Exact and soft boundary conditions in Physics-Informed Neural Networks for the Variable Coefficient Poisson equation

• URL: http://arxiv.org/abs/2310.02548v1
• Date: Wed, 4 Oct 2023 03:16:03 GMT
• Title: Exact and soft boundary conditions in Physics-Informed Neural Networks for the Variable Coefficient Poisson equation
• Authors: Sebastian Barschkis
• Abstract summary: Boundary conditions (BCs) are a key component in every Physics-Informed Neural Network (PINN) BCs constrain the underlying boundary value problem (BVP) that a PINN tries to approximate. This study examines how soft loss-based and exact distance function-based BC imposition approaches differ when applied in PINNs.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Boundary conditions (BCs) are a key component in every Physics-Informed Neural Network (PINN). By defining the solution to partial differential equations (PDEs) along domain boundaries, BCs constrain the underlying boundary value problem (BVP) that a PINN tries to approximate. Without them, unique PDE solutions may not exist and finding approximations with PINNs would be a challenging, if not impossible task. This study examines how soft loss-based and exact distance function-based BC imposition approaches differ when applied in PINNs. The well known variable coefficient Poisson equation serves as the target PDE for all PINN models trained in this work. Besides comparing BC imposition approaches, the goal of this work is to also provide resources on how to implement these PINNs in practice. To this end, Keras models with Tensorflow backend as well as a Python notebook with code examples and step-by-step explanations on how to build soft/exact BC PINNs are published alongside this review.

Related papers

• Robust Physics Informed Neural Networks [0.21756081703275998]
  We introduce a robust version of the Physics-Informed Neural Networks (RPINNs) to approximate the Partial Differential Equations (PDEs) solution. We test our RPINN algorithm on two Laplace problems and one advection-diffusion problem in two spatial dimensions.
  arXiv  Detail & Related papers  (2024-01-04T14:42:29Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating method [0.0]
  Physics-informed neural networks (PINNs) are appealing data-driven tools for solving and inferring solutions to nonlinear partial differential equations (PDEs) This paper explores the use of the Schwarz alternating method as a means to couple PINNs with each other and with conventional numerical models.
  arXiv  Detail & Related papers  (2023-11-01T01:59:28Z)
• Predictive Limitations of Physics-Informed Neural Networks in Vortex Shedding [0.0]
  We look at the flow around a 2D cylinder and find that data-free PINNs are unable to predict vortex shedding. Data-driven PINN exhibits vortex shedding only while the training data is available, but reverts to the steady state solution when the data flow stops. The distribution of the Koopman eigenvalues on the complex plane suggests that PINN is numerically dispersive and diffusive.
  arXiv  Detail & Related papers  (2023-05-31T22:59:52Z)
• $\Delta$-PINNs: physics-informed neural networks on complex geometries [2.1485350418225244]
  Physics-informed neural networks (PINNs) have demonstrated promise in solving forward and inverse problems involving partial differential equations. To date, there is no clear way to inform PINNs about the topology of the domain where the problem is being solved. We propose a novel positional encoding mechanism for PINNs based on the eigenfunctions of the Laplace-Beltrami operator.
  arXiv  Detail & Related papers  (2022-09-08T18:03:19Z)
• Fourier Neural Operator with Learned Deformations for PDEs on General Geometries [75.91055304134258]
  We propose a new framework, viz., geo-FNO, to solve PDEs on arbitrary geometries. Geo-FNO learns to deform the input (physical) domain, which may be irregular, into a latent space with a uniform grid. We consider a variety of PDEs such as the Elasticity, Plasticity, Euler's, and Navier-Stokes equations, and both forward modeling and inverse design problems.
  arXiv  Detail & Related papers  (2022-07-11T21:55:47Z)
• 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)
• Improved Training of Physics-Informed Neural Networks with Model Ensembles [81.38804205212425]
  We propose to expand the solution interval gradually to make the PINN converge to the correct solution. All ensemble members converge to the same solution in the vicinity of observed data. We show experimentally that the proposed method can improve the accuracy of the found solution.
  arXiv  Detail & Related papers  (2022-04-11T14:05:34Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• dNNsolve: an efficient NN-based PDE solver [62.997667081978825]
  We introduce dNNsolve, that makes use of dual Neural Networks to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions.
  arXiv  Detail & Related papers  (2021-03-15T19:14:41Z)
• Physics informed deep learning for computational elastodynamics without labeled data [13.084113582897965]
  We present a physics-informed neural network (PINN) with mixed-variable output to model elastodynamics problems without resort to labeled data. Results show the promise of PINN in the context of computational mechanics applications.
  arXiv  Detail & Related papers  (2020-06-10T19:05:08Z)

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.