A comparison of PINN approaches for drift-diffusion equations on metric graphs

• URL: http://arxiv.org/abs/2205.07195v1
• Date: Sun, 15 May 2022 06:17:33 GMT
• Title: A comparison of PINN approaches for drift-diffusion equations on metric graphs
• Authors: Jan Blechschmidt, Jan-Frederik Pietschman, Tom-Christian Riemer, Martin Stoll, Max Winkler
• Abstract summary: We focus on comparing machine learning approaches for quantum graphs. In our case the differential equation is a drift-diffusion model. We compare several PINN approaches for solving the drift-diffusion on the metric graph.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper we focus on comparing machine learning approaches for quantum graphs, which are metric graphs, i.e., graphs with dedicated edge lengths, and an associated differential operator. In our case the differential equation is a drift-diffusion model. Computational methods for quantum graphs require a careful discretization of the differential operator that also incorporates the node conditions, in our case Kirchhoff-Neumann conditions. Traditional numerical schemes are rather mature but have to be tailored manually when the differential equation becomes the constraint in an optimization problem. Recently, physics informed neural networks (PINNs) have emerged as a versatile tool for the solution of partial differential equations from a range of applications. They offer flexibility to solve parameter identification or optimization problems by only slightly changing the problem formulation used for the forward simulation. We compare several PINN approaches for solving the drift-diffusion on the metric graph.

Related papers

• FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems [41.94295877935867]
  numerical simulation and optimization of technical systems described by partial differential equations is expensive. A comparatively new approach in this context is to combine the good approximation properties of neural networks with the classical finite element method. In this paper, we extend this approach to saddle-point and non-linear fluid dynamics problems, respectively.
  arXiv  Detail & Related papers  (2024-09-06T07:17:01Z)
• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Data [0.0]
  We represent the PDE as a neural network and use an intermediate state representation similar to a Physics-Informed Neural Network (PINN) We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem. Our results on the Burgers' and the Korteweg-De Vreis equations demonstrate that the latter constrained method outperforms the penalty method.
  arXiv  Detail & Related papers  (2024-05-30T01:55:44Z)
• A Physics-driven GraphSAGE Method for Physical Process Simulations Described by Partial Differential Equations [2.1217718037013635]
  A physics-driven GraphSAGE approach is presented to solve problems governed by irregular PDEs. A distance-related edge feature and a feature mapping strategy are devised to help training and convergence. The robust PDE surrogate model for heat conduction problems parameterized by the Gaussian singularity random field source is successfully established.
  arXiv  Detail & Related papers  (2024-03-13T14:25:15Z)
• Physics-Informed Quantum Machine Learning: Solving nonlinear differential equations in latent spaces without costly grid evaluations [21.24186888129542]
  We propose a physics-informed quantum algorithm to solve nonlinear and multidimensional differential equations. By measuring the overlaps between states which are representations of DE terms, we construct a loss that does not require independent sequential function evaluations on grid points. When the loss is trained variationally, our approach can be related to the differentiable quantum circuit protocol.
  arXiv  Detail & Related papers  (2023-08-03T15:38:31Z)
• A deep learning approach to solve forward differential problems on graphs [6.756351172952362]
  We propose a novel deep learning approach to solve one-dimensional non-linear elliptic, parabolic, and hyperbolic problems on graphs. A system of physics-informed neural network (PINN) models is used to solve the differential equations.
  arXiv  Detail & Related papers  (2022-10-07T16:06:42Z)
• Neural Basis Functions for Accelerating Solutions to High Mach Euler Equations [63.8376359764052]
  We propose an approach to solving partial differential equations (PDEs) using a set of neural networks. We regress a set of neural networks onto a reduced order Proper Orthogonal Decomposition (POD) basis. These networks are then used in combination with a branch network that ingests the parameters of the prescribed PDE to compute a reduced order approximation to the PDE.
  arXiv  Detail & Related papers  (2022-08-02T18:27:13Z)
• Learning to Solve PDE-constrained Inverse Problems with Graph Networks [51.89325993156204]
  In many application domains across science and engineering, we are interested in solving inverse problems with constraints defined by a partial differential equation (PDE) Here we explore GNNs to solve such PDE-constrained inverse problems. We demonstrate computational speedups of up to 90x using GNNs compared to principled solvers.
  arXiv  Detail & Related papers  (2022-06-01T18:48:01Z)
• Fourier Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  We formulate a new neural operator by parameterizing the integral kernel directly in Fourier space. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. It is up to three orders of magnitude faster compared to traditional PDE solvers.
  arXiv  Detail & Related papers  (2020-10-18T00:34:21Z)
• Large-scale Neural Solvers for Partial Differential Equations [48.7576911714538]
  Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. Recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. We examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) We discuss the accuracy of GatedPINN with respect to analytical solutions -- as well as state-of-the-art numerical solvers, such as spectral solvers.
  arXiv  Detail & Related papers  (2020-09-08T13:26:51Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.