Related papers: Symplectic Neural Operators for Learning Infinite …
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…
Stochastic partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. Neural Operators, generations of neural networks with capability of learning maps…
This work concerns control-oriented and structure-preserving learning of low-dimensional approximations of high-dimensional physical systems, with a focus on mechanical systems. We investigate the integration of neural autoencoders in model…
Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of…
Hybrid machine learning based on Hamiltonian formulations has recently been successfully demonstrated for simple mechanical systems, both energy conserving and not energy conserving. We introduce a pseudo-Hamiltonian formulation that is a…
We propose explicit symplectic integrators of molecular dynamics (MD) algorithms for rigid-body molecules in the canonical and isothermal-isobaric ensembles. We also present a symplectic algorithm in the constant normal pressure and lateral…
We consider an operator-theoretic approach to linear infinite-dimensional port-Hamiltonian systems. In particular, we use the theory of system nodes by Staffans to formulate a~suitable concept for port-Hamiltonian systems, which allows a…
The rapid advancements in high-dimensional statistics and machine learning have increased the use of first-order methods. Many of these methods can be regarded as instances of the proximal point algorithm. Given the importance of the…
For many-particle systems defined on lattices we investigate the global structure of effective Hamiltonians and observables obtained by means of a suitable basis transformation. We study transformations which lead to effective Hamiltonians…
This article is concerned with analytic Hamiltonian dynamical systems in infinite dimension in a neighborhood of an elliptic fixed point. Given a quadratic Hamiltonian, we consider the set of its analytic higher order perturbations. We…
By embedding physical intuition, network architectures enforce fundamental properties, such as energy conservation laws, leading to plausible predictions. Yet, scaling these models to intrinsically high-dimensional systems remains a…
Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear Hamiltonian systems with highly…
In this paper, we propose Neumann Series Neural Operator (NSNO) to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions. Helmholtz equation is a crucial partial differential…
There has been increasing interest in methodologies that incorporate physics priors into neural network architectures to enhance their modeling capabilities. A family of these methodologies that has gained traction are Hamiltonian neural…
Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit…
This article considers Hamiltonian mechanical systems with potential functions admitting jump discontinuities. The focus is on accurate and efficient numerical approximations of their solutions, which will be defined via the laws of…
This work develops a symplectic framework for quantum computing to be applied to classical Hamiltonian systems, exploiting the intrinsic geometric compatibility between unitary quantum evolution and symplectic phase-space dynamics in a…
There has been a wave of interest in applying machine learning to study dynamical systems. We present a Hamiltonian neural network that solves the differential equations that govern dynamical systems. This is an equation-driven machine…
While symplectic integration methods based on operator splitting are well established in many branches of science, high order methods for Hamiltonian systems that split in more than two parts have not been studied in great detail. Here, we…
Operator learning based on neural operators has emerged as a promising paradigm for the data-driven approximation of operators, mapping between infinite-dimensional Banach spaces. Despite significant empirical progress, our theoretical…