Related papers: Pioneer: Physics-informed Riemannian Graph ODE for…
The Generalized Riemann Problems (GRP) for nonlinear hyperbolic systems of balance laws in one space dimension are now well-known and can be formulated as follows: Given initial-data which are smooth on two sides of a discontinuity,…
Entropic dynamics is a framework in which the laws of dynamics are derived as an application of entropic methods of inference. Its successes include the derivation of quantum mechanics and quantum field theory from probabilistic principles.…
Macroscopic traffic flow is stochastic, but the physics-informed deep learning methods currently used in transportation literature embed deterministic PDEs and produce point-valued outputs; the stochasticity of the governing dynamics plays…
Dissipative partial differential equations that exhibit chaotic dynamics tend to evolve to attractors that exist on finite-dimensional manifolds. We present a data-driven reduced order modeling method that capitalizes on this fact by…
In this paper we present a deep learning method to predict the temporal evolution of dissipative dynamic systems. We propose using both geometric and thermodynamic inductive biases to improve accuracy and generalization of the resulting…
Dynamical systems appear in nearly every aspect of the physical world. As such, understanding the properties of dynamical systems is of great importance. Typically, a dynamical system is described by a system of ordinary differential…
We present a novel method to estimate the dominant eigenvalue and eigenvector pair of any non-negative real matrix via graph infection. The key idea in our technique lies in approximating the solution to the first-order matrix ordinary…
We introduce a combinatorial topological framework for characterizing the global dynamics of ordinary differential equations (ODEs). The approach is motivated by the study of gene regulatory networks, which are often modeled by ODEs that…
This paper presents a unified geometric framework for Brownian motion on manifolds, encompassing intrinsic Riemannian manifolds, embedded submanifolds, and Lie groups. The approach constructs the stochastic differential equation by…
Time-continuous dynamical systems defined on graphs are often used to model complex systems with many interacting components in a non-spatial context. In the reverse sense attaching meaningful dynamics to given 'interaction diagrams' is a…
In many robot control problems, factors such as stiffness and damping matrices and manipulability ellipsoids are naturally represented as symmetric positive definite (SPD) matrices, which capture the specific geometric characteristics of…
Developing the proper representations for simulating high-speed flows with strong shock waves, rarefactions, and contact discontinuities has been a long-standing question in numerical analysis. Herein, we employ neural operators to solve…
Standard diffusion models for graph generation typically rely on uniform time-stepping, an approach that overlooks the non-homogeneous dynamics of distributional evolution on complex manifolds. In this paper, we present an…
Deep generative models such as flow matching and diffusion models have shown great potential in learning complex distributions and dynamical systems, but often act as black-boxes, neglecting underlying physics. In contrast, physics-based…
A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on $R^n$ modulo the action of a finite group. Orbifolds have proven interesting in a variety of settings. Spectral geometers have…
The physical phenomena are described by physical quantities related by specific physical laws. In the context of a Physical Theory, the physical quantities and the physical laws are described, respectively, by suitable geometrical objects…
We consider two Riemannian geometries for the manifold $\mathcal{M}(p,m\times n)$ of all $m\times n$ matrices of rank $p$. The geometries are induced on $\mathcal{M}(p,m\times n)$ by viewing it as the base manifold of the submersion…
Entropic Dynamics is a framework in which quantum theory is derived as an application of entropic methods of inference. There is no underlying action principle. Instead, the dynamics is driven by entropy subject to the appropriate…
By interpreting the forward dynamics of the latent representation of neural networks as an ordinary differential equation, Neural Ordinary Differential Equation (Neural ODE) emerged as an effective framework for modeling a system dynamics…
In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the…