Related papers: BEACONS: Bounded-Error, Algebraically-Composable N…

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…

A large class of hyperbolic and advection-dominated PDEs can have solutions with discontinuities. This paper investigates, both theoretically and empirically, the operator learning of PDEs with discontinuous solutions. We rigorously prove,…

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 study, we present and validate the predictive capability of the Physics-Informed Neural Networks (PINNs) methodology for solving a variety of engineering and biological dynamical systems governed by ordinary differential equations…

Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or…

Elliptic partial differential equations (PDEs) are a major class of time-independent PDEs that play a key role in many scientific and engineering domains such as fluid dynamics, plasma physics, and solid mechanics. Recently, neural…

Deep models have recently emerged as promising tools to solve partial differential equations (PDEs), known as neural PDE solvers. While neural solvers trained from either simulation data or physics-informed loss can solve PDEs reasonably…

Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks…

Verification of Neural Networks (NNs) that approximate the solution of Partial Differential Equations (PDEs) is a major milestone towards enhancing their trustworthiness and accelerating their deployment, especially for safety-critical…

Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that…

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…

In this paper, we investigate the applications of operator learning, specifically DeepONet, for solving nonlinear partial differential equations (PDEs). Unlike conventional function learning methods that require training separate neural…

This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations…

In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as…

Deep neural networks are powerful tools for solving nonlinear problems in science and engineering, but training highly accurate models becomes challenging as problem complexity increases. Non-convex optimization and sensitivity to…

DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with…

This paper explores the difficulties in solving partial differential equations (PDEs) using physics-informed neural networks (PINNs). PINNs use physics as a regularization term in the objective function. However, a drawback of this approach…

Partial differential equations (PDEs) serve as the cornerstone of mathematical physics. In recent years, Physics-Informed Neural Networks (PINNs) have significantly reduced the dependence on large datasets by embedding physical laws…

We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the…

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we…