arXiv research

Develops methods for conditional symmetries of higher-order PDEs, removing unnecessary assumptions.

In this paper, a standard PDE for the pricing of arithmetic average strike Asian call option is presented. A Crank-Nicolson Implicit Method and a Higher Order Compact finite difference scheme for this pricing problem is derived. Both these schemes were implemented for various values of risk free rate and volatility. Th…

Paper tackles stochastic control with mean and higher-order moments, finding Nash equilibria.

New framework trains large SciML models solving PDEs in reasonable time.

In this paper we study the reductions of evolutionary PDEs on the manifold of the stationary points of time--dependent symmetries. In particular we describe how that the finite dimensional Hamiltonian structure of the reduced system is obtained from the Hamiltonian structure of the initial PDE and we construct the time…

In this paper we derive the symplectic framework for field theories defined by higher-order Lagrangians. The construction is based on the symplectic reduction of suitable spaces of iterated jets. The possibility of reducing a higher-order system of PDEs to a constrained first-order one, the symplectic structures natura…

First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this …

Graph neural networks learn PDEs from sparse, irregular data.

In this paper we extend Buchen's method to develop a new technique for pricing of some exotic options with several expiry dates(more than 3 expiry dates) using a concept of higher order binary option. At first we introduce the concept of higher order binary option and then provide the pricing formulae of $n$-th order b…

In this work, we have presented a simple analytical approximation scheme for generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown that it is possible to carry out a recursive approximation to an arbitrarily highe…

We study higher-order conservation laws of the non-linearizable elliptic Poisson equation $\frac{{\partial}^2 u}{\partial z \partial \bar{z}} = -f(u)$ as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for diffe…

Characterizes surfaces enveloped by rotating cones for CNC machining.

Jet bundles as higher-order polarised $k$-contact manifolds

HTE improves PINNs for high-dimensional, high-order PDEs by reducing computational cost and memory usage.

DeepSVM learns SVMs without PDE solving, achieving high pricing accuracy.

In this article, we consider a 2 factors-model for pricing defaultable bond with discrete default intensity and barrier where the 2 factors are stochastic risk free short rate process and firm value process. We assume that the default event occurs in an expected manner when the firm value reaches a given default barrie…

We use methods from exterior differential systems (EDS) to develop a geometric theory of scalar, first-order Lagrangian functionals and their associated Euler-Lagrange PDEs, subject to contact transformations. The first chapter contains an introduction of the classical Poincare-Cartan form in the context of EDS, follow…

New Lie systems defined on $k$-contact manifolds, with applications.

New ADANNs improve PDE approximations.

A weak law of large numbers is established for a sequence of systems of N classical point particles with logarithmic pair potential in $\bbR^n$, or $\bbS^n$, $n\in \bbN$, which are distributed according to the configurational microcanonical measure $δ(E-H)$, or rather some regularization thereof, where H is the configu…

The paper develops a new approach to solve vector-valued PDEs on manifolds with minimal regularity.

The theory of multidimensional Poisson vertex algebras (mPVAs) provides a completely algebraic formalism to study the Hamiltonian structure of PDEs, for any number of dependent and independent variables. In this paper, we compute the cohomology of the PVAs associated with two-dimensional, two-components Poisson bracket…

This thesis advances algorithms and software for QMC, GP, and sciML.

We consider the projective Finsler metrizability problem: under what conditions the solutions of a given system of second-order ordinary differential equations (SODE) coincide with the geodesics of a Finsler metric, as oriented curves. SODEs with isotropic curvature have already been thoroughly studied in the literatur…

In this article, we study the problem of pricing defaultable bond with discrete default intensity and barrier under constant risk free short rate using higher order binary options and their integrals. In our credit risk model, the risk free short rate is a constant and the default event occurs in an expected manner whe…

A new method infers parameters from PDEs using Gaussian processes.

Study heat content in sub-Riemannian structures, proving asymptotic series existence and coefficients.

Neural-PDE learns PDEs from data using LSTM, outperforming traditional methods.

PDE-Net 2.0 learns PDEs from data without prior knowledge.

Neural Q-learning tackles high-dimensional PDEs.

PDE-DKL combines NNs and GPs for high-dimensional PDE problems.

Solves second-order PDEs using quotients and differential invariants.

Graphs improve theorem proving in higher-order logic.

We study an intrinsic distribution, called polar, on the space of $l$-dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle…

PRISMA uses PDE residuals for fast, robust, and accurate inference.

Convert PDEs into Pfaffian fibrations for easier study.

Paper introduces techniques to learn higher-order programs, improving predictive accuracy and reducing learning times.

Meta-learning base distributions for efficient PDE solutions.

For certain classes of knots we define geometric invariants called higher-order genera. Each of these invariants is a refinement of the slice genus of a knot. We find lower bounds for the higher-order genera in terms of certain von Neumann $ρ$-invariants, which we call higher-order signatures. The higher-order genera o…

Paper uses deep learning to solve PDEs without supervision.

Kernel method learns PDEs from noisy data.

Solves a PDE for Landsberg surfaces using new Finsler surface insights.

A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a triangle in the network. However, higher-order cliques beyond triangles are crucia…

Survey on conservation laws for geometric PDEs.

Using the theory of the symmetry group for PDEs [15, 17], we derive the symmetry group G associated to surfaces PDE. Several group invariant solutions of the surfaces PDE are given by solving a reduced system of partial differential equations.

Stability of capillary hypersurfaces with higher order mean curvature.

PDE-based G-CNNs add geometric symmetries to CNNs without augmentation.

Unified framework solves nonlinear PDEs and IPs using Gaussian processes.