PDGM: a Neural Network Approach to Solve Path-Dependent Partial
Differential Equations
- URL: http://arxiv.org/abs/2003.02035v2
- Date: Fri, 3 Apr 2020 19:18:44 GMT
- Title: PDGM: a Neural Network Approach to Solve Path-Dependent Partial
Differential Equations
- Authors: Yuri F. Saporito and Zhaoyu Zhang
- Abstract summary: We propose a novel numerical method for Path-Dependent Partial Differential Equations (PPDEs)
The method consists of using a combination of feed-forward and Long-Term Memory architectures to model the solution of the PPDE.
We analyze several numerical examples, many from the Financial Mathematics literature, that show the capabilities of the method under very different situations.
- Score: 4.1499725848998965
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: In this paper, we propose a novel numerical method for Path-Dependent Partial
Differential Equations (PPDEs). These equations firstly appeared in the seminal
work of Dupire [2009], where the functional It\^o calculus was developed to
deal with path-dependent financial derivatives contracts. More specificaly, we
generalize the Deep Galerking Method (DGM) of Sirignano and Spiliopoulos [2018]
to deal with these equations. The method, which we call Path-Dependent DGM
(PDGM), consists of using a combination of feed-forward and Long Short-Term
Memory architectures to model the solution of the PPDE. We then analyze several
numerical examples, many from the Financial Mathematics literature, that show
the capabilities of the method under very different situations.
Related papers
- Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations.
We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
arXiv Detail & Related papers (2024-06-05T17:59:22Z) - Towards Efficient Time Stepping for Numerical Shape Correspondence [55.2480439325792]
Methods based on partial differential equations (PDEs) have been established, encompassing e.g. the classic heat kernel signature.
We consider here several time stepping schemes. The goal of this investigation is to assess, if one may identify a useful property of methods for time integration for the shape analysis context.
arXiv Detail & Related papers (2023-12-21T13:40:03Z) - Optimizing Solution-Samplers for Combinatorial Problems: The Landscape
of Policy-Gradient Methods [52.0617030129699]
We introduce a novel theoretical framework for analyzing the effectiveness of DeepMatching Networks and Reinforcement Learning methods.
Our main contribution holds for a broad class of problems including Max-and Min-Cut, Max-$k$-Bipartite-Bi, Maximum-Weight-Bipartite-Bi, and Traveling Salesman Problem.
As a byproduct of our analysis we introduce a novel regularization process over vanilla descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.
arXiv Detail & Related papers (2023-10-08T23:39:38Z) - A Bayesian Framework for learning governing Partial Differential
Equation from Data [0.0]
We present a new approach to discovering partial differential equations (PDEs) by combining variational Bayes and sparse linear regression.
Our method offers a promising avenue for discovering PDEs from data and has potential applications in fields such as physics, engineering, and biology.
arXiv Detail & Related papers (2023-06-08T02:48:37Z) - A Neural RDE-based model for solving path-dependent PDEs [5.6293920097580665]
The concept of the path-dependent partial differential equation (PPDE) was first introduced in the context of path-dependent derivatives in financial markets.
Compared to the classical PDE, the solution of a PPDE involves an infinite-dimensional spatial variable.
We propose a rough neural differential equation (NRDE)-based model to learn PPDEs, which effectively encodes the path information through the log-signature feature.
arXiv Detail & Related papers (2023-06-01T20:19:41Z) - D-CIPHER: Discovery of Closed-form Partial Differential Equations [80.46395274587098]
We propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations.
We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently.
arXiv Detail & Related papers (2022-06-21T17:59:20Z) - An application of the splitting-up method for the computation of a
neural network representation for the solution for the filtering equations [68.8204255655161]
Filtering equations play a central role in many real-life applications, including numerical weather prediction, finance and engineering.
One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method.
We combine this method with a neural network representation to produce an approximation of the unnormalised conditional distribution of the signal process.
arXiv Detail & Related papers (2022-01-10T11:01:36Z) - Interpolating between BSDEs and PINNs -- deep learning for elliptic and
parabolic boundary value problems [1.52292571922932]
High-dimensional partial differential equations are a recurrent challenge in economics, science and engineering.
We suggest a methodology based on the novel $textitdiffusion loss$ that interpolates between BSDEs and PINNs.
Our contribution opens the door towards a unified understanding of numerical approaches for high-dimensional PDEs.
arXiv Detail & Related papers (2021-12-07T15:01:24Z) - q-Paths: Generalizing the Geometric Annealing Path using Power Means [51.73925445218366]
We introduce $q$-paths, a family of paths which includes the geometric and arithmetic mixtures as special cases.
We show that small deviations away from the geometric path yield empirical gains for Bayesian inference.
arXiv Detail & Related papers (2021-07-01T21:09:06Z) - Solving non-linear Kolmogorov equations in large dimensions by using
deep learning: a numerical comparison of discretization schemes [16.067228939231047]
Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena.
Deep learning has been introduced to solve these equations in high-dimensional regimes.
We show that, for some discretization schemes, improvements in the accuracy are possible without affecting the observed computational complexity.
arXiv Detail & Related papers (2020-12-09T07:17:26Z) - Numerically Solving Parametric Families of High-Dimensional Kolmogorov
Partial Differential Equations via Deep Learning [8.019491256870557]
We present a deep learning algorithm for the numerical solution of parametric families of high-dimensional linear Kolmogorov partial differential equations (PDEs)
Our method is based on reformulating the numerical approximation of a whole family of Kolmogorov PDEs as a single statistical learning problem using the Feynman-Kac formula.
We show that a single deep neural network trained on simulated data is capable of learning the solution functions of an entire family of PDEs on a full space-time region.
arXiv Detail & Related papers (2020-11-09T17:57:11Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.