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In this paper, we propose DeepMartNet - a Martingale based deep neural network learning method for solving Dirichlet boundary value problems (BVPs) and eigenvalue problems for elliptic partial differential equations (PDEs) in high…
Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed-form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the…
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with…
A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…
Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here,…
We present a parallel algorithm for solving backward stochastic differential equations (BSDEs in short) which are very useful theoretic tools to deal with many financial problems ranging from option pricing option to risk management. Our…
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…
Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when…
Partial differential equations (PDEs) involving high contrast and oscillating coefficients are common in scientific and industrial applications. Numerical approximation of these PDEs is a challenging task that can be addressed, for example,…
Techniques from deep learning play a more and more important role for the important task of calibration of financial models. The pioneering paper by Hernandez [Risk, 2017] was a catalyst for resurfacing interest in research in this area. In…
A new deep-learning neural network architecture based on high-order weak approximation algorithms for stochastic differential equations (SDEs) is proposed. The architecture enables the efficient learning of martingales by deep learning…
This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and…
High-dimensional partial-differential equations (PDEs) arise in a number of fields of science and engineering, where they are used to describe the evolution of joint probability functions. Their examples include the Boltzmann and…
Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise…
Many physical processes such as weather phenomena or fluid mechanics are governed by partial differential equations (PDEs). Modelling such dynamical systems using Neural Networks is an active research field. However, current methods are…
We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network,…
Partial differential equations (PDEs) are a model candidate for soft sensors in industrial processes with spatiotemporal dependence. Although physics-informed neural networks (PINNs) are a promising machine learning method for solving PDEs,…
Existing deep learning-based calibration scheme for rough volatility models predominantly rely on supervised learning frameworks, which incur significant computational costs due to the necessity of generating massive synthetic training…
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a…
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering…