Multilayer heat equations and their solutions via oscillating integral transforms
By expanding the Dirac delta function in terms of the eigenfunctions of the corresponding Sturm-Liouville problem, we construct some new (oscillating) integral transforms. These transforms are then used to solve various finance, physics, and mathematics problems, which could be characterized by the existence of a multilayer spatial structure and moving (time-dependent) boundaries (internal interfaces) between the layers. Thus, constructed solutions are semi-analytical and extend the authors' previous work (Itkin, Lipton, Muravey, Multilayer heat equations: application to finance, FMF, 1, 2021). However, our new method doesn't duplicate the previous one but provides alternative representations of the solution which have different properties and serve other purposes.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Multilayer heat equations: application to finance
In this paper, we develop a Multilayer (ML) method for solving one-factor parabolic equations. Our approach provides a powerful alternative to the well-known finite difference and Monte Carlo methods. We discuss various …
Learning Heat-based Equations in Self-similar variables
We study solution learning for heat-based equations in self-similar variables (SSV). We develop an SSV training framework compatible with standard neural-operator training. We instantiate this framework on the two-dimens…
On Theory-training Neural Networks to Infer the Solution of Highly Coupled Differential Equations
Deep neural networks are transforming fields ranging from computer vision to computational medicine, and we recently extended their application to the field of phase-change heat transfer by introducing theory-trained neu…
An unsupervised learning approach to solving heat equations on chip based on Auto Encoder and Image Gradient
Solving heat transfer equations on chip becomes very critical in the upcoming 5G and AI chip-package-systems. However, batches of simulations have to be performed for data driven supervised machine learning models. Data …
Some observations on high-dimensional partial differential equations with Barron data
We use explicit representation formulas to show that solutions to certain partial differential equations lie in Barron spaces or multilayer spaces if the PDE data lie in such function spaces. Consequently, these solution…
Vocal Bursts Intensity Prediction