In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation.
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48 results for “Time-Dependent Equations”
Paper shows how scattering maps of Schrödinger equations relate to metrics.
problem Relating scattering maps of time-dependent Schrödinger equations to metrics.
method Analyzes scattering maps for specific classes of metrics and diffeomorphisms.
result Scattering maps differ by a compact operator if and only if metrics are related by diffeomorphism.
The time-dependent maximum principle for systems of parabolic equations subject to an avoidance setmath.DG
In this paper we prove two extensions of Hamilton's maximal principle for systems pf parabolic equations which sould be useful for the study of the Ricci flow and some other geometric evolution equations. One extension is a time-dependent maximum principle and the other is a time-dependent maximum principle subject to …
Mathematical models with time dependent parameters are of great interest in financial Mathematics because they capture real life scenarios in the financial market. In this study, via the Lie group technique, we analyse evolution-type equations with time dependent parameters and give the general symmetry structure of th…
Paper develops semi-analytic method for American options in time-dependent jump-diffusion models.
problem Pricing American options in models with time-dependent and exponential jumps.
method Generalizes existing methods for barrier and American options to handle arbitrary time dependencies and solves the problem through algebraic and Fredholm-Volterra equations.
result Presents a semi-analytic solution for American options in time-dependent jump-diffusion models with exponential jumps.
The paper analyzes convergence of Langevin dynamics with time-dependent metrics.
problem Analyzing convergence of Langevin dynamics with time-dependent metrics.
method Formulated a modified gradient flow of the Kullback-Leibler divergence, selected a time-dependent relative Fisher information functional, and developed a time-dependent Hessian matrix condition.
result Proved convergence conditions for various Langevin dynamics.
Develops methods to learn correlation potentials for time-dependent Kohn-Sham systems.
problem Learning the correlation potential for time-dependent Kohn-Sham systems.
method Optimizing a least-squares objective subject to the TDKS equation using adjoints.
result Learned correlation potential models match ground truth electron densities and can have memory.
Study of time-dependent metrics and connections in geometry.
problem Understanding geodesics and connections in time-dependent Riemannian manifolds.
method Examine connections on product manifolds, explore parallel transport, geodesics, and torsion.
result Define the derivative of a one-parameter family of connections.
We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the finite maturity American put option. The optimal exercise boundary satisfies the nonlinear integral equation of Volterra type. We …
We consider Noether symmetries of the equations defined by the sections of characteristic line bundles of nondegenerate 1-forms and of the associated perturbed systems. It appears that this framework can be used for time-dependent systems with constraints and nonconservative forces, allowing a quite simple and transpar…
Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationsstat.ML
We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where: (1) all we observe are …
New method for pricing American options in time-dependent models, improving accuracy and efficiency.
problem Pricing American options in time-dependent models with improved accuracy and efficiency.
method Semi-analytical pricing using a nonlinear Volterra integral equation and numerical methods.
result Improved accuracy and efficiency in pricing American options compared to forward finite difference solvers.
We study the heat equation on time-dependent metric measure spaces (as well as the dual and the adjoint heat equation) and prove existence, uniqueness and regularity. Of particular interest are properties which characterize the underlying space as a super Ricci flow as previously introduced by the second author. Our ma…
Paper explores solving HJB equations using neural networks.
problem Solving high-dimensional time-dependent HJB equations.
method Neural Galerkin methods with nonlinearly parametrized trial functions.
result Closed-form solutions for trial functions.
Develops semi-closed form solutions for barrier and American options on time-dependent OU process.
problem Valuation of barrier and American options on a time-dependent Ornstein-Uhlenbeck process.
method Semi-closed form solutions involving numerical solution of Fredholm equations and integration of Jacobi theta functions.
result Method is more efficient than backward finite difference method and can be as efficient as forward finite difference solver with better accuracy and stability.
