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In this paper, we consider the composition of two independent processes : one process corresponds to position and the other one to time. Such processes will be called iterated processes. We first propose an algorithm based on the Euler…
Since the seminal work of [9] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for…
We propose a Deep-Picard iteration framework for high-dimensional nonlinear space-time fractional diffusion equations.The method is based on a nonlinear fractional Feynman--Kac fixed-point formulation, which replaces direct discretization…
Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed…
Deep learning has achieved remarkable success in diverse applications; however, its use in solving partial differential equations (PDEs) has emerged only recently. Here, we present an overview of physics-informed neural networks (PINNs),…
In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and the finite element method called Phi-FEM. For that, we use the Fourier Neural Operator (FNO), a learning mapping…
We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral…
Neural network-based methods have emerged as powerful tools for solving partial differential equations (PDEs) in scientific and engineering applications, particularly when handling complex domains or incorporating empirical data. These…
Backward stochastic differential equation (BSDE) provides probabilistic solutions for a class of parabolic partial differential equations (PDEs). DeepBSDE and FBSNN are two deep learning approaches for solving high-dimensional PDEs through…
Physics-informed neural network (PINN) has shown great potential in solving partial differential equations. However, it faces challenges when dealing with problems involving steep gradients. The solutions to singularly perturbed…
Deep learning has been shown to be an effective tool in solving partial differential equations (PDEs) through physics-informed neural networks (PINNs). PINNs embed the PDE residual into the loss function of the neural network, and have been…
We propose a new multistep deep learning-based algorithm for the resolution of moderate to high dimensional nonlinear backward stochastic differential equations (BSDEs) and their corresponding parabolic partial differential equations (PDE).…
Fast and accurate solutions of time-dependent partial differential equations (PDEs) are of pivotal interest to many research fields, including physics, engineering, and biology. Generally, implicit/semi-implicit schemes are preferred over…
As the dimension of a system increases, traditional methods for control and differential games rapidly become intractable, making the design of safe autonomous agents challenging in complex or team settings. Deep-learning approaches avoid…
We propose a trainable-by-parts surrogate model for solving forward and inverse parameterized nonlinear partial differential equations. Like several other surrogate and operator learning models, the proposed approach employs an encoder to…
We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms…
In this paper, we propose a novel supervised learning method that is called Deep Embedding Kernel (DEK). DEK combines the advantages of deep learning and kernel methods in a unified framework. More specifically, DEK is a learnable kernel…
Estimating the likelihood, timing, and nature of events is a major goal of modeling stochastic dynamical systems. When the event is rare in comparison with the timescales of simulation and/or measurement needed to resolve the elemental…
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…
The neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility in representing the solution of the partial differential equation. In training a neural…