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Partial Differential Equations (PDEs) are used to model a variety of dynamical systems in science and engineering. Recent advances in deep learning have enabled us to solve them in a higher dimension by addressing the curse of…

Neural Ordinary Differential Equations model dynamical systems with ODEs learned by neural networks. However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in…

Neural networks inspired by differential equations have proliferated for the past several years. Neural ordinary differential equations (NODEs) and neural controlled differential equations (NCDEs) are two representative examples of them. In…

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…

Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we…

As we know differential equations are very useful for electrical engineers to solve a variety of problems like: voltage across a capacitor, input versus output voltage, etc. Therefore, the goal of this paper is to find the solutions of…

Due to the curse of dimensionality, solving high dimensional parabolic partial differential equations (PDEs) has been a challenging problem for decades. Recently, a weak adversarial network (WAN) proposed in (Y.Zang et al., 2020) offered a…

When using Neural Networks as trial functions to numerically solve PDEs, a key choice to be made is the loss function to be minimised, which should ideally correspond to a norm of the error. In multiple problems, this error norm coincides…

We propose a method combining boundary integral equations and neural networks (BINet) to solve partial differential equations (PDEs) in both bounded and unbounded domains. Unlike existing solutions that directly operate over original PDEs,…

The linear inverse problem is fundamental to the development of various scientific areas. Innumerable attempts have been carried out to solve different variants of the linear inverse problem in different applications. Nowadays, the rapid…

A novel way of using neural networks to learn the dynamics of time delay systems from sequential data is proposed. A neural network with trainable delays is used to approximate the right hand side of a delay differential equation. We relate…

Neural networks have been applied to control problems, typically by combining data, differential equation residuals, and objective costs in the training loss or by incorporating auxiliary architectural components. Instead, we propose a…

Deep Learning (DL), in particular deep neural networks (DNN), by default is purely data-driven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering…

Recently proposed numerical algorithms for solving high-dimensional nonlinear partial differential equations (PDEs) based on neural networks have shown their remarkable performance. We review some of them and study their convergence…

We propose a new approach for large-scale high-dynamic range computational imaging. Deep Neural Networks (DNNs) trained end-to-end can solve linear inverse imaging problems almost instantaneously. While unfolded architectures provide…

The physical world is governed by the laws of physics, often represented in form of nonlinear partial differential equations (PDEs). Unfortunately, solution of PDEs is non-trivial and often involves significant computational time. With…

Deep neural networks have proven extremely efficient at solving a wide rangeof inverse problems, but most often the uncertainty on the solution they provideis hard to quantify. In this work, we propose a generic Bayesian framework…

This paper proposes the Nerual Energy Descent (NED) via neural network evolution equations for a wide class of deep learning problems. We show that deep learning can be reformulated as the evolution of network parameters in an evolution…

We propose a new deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and forward-backward stochastic differential equations with jumps (FBSDEJs). This novel algorithm can be viewed as an…

Utilizing physics-informed neural networks (PINN) to solve partial differential equations (PDEs) becomes a hot issue and also shows its great powers, but still suffers from the dilemmas of limited predicted accuracy in the sampling domain…