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Physics-Informed Neural Networks (PINNs) are a class of deep learning neural networks that learn the response of a physical system without any simulation data, and only by incorporating the governing partial differential equations (PDEs) in…

Physics-informed neural networks (PINNs) have shown remarkable prospects in the solving the forward and inverse problems involving partial differential equations (PDEs). The method embeds PDEs into the neural network by calculating PDE loss…

Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…

The recently developed physics-informed machine learning has made great progress for solving nonlinear partial differential equations (PDEs), however, it may fail to provide reasonable approximations to the PDEs with discontinuous…

Distributed optimization is fundamental to modern machine learning applications like federated learning, but existing methods often struggle with ill-conditioned problems and face stability-versus-speed tradeoffs. We introduce fractional…

Accurate and efficient solutions of spatiotemporal partial differential equations (PDEs), such as phase-field models, are fundamental for understanding interfacial dynamics and microstructural evolution in materials science and fluid…

Computational Fluid Dynamics (CFD) simulation by the numerical solution of the Navier-Stokes equations is an essential tool in a wide range of applications from engineering design to climate modeling. However, the computational cost and…

Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by…

The rise of Physics-Informed Neural Networks (PINNs) has popularized the concept of solving differential equations via residual minimization. However, neural networks are often viewed as ``black boxes" requiring significant computational…

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their…

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose…

Computational fluid dynamics (CFD) is a specialised branch of fluid mechanics that utilises numerical methods and algorithms to solve and analyze fluid-flow problems. One promising avenue to enhance CFD is the use of quantum computing,…

Simulating spatiotemporal turbulence with high fidelity remains a cornerstone challenge in computational fluid dynamics (CFD) due to its intricate multiscale nature and prohibitive computational demands. Traditional approaches typically…

Spatiotemporal partial differential equations (PDEs) find extensive application across various scientific and engineering fields. While numerous models have emerged from both physics and machine learning (ML) communities, there is a growing…

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or accurate resolution of their solutions may be intractable. Instead,…

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…

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for…

Computational cardiovascular flow modeling plays a crucial role in understanding blood flow dynamics. While 3D models provide acute details, they are computationally expensive, especially with fluid-structure interaction (FSI) simulations.…

Physics Informed Neural Networks (PINNs) often exhibit failure modes in which the PDE residual loss converges while the solution error stays large, a phenomenon traditionally blamed on local optima separated from the true solution by steep…

In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a…