Probabilistic method combines space and time uncertainties in PDEs.
problem Separate treatment of space and time in PDE solvers obscures interactions and error quantification.
method Gaussian process interpretation of finite difference methods interacting with probabilistic ODE solvers.
result Joint quantification of space- and time-uncertainty possible without sacrificing ODE solver performance.
New quantum algorithm simplifies complex financial derivatives pricing.
problem Complex financial derivatives pricing with high dimensionality.
method Quantum-inspired variational algorithms combined with neural-network quantum states.
result Simplified pricing of European options with many correlated assets.
We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a generic set of particular solutions of the system and some constants. We next study…
Optimizes electric field to control molecule states in Hartree-Fock theory.
problem Optimizing electric field to drive molecule from initial to target state.
method Trust region optimization with gradients from adjoint state method.
result Achieves desired target states with minimal control effort.
New method for pricing barrier options in time-dependent λ-SABR model.
problem Pricing barrier options in the time-dependent λ-SABR model.
method Modified integral transform method and Fourier-Bessel series solution.
result Semi-analytical solution for barrier options in λ-SABR model.
In this paper, the author discusses the elliptic type gradient estimate for the solution of the time-dependent Schrödinger equations on noncompact manifolds. As its application, the dimension-free Harnack inequality and the Liouville type theorem for the Schrödinger equation are proved.
A new RNN model tackles long-time dependencies with fast, invertible, and memory-efficient hidden states.
problem Challenges in processing sequential inputs with long-time dependencies in RNNs.
method A novel RNN architecture based on a Hamiltonian system of oscillators.
result The proposed RNN mitigates exploding and vanishing gradient problems, providing state-of-the-art performance.
MAntRA combines machine learning and Bayesian methods for time-dependent reliability analysis of unknown systems.
problem Time-dependent reliability analysis of systems with unknown governing physics.
method Combines machine learning, Bayesian statistics, and stochastic integration to discover and analyze SDEs from data.
result Demonstrates the effectiveness of MAntRA on three numerical examples, indicating its potential for in-situ and heritage structure analysis.
GrADE: A graph based data-driven solver for time-dependent nonlinear partial differential equationsstat.ML
GrADE uses graph neural networks and Neural ODE for solving time-dependent nonlinear PDEs efficiently.
problem Solving time-dependent nonlinear PDEs is computationally challenging and time-consuming.
method GrADE combines graph neural networks for spatial modeling and Neural ODE for temporal modeling, using attention mechanisms.
result GrADE efficiently solves PDEs, demonstrating scalability and better accuracy compared to existing methods.
A new method uses deep learning to efficiently solve complex physics equations in high dimensions.
problem Efficiently solving high-dimensional time-dependent PDEs with dynamic solutions.
method Deep adaptive sampling framework for PINNs extended to spacetime domains using normalizing flows.
result The method effectively identifies and tracks high-residual regions in both space and time.
On the Hamiltonian and Lagrangian structures of time-dependent reductions of evolutionary PDEsmath.DG
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 work, we present a machine learning approach for reducing the error when numerically solving time-dependent partial differential equations (PDE). We use a fully convolutional LSTM network to exploit the spatiotemporal dynamics of PDEs. The neural network serves to enhance finite-difference and finite-volume met…
New method uses randomized sparse neural networks to solve time-dependent PDEs more accurately and efficiently.
problem Numerical challenges in training neural networks sequentially in time to solve time-dependent PDEs.
method Introduces Neural Galerkin schemes that update randomized sparse subsets of network parameters at each time step.
result Up to two orders of magnitude more accurate and two orders of magnitude faster than dense update schemes.
The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.
problem Finding general solutions for Lie systems.
method Develops mixed superposition rules for Lie systems with imprimitive Lie algebras and semidirect sums.
result Extends coalgebra method to Lie systems of partial differential equations.
Quantum mechanics applied to option pricing with a time-dependent bubble.
problem Option pricing with a time-dependent arbitrage bubble.
method Application of Dirac's interaction picture to the Black-Scholes equation.
result Exact and approximate solutions for option pricing with a square bubble.
