Related papers: Signature Kernel and Schwinger-Dyson Kernel Equati…
Quadrature formulas for spheres, the rotation group, and other compact, homogeneous manifolds are important in a number of applications and have been the subject of recent research. The main purpose of this paper is to study coordinate…
Evaluating whether data streams are drawn from the same distribution is at the heart of various machine learning problems. This is particularly relevant for data generated by dynamical systems since such systems are essential for many…
This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and…
We analyze feature learning in infinite-width neural networks trained with gradient flow through a self-consistent dynamical field theory. We construct a collection of deterministic dynamical order parameters which are inner-product kernels…
In this paper, we investigate reflected backward stochastic differential equations driven by rough paths (rough RBSDEs), which can be viewed as probabilistic representations of nonlinear rough partial differential equations (rough PDEs) or…
A numerical scheme is presented for solving the Helmholtz equation with Dirichlet or Neumann boundary conditions on piecewise smooth open curves, where the curves may have corners and multiple junctions. Existing integral equation methods…
Resonant solutions of the quantum Schr\"odinger equation occur at complex energies where the S-matrix becomes singular. Knowledge of such resonances is important in the study of the underlying physical system. Often the Schr\"odinger…
The dymamical chiral symmetry breaking in higher- derivative quantum gravity has been investigated on the flat background. The Schwinger- Dyson equations numerical solutions have been found in the ladder approximation. Both two- and four-…
The Schwinger-Dyson equation of fermion self-energy in the linearization approximation is solved exactly in a theory with gauge and effective four-fermion interactions. Different expressions for the indepedent solutions which respectively…
Feynman's path integral formulation arose from his attempt to incorporate the Lagrangian framework into quantum mechanics, offering what he regarded as a more fundamental perspective than the Hamiltonian approach, particularly in the…
Physics-Informed Neural Networks (PINNs) are a powerful class of numerical solvers for partial differential equations, employing deep neural networks with successful applications across a diverse set of problems. However, their…
The signature kernel is a positive definite kernel for sequential data. It inherits theoretical guarantees from stochastic analysis, has efficient algorithms for computation, and shows strong empirical performance. In this short survey…
We develop a data-driven machine learning approach to identifying parameters with steady-state solutions, locating such solutions, and determining their linear stability for systems of ordinary differential equations and dynamical systems…
Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations.…
This research explores neural network-based numerical approximation of two-dimensional convection-dominated singularly perturbed problems on square, circular, and elliptic domains. Singularly perturbed boundary value problems pose…
A path integral is presented that solves a general class of linear second order partial differential equations with Dirichlet/Neumann boundary conditions. Elementary kernels are constructed for both Dirichlet and Neumann boundary…
In this work, we propose a simple kernel ridge regression (KRR) framework with a dynamic-aware validation strategy for long-term prediction of complex dynamical systems. By employing a data-driven kernel derived from diffusion maps, the…
Accurate and efficient environment representation is crucial for robotic applications such as motion planning, manipulation, and navigation. Signed distance functions (SDFs) have emerged as a powerful representation for encoding distance to…
Parallel transport, or path development, provides a rich characterization of paths which preserves the underlying algebraic structure of concatenation. The path signature is universal among such maps: any (translation-invariant) parallel…
We extend the adjoint method to complex-valued PDEs and introduce the Fr\'echet root sensitivity kernel, as the most fundamental kernel from which all other material-sensitivity kernels can be derived. We apply this framework to four…