Related papers: Solving Partial Differential Equations in Differen…
Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate…
Neural ordinary differential equations (NODEs), one of the most influential works of the differential equation-based deep learning, are to continuously generalize residual networks and opened a new field. They are currently utilized for…
We proposed the boundary-integral type neural networks (BINN) for the boundary value problems in computational mechanics. The boundary integral equations are employed to transfer all the unknowns to the boundary, then the unknowns are…
We propose a very general framework for deriving rigorous bounds on the approximation error for physics-informed neural networks (PINNs) and operator learning architectures such as DeepONets and FNOs as well as for physics-informed operator…
Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into…
Random ordinary differential equations (RODEs), i.e. ODEs with random parameters, are often used to model complex dynamics. Most existing methods to identify unknown governing RODEs from observed data often rely on strong prior knowledge.…
We develop a novel physics informed deep learning approach for solving nonlinear drift-diffusion equations on metric graphs. These models represent an important model class with a large number of applications in areas ranging from transport…
Recent advances in the theory of Neural Operators (NOs) have enabled fast and accurate computation of the solutions to complex systems described by partial differential equations (PDEs). Despite their great success, current NO-based…
Current physics-informed (standard or deep operator) neural networks still rely on accurately learning the initial and/or boundary conditions of the system of differential equations they are solving. In contrast, standard numerical methods…
Developing neural operators that accurately predict the behavior of systems governed by partial differential equations (PDEs) across unseen parameter regimes is crucial for robust generalization in scientific and engineering applications.…
In this paper, we introduce the Spectral Coefficient Learning via Operator Network (SCLON), a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The…
Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed…
Deep Operator Networks (DeepONets) have become a central tool in data-driven operator learning, providing flexible surrogates for nonlinear mappings arising in partial differential equations (PDEs). However, the standard trunk design based…
Modern power systems require fast and accurate dynamic simulations for stability assessment, digital twins, and real-time control, but classical ODE solvers are often too slow for large-scale or online applications. We propose a…
There has been an arising trend of adopting deep learning methods to study partial differential equations (PDEs). In this paper, we introduce a deep recurrent framework for solving time-dependent PDEs without generating large scale data…
In this paper, we propose a a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. For linear…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE…
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…
Learning and solving governing equations of a physical system, represented by partial differential equations (PDEs), from data is a central challenge in a variety of areas of science and engineering. Traditional numerical methods for…