Related papers: Differential Euler: Designing a Neural Network app…
Deep neural networks proved to be a very useful and powerful tool with many practical applications. They especially excel at learning from large data sets with labeled samples. However, in order to achieve good learning results, the network…
Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…
This monograph describes a Riemannian geometric reduction approach to the three-body problem. The fundamental theorems are presented in the introductory part, whereas their proofs are provided in later chapters where specific topics are…
We prove the existence of periodic solutions of the N=(n+1)-body problem starting with n bodies whose reduced motion is close to a non-degenerate central configuration and replacing one of them by the center of mass of a pair of bodies…
This paper presents a study of the isosceles problem resulting by a perturbation of Euler's collinear solution under Newtonian gravitational attraction of three bodies in space. After the Hamiltonian was obtained, a circumference of…
We develop a class of emulators for solving quantum three-body scattering problems. They are based on combining the variational method for scattering observables and the recently proposed eigenvector continuation concept. The emulators are…
We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with…
We propose a Newton-based scheme, initialized by neural operator predictions, to accelerate the parametric solution of nonlinear problems in computational solid mechanics. First, a physics informed conditional neural field is trained to…
Neural networks are a promising tool for characterizing intermediate-scale quantum devices from limited amounts of measurement data. A challenging problem in this area is to learn the action of an unknown quantum process on an ensemble of…
In the Newtonian 3-body problem, for any choice of the three masses, there are exactly three Euler configurations (also known as the three Euler points). In Helmholtz' problem of 3 point vortices in the plane, there are at most three…
This paper proposes a two-time scale neurodynamic duplex approach to solve distributionally robust geometric joint chance-constrained optimization problems. The probability distributions of the row vectors are not known in advance and…
The neural network method of solving differential equations is used to approximate the electric potential and corresponding electric field in the slit-well microfluidic device. The device's geometry is non-convex, making this a challenging…
We present an efficient variational integrator for multibody systems. Variational integrators reformulate the equations of motion for multibody systems as discrete Euler-Lagrange (DEL) equations, transforming forward integration into a…
Simulators based on neural networks offer a path to orders-of-magnitude faster electromagnetic wave simulations. Existing models, however, only address narrowly tailored classes of problems and only scale to systems of a few dozen degrees…
Two neural-network-based numerical schemes are proposed to solve the classical obstacle problems. The schemes are based on the universal approximation property of neural networks, and the cost functions are taken as the energy minimization…
In this paper, we study the challenging problem of predicting the dynamics of objects in static images. Given a query object in an image, our goal is to provide a physical understanding of the object in terms of the forces acting upon it…
Constrained optimization problems appear in a wide variety of challenging real-world problems, where constraints often capture the physics of the underlying system. Classic methods for solving these problems rely on iterative algorithms…
The interaction of neural networks with physical equations offers a wide range of applications. We provide a method which enables a neural network to transform objects subject to given physical constraints. Therefore an U-Net architecture…
The plane case of central configurations with four different masses is analyzed theoretically and is computed numerically. We follow Dziobek's approach to four body central configurations with a direct implicit method of our own in which…
Understanding the uncertainty of a neural network's (NN) predictions is essential for many purposes. The Bayesian framework provides a principled approach to this, however applying it to NNs is challenging due to large numbers of parameters…