Related papers: Symplectic Neural Flows for Modeling and Discovery

We introduce Neural Flow Maps, a novel simulation method bridging the emerging paradigm of implicit neural representations with fluid simulation based on the theory of flow maps, to achieve state-of-the-art simulation of inviscid fluid…

We propose a neural physics system for real-time, interactive fluid simulations. Traditional physics-based methods, while accurate, are computationally intensive and suffer from latency issues. Recent machine-learning methods reduce…

Integrals of the Liouville $1$-form, known as the first Poincar\'e integral invariant, provide a computable figure of merit for monitoring the conservation of symplecticity in the numerical integration of Hamiltonian systems. These…

We develop inductive biases for the machine learning of complex physical systems based on the port-Hamiltonian formalism. To satisfy by construction the principles of thermodynamics in the learned physics (conservation of energy,…

Hamiltonian dynamics can be used to produce distant proposals for the Metropolis algorithm, thereby avoiding the slow exploration of the state space that results from the diffusive behaviour of simple random-walk proposals. Though…

This paper presents a "historical" formalism for dynamical systems, in its Hamiltonian version (Lagrangian version was presented in a previous paper). It is universal, in the sense that it applies equally well to time dynamics and to field…

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…

Topological techniques have become a popular tool for studying information flows in neural networks. In particular, simplicial homology theory is used to analyze how cognitive representations of space emerge from large conglomerates of…

We detect, by using symplectic topology, invariant measures with large rotation vectors for a class of Hamiltonian flows.

Conformally symplectic diffeomorphisms $f:M \rightarrow M$ transform a symplectic form $\omega$ on a manifold M into a multiple of itself, $f^* \omega = \eta \omega$. We assume $\omega$ is bounded, as some of the results may fail otherwise.…

A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry structure of…

Concise, accurate descriptions of physical systems through their conserved quantities abound in the natural sciences. In data science, however, current research often focuses on regression problems, without routinely incorporating…

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete…

We introduce a recent symplectic integration scheme derived for solving physically motivated systems with non-separable Hamiltonians. We show its relevance to Riemannian manifold Hamiltonian Monte Carlo (RMHMC) and provide an alternative to…

Symplectic integrators with long-term preservation of integrals of motion are introduced for the guiding-center model of plasma particles in toroidal magnetic fields of general topology. An efficient transformation to canonical coordinates…

In a previous paper [3] we have studied flows defined on polytopes, presenting a new method to encapsulate its asymptotic dynamics along the edge-vertex heteroclinic network. These results apply to the class of polymatrix replicator…

We study the non-canonical symplectic structure, or K-symplectic structure inherited by the charged particle dynamics. Based on the splitting technique, we construct non-canonical symplectic methods which is explicit and stable for the…

We study in detail the dynamics of conformal Hamiltonian flows that are defined on a conformal symplectic manifold (this notion was popularized by Vaisman in 1976). We show that they exhibit some conservative and dissipative behaviours. We…

The Schrieffer-Wolff transformation (SWT) is a foundational perturbative method for deriving effective Hamiltonians in quantum systems by systematically eliminating couplings between pairs of energy distant subspaces. Despite recent…

Accurate and efficient fluid flow models are essential for applications relating to many physical phenomena including geophysical, aerodynamic, and biological systems. While these flows may exhibit rich and multiscale dynamics, in many…