A Deep Learning Framework for Solving Hyperbolic Partial Differential Equations: Part I
Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when trying to approximate PDEs with dominant hyperbolic character. This research focuses on the development of a physics informed deep learning framework to approximate solutions to nonlinear PDEs that can develop shocks or discontinuities without any a-priori knowledge of the solution or the location of the discontinuities. The work takes motivation from finite element method that solves for solution values at nodes in the discretized domain and use these nodal values to obtain a globally defined solution field. Built on the rigorous mathematical foundations of the discontinuous Galerkin method, the framework naturally handles imposition of boundary conditions (Neumann/Dirichlet), entropy conditions, and regularity requirements. Several numerical experiments and validation with analytical solutions demonstrate the accuracy, robustness, and effectiveness of the proposed framework.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations
Graph neural networks (GNNs) leverage message passing mechanisms to learn the topological features of graph data. Traditional GNNs learns node features in a spatial domain unrelated to the topology, which can hardly ensu…
Physics-informed attention-based neural network for solving non-linear partial differential equations
Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs). PINNs are based on simple architectures, and learn the be…
SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs
We introduce a class of Sparse, Physics-based, and partially Interpretable Neural Networks (SPINN) for solving ordinary and partial differential equations (PDEs). By reinterpreting a traditional meshless representation o…
Hyena Neural Operator for Partial Differential Equations
Numerically solving partial differential equations typically requires fine discretization to resolve necessary spatiotemporal scales, which can be computationally expensive. Recent advances in deep learning have provided…
A data-driven learned discretization approach in finite volume schemes for hyperbolic conservation laws and varying boundary conditions
This paper presents a data-driven finite volume method for solving 1D and 2D hyperbolic partial differential equations. This work builds upon the prior research incorporating a data-driven finite-difference approximation…