Jet Methods in Time-Dependent Lagrangian Biomechanicsphysics.bio-ph
In this paper we propose the time-dependent generalization of an `ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds associated to the extended musculo-skeletal configuration…
We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex if the metric evolves under super Ricci flow (which includes Ricci flow and fixed…
The usual formulations of time-dependent mechanics start from a given splitting of the coordinate bundle . From physical viewpoint, this splitting means that a reference frame has been chosen. Obviously, such a splitting is broken under reference frame transformations and time-dependent canonical …
Time dependent quantum scattering theory on complete manifolds with a corner of codimension 2math.DG
We show the existence and orthogonality of wave operators naturally associated to a compatible Laplacian on a complete manifold with a corner of codimension 2. In fact, we prove asymptotic completeness i.e. that the image of these wave operators is equal to the space of absolutely continuous states of the compatible La…
In the current paper,under the transverse Ricci flow on a totally geodesic Riemannian foliation, we prove two types of differential Harnack inequalities (Li-Yau gradient estimate) for the positive solutions of the heat equation associated with the time dependent horizontal Laplacian operators. We also get a time depend…
Dynamic Structural Causal Models handle time-dependent systems with cycles and latent confounding.
problem Representing and analyzing systems of Stochastic Differential Equations (SDEs) with DSCMs.
method Define time-splitting and subsampling operations to analyze DSCMs of SDEs, and apply existing causal discovery algorithms to time-series data.
result DSCMs provide a graphical Markov property for SDEs and enable identification of time-dependent causal effects.
Deep learning estimates time-varying Markov model parameters.
problem Estimating time-dependent parameters in Markov models.
method Reframes parameter estimation as an optimization problem using maximum likelihood.
result Real solution close to SDE with neural network-derived parameters under specific conditions.
There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labelled as Physics Informed Neural Networks (PINNs). In this paper, we develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to s…
In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show t…
Study shows how heat leaks from material sets in low diffusivity scenarios.
problem Understanding heat leakage from material sets in low diffusivity limits.
method Generalized leading-order asymptotics for time-dependent diffusion processes.
result Diffusive transport out of a material set is proportional to the surface area of the set boundary.
A time-dependent double-barrier option is a derivative security that delivers the terminal value at expiry if neither of the continuous time-dependent barriers $b_\pm:[0,T]\to \RR_+$ have been hit during the time interval . Using a probabilistic approach we obtain a decomposition of the barrier opti…
Derives semi-closed form prices for barrier options in the Hull-White model.
problem Calculating prices of barrier options in the Hull-White model with time-dependent parameters.
method Applies generalized integral transform and heat potentials to solve linear Volterra equations of the first kind.
result The method provides more efficient and accurate solutions compared to finite difference methods.
Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved estimate with time-dependent Gaussian weight.
result Uniform bounds for heat equation along Ricci flow.
This project attempts to address the problem of asset pricing in a financial market, where the interest rates and volatilities exhibit regime switching. This is an extension of the Black-Scholes model. Studies of Markov-modulated regime switching models have been well-documented. This project extends that notion to a c…
The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility depending on the option price. We review a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. R…
A notion of implicit difference equation on a Lie groupoid is introduced and an algorithm for extracting the integrable part (backward or/and forward) is formulated. As an application, we prove that discrete Lagrangian dynamics on a Lie groupoid may be described in terms of Lagrangian implicit difference equations …
Researchers find the optimal exercise time for American options using a specific type of diffusion process.
problem Finding the optimal time to exercise American options with a time-dependent Ornstein-Uhlenbeck process.
method Optimal stopping problem, probabilistic arguments, non-linear Volterra-type integral equation, Picard iteration algorithm.
result They derive a non-linear Volterra-type integral equation and prove the exercise boundary's Lipschitz continuity and differentiability almost everywhere.
While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from {\em small} data. In particular, we introduce \emph{hidden physics models}, which are essentially data-efficien…