Towards Model Discovery Using Domain Decomposition and PINNs

• URL: http://arxiv.org/abs/2410.01599v1
• Date: Wed, 2 Oct 2024 14:38:37 GMT
• Title: Towards Model Discovery Using Domain Decomposition and PINNs
• Authors: Tirtho S. Saha, Alexander Heinlein, Cordula Reisch,
• Abstract summary: The study evaluates the performance of two approaches, namely Physics-Informed Neural Networks (PINNs) and Finite Basis Physics-Informed Neural Networks (FBPINNs) We find a better performance for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.
• Score: 44.99833362998488
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We enhance machine learning algorithms for learning model parameters in complex systems represented by ordinary differential equations (ODEs) with domain decomposition methods. The study evaluates the performance of two approaches, namely (vanilla) Physics-Informed Neural Networks (PINNs) and Finite Basis Physics-Informed Neural Networks (FBPINNs), in learning the dynamics of test models with a quasi-stationary longtime behavior. We test the approaches for data sets in different dynamical regions and with varying noise level. As results, we find a better performance for the FBPINN approach compared to the vanilla PINN approach, even in cases with data from only a quasi-stationary time domain with few dynamics.

Related papers

• Neural Port-Hamiltonian Differential Algebraic Equations for Compositional Learning of Electrical Networks [20.12750360095627]
  We develop compositional learning algorithms for coupled dynamical systems. We use neural networks to parametrize unknown terms in differential and algebraic components of a port-Hamiltonian DAE. We train individual N-PHDAE models for separate grid components, before coupling them to accurately predict the behavior of larger-scale networks.
  arXiv  Detail & Related papers  (2024-12-15T15:13:11Z)
• Adapting Physics-Informed Neural Networks to Improve ODE Optimization in Mosquito Population Dynamics [0.019972837513980313]
  We propose a PINN framework with several improvements for forward and inverse problems for ODE systems. The framework tackles the gradient imbalance and stiff problems posed by mosquito ordinary differential equations. Preliminary results indicate that physics-informed machine learning holds significant potential for advancing the study of ecological systems.
  arXiv  Detail & Related papers  (2024-06-07T17:40:38Z)
• Transfer Learning with Physics-Informed Neural Networks for Efficient Simulation of Branched Flows [1.1470070927586016]
  Physics-Informed Neural Networks (PINNs) offer a promising approach to solving differential equations. We adopt a recently developed transfer learning approach for PINNs and introduce a multi-head model. We show that our methods provide significant computational speedups in comparison to standard PINNs trained from scratch.
  arXiv  Detail & Related papers  (2022-11-01T01:50:00Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs [70.51212431290611]
  Partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. We propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic Phi41 model and the 2d Navier-Stokes equation.
  arXiv  Detail & Related papers  (2022-04-13T08:53:41Z)
• Characterizing possible failure modes in physics-informed neural networks [55.83255669840384]
  Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs. We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
  arXiv  Detail & Related papers  (2021-09-02T16:06:45Z)
• Stochastic analysis of heterogeneous porous material with modified neural architecture search (NAS) based physics-informed neural networks using transfer learning [0.0]
  modified neural architecture search method (NAS) based physics-informed deep learning model is presented. A three dimensional flow model is built to provide a benchmark to the simulation of groundwater flow in highly heterogeneous aquifers.
  arXiv  Detail & Related papers  (2020-10-03T19:57:54Z)
• DyNODE: Neural Ordinary Differential Equations for Dynamics Modeling in Continuous Control [0.0]
  We present a novel approach that captures the underlying dynamics of a system by incorporating control in a neural ordinary differential equation framework. Results indicate that a simple DyNODE architecture when combined with an actor-critic reinforcement learning algorithm outperforms canonical neural networks.
  arXiv  Detail & Related papers  (2020-09-09T12:56:58Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• An Ode to an ODE [78.97367880223254]
  We present a new paradigm for Neural ODE algorithms, called ODEtoODE, where time-dependent parameters of the main flow evolve according to a matrix flow on the group O(d) This nested system of two flows provides stability and effectiveness of training and provably solves the gradient vanishing-explosion problem.
  arXiv  Detail & Related papers  (2020-06-19T22:05:19Z)

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.