpretability and prediction accuracy. While previous works are mostly limited to the Euclidean spaces, their…
An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The…
The inductive biases of graph representation learning algorithms are often encoded in the background geometry of their embedding space. In this paper, we show that general directed graphs can be effectively represented by an embedding model…
Generating physically feasible dynamics in a data-driven context is challenging, especially when adhering to physical priors expressed in specific equations or formulas. Existing methodologies often overlook the integration of physical…
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…
In this paper, we propose RiemannianFlow, a deep generative model that allows robots to learn complex and stable skills evolving on Riemannian manifolds. Examples of Riemannian data in robotics include stiffness (symmetric and positive…
Newtonian, Lagrangian, and Hamiltonian dynamical systems are well formalized mathematically. They give rise to geometric structures describing motion of a point in smooth manifolds. Riemannian metric is a different geometric structure…
Computational optimal transport (OT) offers a principled framework for generative modeling. Neural OT methods, which use neural networks to learn an OT map (or potential) from data in an amortized way, can be evaluated out of sample after…
We show that the standard discrete update rule of transformer layers can be naturally interpreted as a forward Euler discretization of a continuous dynamical system. Our Transformer Flow Approximation Theorem demonstrates that, under…
Decentralized optimization on Riemannian manifolds is foundational for many modern machine learning and signal processing applications in which data are non-Euclidean and generated and processed in a distributed manner. Although intrinsic…
In part III is realized the consistent development of Heisenberg--Dyson's two-layer matrix approximation to the graph formalism for postulating discrete physical objects (DPO) introduced in parts I-II in the form of discrete sets of…
Grasping objects of different shapes and sizes - a foundational, effortless skill for humans - remains a challenging task in robotics. Although model-based approaches can predict stable grasp configurations for known object models, they…