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Numerical solutions of partial differential equations (PDEs) require expensive simulations, limiting their application in design optimization, model-based control, and large-scale inverse problems. Surrogate modeling techniques seek to…

Deep learning has been proposed as an efficient alternative for the numerical approximation of PDE solutions, offering fast, iterative simulation of PDEs through the approximation of solution operators. However, deep learning solutions have…

Data augmentation is a popular technique which helps improve generalization capabilities of deep neural networks. It plays a pivotal role in remote-sensing scenarios in which the amount of high-quality ground truth data is limited, and…

Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…

Many engineering and scientific fields have recently become interested in modeling terms in partial differential equations (PDEs) with neural networks, which requires solving the inverse problem of learning neural network terms from…

Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability.…

In this article, we propose a new numerical approach to high-dimensional partial differential equations (PDEs) arising in the valuation of exotic derivative securities. The proposed method is extended from Reisinger and Wittum (2007) and…

Simulating the time evolution of Partial Differential Equations (PDEs) of large-scale systems is crucial in many scientific and engineering domains such as fluid dynamics, weather forecasting and their inverse optimization problems.…

High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in…

This paper develops a probabilistic numerical method for solution of partial differential equations (PDEs) and studies application of that method to PDE-constrained inverse problems. This approach enables the solution of challenging inverse…

Coupled 3D-1D problems arise in many practical applications, in an attempt to reduce the computational burden in simulations where cylindrical inclusions with a small section are embedded in a much larger domain. Nonetheless the resolution…

We investigate the inverse problem for Partial Differential Equations (PDEs) in scenarios where the parameters of the given PDE dynamics may exhibit changepoints at random time. We employ Physics-Informed Neural Networks (PINNs) - universal…

Solving Partial Differential Equations (PDEs) is a cornerstone of engineering and scientific research. Traditional methods for PDE solving are cumbersome, relying on manual setup and domain expertise. While Physics-Informed Neural Network…

Partial differential equations (PDEs) are central to describing complex physical system simulations. Their expensive solution techniques have led to an increased interest in deep neural network based surrogates. However, the practical…

Multiphysics problems such as multicomponent diffusion, phase transformations in multiphase systems and alloy solidification involve numerical solution of a coupled system of nonlinear partial differential equations (PDEs). Numerical…

Large-batch SGD is important for scaling training of deep neural networks. However, without fine-tuning hyperparameter schedules, the generalization of the model may be hampered. We propose to use batch augmentation: replicating instances…

One of the main challenges in solving time-dependent partial differential equations is to develop computationally efficient solvers that are accurate and stable. Here, we introduce a graph neural network approach to finding efficient PDE…

Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion…

This paper proposes a Kolmogorov high order deep neural network (K-HOrderDNN) for solving high-dimensional partial differential equations (PDEs), which improves the high order deep neural networks (HOrderDNNs). HOrderDNNs have been…