arXiv research

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.

Probabilistic method combines space and time uncertainties in PDEs.

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 …

The paper analyzes convergence of Langevin dynamics with time-dependent metrics.

Study of time-dependent metrics and connections in geometry.

New quantum algorithm simplifies complex financial derivatives pricing.

Paper explores solving HJB equations using neural networks.

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…

Paper develops semi-analytic method for American options in time-dependent jump-diffusion models.

A new method uses deep learning to efficiently solve complex physics equations in high dimensions.

GrADE uses graph neural networks and Neural ODE for solving time-dependent nonlinear PDEs efficiently.

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…

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…

The paper develops methods to derive mixed superposition rules for Lie systems and applies them to various physical systems.

MAntRA combines machine learning and Bayesian methods for time-dependent reliability analysis of unknown systems.

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…

New method uses randomized sparse neural networks to solve time-dependent PDEs more accurately and efficiently.

Dynamic Structural Causal Models handle time-dependent systems with cycles and latent confounding.

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…

A new RNN model tackles long-time dependencies with fast, invertible, and memory-efficient hidden states.

Deep learning estimates time-varying Markov model parameters.

LFIS uses a time-dependent velocity field to sample from complex distributions.

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 $G$ may be described in terms of Lagrangian implicit difference equations …

Paper shows how scattering maps of Schrödinger equations relate to metrics.

A new model adapts Hurst parameter in real-time for volatility forecasting.

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…

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 …

The usual formulations of time-dependent mechanics start from a given splitting $Y=R\times M$ of the coordinate bundle $Y\to R$. 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 …

Study methods to recover unknown processes in PDEs from data.

We propose a neural network based approach for extracting models from dynamic data using ordinary and partial differential equations. In particular, given a time-series or spatio-temporal dataset, we seek to identify an accurate governing system which respects the intrinsic differential structure. The unknown governing…

Paper introduces a modified Allen-Cahn equation for better energy equipartition.

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…

Researchers find the optimal exercise time for American options using a specific type of diffusion process.

We present a reformulation of the inverse problem of the calculus of variations for time dependent systems of second order ordinary differential equations using the Frölicher-Nijenhuis theory on the first jet bundle, $J^1π$. We prove that a system of time dependent SODE, identified with a semispray $S$, is Lagrangian i…

Stochastic delay differential equations (SDDE's) have been used for financial modeling. In this article, we study a SDDE obtained by the equation of a CIR process, with an additional fixed delay term in drift; in particular, we prove that there exists a unique strong solution (positive and integrable) which we call fix…

If $U:[0,+\infty[\times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $\partial_tU+ H(x,\partial_xU)=0,$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we prove the set $Σ(U)$, of points where $U$ is not differentiable, is locally contrac…

A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize Lie algebras and Lie algebroids. This scheme covers and unifies various geometrical approaches to mechanics in the Lagrangian and Hamilt…

LR-EDNN reduces PDE solver complexity by limiting network weights to low-rank subspace.

RandNet-Parareal uses neural networks to speed up time-parallel PDE solving.

A new method models continuous-time counterfactual outcomes using neural controlled differential equations.

A new kernel framework analyzes spatio-temporal data from dynamic equations.

Develops methods to learn correlation potentials for time-dependent Kohn-Sham systems.

Let $(M, g(t))$, $t\in[0,T)$ be a closed Riemannian $n$-manifold whose Riemannian metric $g(t)$ evolves by the geometric flow $\frac{\partial }{\partial t} g_{ij}=-2S_{ij}$, where $S_{ij}(t)$ is a symmetric two-tensor on $(M,g(t))$. We discuss differential Harnack estimates for positive solution to the porous medium …

Generalization bounds derived for neural ODEs and deep residual networks.

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 …

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

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…

New method for pricing American options in time-dependent models, improving accuracy and efficiency.