Solving Differential Equations using Physics-Informed Deep Equilibrium Models

• URL: http://arxiv.org/abs/2406.03472v2
• Date: Fri, 28 Jun 2024 17:44:28 GMT
• Title: Solving Differential Equations using Physics-Informed Deep Equilibrium Models
• Authors: Bruno Machado Pacheco, Eduardo Camponogara,
• Abstract summary: This paper introduces Physics-Informed Deep Equilibrium Models (PIDEQs) for solving initial value problems (IVPs) of ordinary differential equations (ODEs) By bridging deep learning and physics-based modeling, this work advances computational techniques for solving IVPs, with implications for scientific computing and engineering applications.
• Score: 4.237218036051422
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper introduces Physics-Informed Deep Equilibrium Models (PIDEQs) for solving initial value problems (IVPs) of ordinary differential equations (ODEs). Leveraging recent advancements in deep equilibrium models (DEQs) and physics-informed neural networks (PINNs), PIDEQs combine the implicit output representation of DEQs with physics-informed training techniques. We validate PIDEQs using the Van der Pol oscillator as a benchmark problem, demonstrating their efficiency and effectiveness in solving IVPs. Our analysis includes key hyperparameter considerations for optimizing PIDEQ performance. By bridging deep learning and physics-based modeling, this work advances computational techniques for solving IVPs, with implications for scientific computing and engineering applications.

Related papers

• Stiff Transfer Learning for Physics-Informed Neural Networks [1.5361702135159845]
  We propose a novel approach, stiff transfer learning for physics-informed neural networks (STL-PINNs) to tackle stiff ordinary differential equations (ODEs) and partial differential equations (PDEs) Our methodology involves training a Multi-Head-PINN in a low-stiff regime, and obtaining the final solution in a high stiff regime by transfer learning. This addresses the failure modes related to stiffness in PINNs while maintaining computational efficiency by computing "one-shot" solutions.
  arXiv  Detail & Related papers  (2025-01-28T20:27:38Z)
• Advancing Generalization in PINNs through Latent-Space Representations [71.86401914779019]
  Physics-informed neural networks (PINNs) have made significant strides in modeling dynamical systems governed by partial differential equations (PDEs) We propose PIDO, a novel physics-informed neural PDE solver designed to generalize effectively across diverse PDE configurations. We validate PIDO on a range of benchmarks, including 1D combined equations and 2D Navier-Stokes equations.
  arXiv  Detail & Related papers  (2024-11-28T13:16:20Z)
• Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods [14.791541465418263]
  We propose learning a solver, i.e., solving partial differential equations (PDEs) using a physics-informed iterative algorithm trained on data. Our method learns to condition a gradient descent algorithm that automatically adapts to each PDE instance. We demonstrate the effectiveness of our method through empirical experiments on multiple datasets.
  arXiv  Detail & Related papers  (2024-10-09T12:28:32Z)
• 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)
• Discovering Interpretable Physical Models using Symbolic Regression and Discrete Exterior Calculus [55.2480439325792]
  We propose a framework that combines Symbolic Regression (SR) and Discrete Exterior Calculus (DEC) for the automated discovery of physical models. DEC provides building blocks for the discrete analogue of field theories, which are beyond the state-of-the-art applications of SR to physical problems. We prove the effectiveness of our methodology by re-discovering three models of Continuum Physics from synthetic experimental data.
  arXiv  Detail & Related papers  (2023-10-10T13:23:05Z)
• NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition [67.46012350241969]
  This paper proposes a general acceleration methodology called NeuralStagger. It decomposing the original learning tasks into several coarser-resolution subtasks. We demonstrate the successful application of NeuralStagger on 2D and 3D fluid dynamics simulations.
  arXiv  Detail & Related papers  (2023-02-20T19:36:52Z)
• 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)
• 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)
• PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations [2.741266294612776]
  We propose a new class of physics-informed neural networks, called physics-informed Variational Autoencoder (PI-VAE) PI-VAE consists of a variational autoencoder (VAE), which generates samples of system variables and parameters. The satisfactory accuracy and efficiency of the proposed method are numerically demonstrated in comparison with physics-informed generative adversarial network (PI-WGAN)
  arXiv  Detail & Related papers  (2022-03-21T21:51:19Z)
• Physics Informed RNN-DCT Networks for Time-Dependent Partial Differential Equations [62.81701992551728]
  We present a physics-informed framework for solving time-dependent partial differential equations. Our model utilizes discrete cosine transforms to encode spatial and recurrent neural networks. We show experimental results on the Taylor-Green vortex solution to the Navier-Stokes equations.
  arXiv  Detail & Related papers  (2022-02-24T20:46:52Z)
• Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems [5.885020100736158]
  We propose an extension of neural ordinary differential equations (NODEs) by introducing an additional set of ODE input parameters to NODEs. This extension allows NODEs to learn multiple dynamics specified by the input parameter instances. We demonstrate the effectiveness of PNODEs with important benchmark problems from computational physics.
  arXiv  Detail & Related papers  (2020-10-28T00:41:28Z)

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.