Related papers: Direct Poisson neural networks: Learning non-sympl…

Pseudo-Hamiltonian neural networks (PHNN) were recently introduced for learning dynamical systems that can be modelled by ordinary differential equations. In this paper, we extend the method to partial differential equations. The resulting…

This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system…

Many dynamical systems -- from robots interacting with their surroundings to large-scale multiphysics systems -- involve a number of interacting subsystems. Toward the objective of learning composite models of such systems from data, we…

Many applications, such as optimization, uncertainty quantification and inverse problems, require repeatedly performing simulations of large-dimensional physical systems for different choices of parameters. This can be prohibitively…

Hamilton's equations are fundamental for modeling complex physical systems, where preserving key properties such as energy and momentum is crucial for reliable long-term simulations. Geometric integrators are widely used for this purpose,…

The interpretation of deep learning as a dynamical system has gained a considerable attention in recent years as it provides a promising framework. It allows for the use of existing ideas from established fields of mathematics for studying…

Non-holonomic mechanical systems can be described by a degenerate almost-Poisson structure (dropping the Jacobi identity) in the constrained space. If enough symmetries transversal to the constraints are present, the system reduces to a…

We propose locally-symplectic neural networks LocSympNets for learning the flow of phase volume-preserving dynamics. The construction of LocSympNets stems from the theorem of the local Hamiltonian description of the divergence-free vector…

The rapid growth of research in exploiting machine learning to predict chaotic systems has revived a recent interest in Hamiltonian Neural Networks (HNNs) with physical constraints defined by the Hamilton's equations of motion, which…

This paper presents a method for learning Hamiltonian dynamics from a limited set of data points. The Hamiltonian vector field is found by regularized optimization over a reproducing kernel Hilbert space of vector fields that are inherently…

We propose a new method of quantization of a wide class of dynamical systems that originates directly from the equations of motion. The method is based on the correspondence between the classical and the quantum Poisson brackets, postulated…

Even though neural networks enjoy widespread use, they still struggle to learn the basic laws of physics. How might we endow them with better inductive biases? In this paper, we draw inspiration from Hamiltonian mechanics to train models…

In this article we study the possibilities of recovering the structure of port-Hamiltonian systems starting from ``unlabelled'' ordinary differential equations describing mechanical systems. The algorithm we suggest solves the problem in…

Hamiltonian Monte Carlo is typically based on the assumption of an underlying canonical symplectic structure. Numerical integrators designed for the canonical structure are incompatible with motion generated by non-canonical dynamics. These…

We introduce a notion of noncommutative Poisson-Nijenhuis structure on the path algebra of a quiver. In particular, we focus on the case when the Poisson bracket arises from a noncommutative symplectic form. The formalism is then applied to…

Reversible evolution of macroscopic and mesoscopic systems can be conveniently constructed from two ingredients: an energy functional and a Poisson bracket. The goal of this paper is to elucidate how the Poisson brackets can be constructed…

A simple Hamiltonian modeling framework for general models in nonlinear optics is given. This framework is specialized to describe the Hamiltonian structure of electromagnetic phenomena in cubicly nonlinear optical media. The model has a…

The metriplectic formalism couples Poisson brackets of the Hamiltonian description with metric brackets for describing systems with both Hamiltonian and dissipative components. The construction builds in asymptotic convergence to a…

To accurately compute data-based prediction of Hamiltonian systems, especially the long-term evolution of such systems, it is essential to utilize methods that preserve the structure of the equations over time. We consider a case that is…

Evolutionary partial differential equations play a crucial role in many areas of science and engineering. Spatial discretization of these equations leads to a system of ordinary differential equations which can then be solved by numerical…