Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs

• URL: http://arxiv.org/abs/2203.05337v2
• Date: Fri, 11 Mar 2022 11:35:05 GMT
• Title: Parsimonious Physics-Informed Random Projection Neural Networks for Initial-Value Problems of ODEs and index-1 DAEs
• Authors: Gianluca Fabiani, Evangelos Galaris, Lucia Russo, Constantinos Siettos
• Abstract summary: We address a physics-informed neural network based on random projections for the numerical solution of IVPs of nonlinear ODEs in linear-implicit form and index-1 DAEs. Based on previous works on random projections, we prove the approximation capability of the scheme for ODEs in the canonical form and index-1 DAEs in the semiexplicit form.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We address a physics-informed neural network based on the concept of random projections for the numerical solution of IVPs of nonlinear ODEs in linear-implicit form and index-1 DAEs, which may also arise from the spatial discretization of PDEs. The scheme has a single hidden layer with appropriately randomly parametrized Gaussian kernels and a linear output layer, while the internal weights are fixed to ones. The unknown weights between the hidden and output layer are computed by Newton's iterations, using the Moore-Penrose pseudoinverse for low to medium, and sparse QR decomposition with regularization for medium to large scale systems. To deal with stiffness and sharp gradients, we propose a variable step size scheme for adjusting the interval of integration and address a continuation method for providing good initial guesses for the Newton iterations. Based on previous works on random projections, we prove the approximation capability of the scheme for ODEs in the canonical form and index-1 DAEs in the semiexplicit form. The optimal bounds of the uniform distribution are parsimoniously chosen based on the bias-variance trade-off. The performance of the scheme is assessed through seven benchmark problems: four index-1 DAEs, the Robertson model, a model of five DAEs describing the motion of a bead, a model of six DAEs describing a power discharge control problem, the chemical Akzo Nobel problem and three stiff problems, the Belousov-Zhabotinsky, the Allen-Cahn PDE and the Kuramoto-Sivashinsky PDE. The efficiency of the scheme is compared with three solvers ode23t, ode23s, ode15s of the MATLAB ODE suite. Our results show that the proposed scheme outperforms the stiff solvers in several cases, especially in regimes where high stiffness or sharp gradients arise in terms of numerical accuracy, while the computational costs are for any practical purposes comparable.

Related papers

• A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equations [9.588717577573684]
  We propose a scalable preconditioned primal hybrid gradient algorithm for solving partial differential equations (PDEs) We compare the performance of the proposed method with several commonly used deep learning algorithms. The numerical results suggest that the proposed method performs efficiently and robustly and converges more stably.
  arXiv  Detail & Related papers  (2024-11-09T20:39:10Z)
• On the Trajectory Regularity of ODE-based Diffusion Sampling [79.17334230868693]
  Diffusion-based generative models use differential equations to establish a smooth connection between a complex data distribution and a tractable prior distribution. In this paper, we identify several intriguing trajectory properties in the ODE-based sampling process of diffusion models.
  arXiv  Detail & Related papers  (2024-05-18T15:59:41Z)
• Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker-Planck Equations [27.164040990410065]
  We propose a score-based solver to fit the score function in SDEs. The proposed score-based SDE solver operates in two stages: first, employing SM, SSM, or Score-PINN to acquire the score; and second, solving the LL via an ODE. The numerical results demonstrate the score-based SDE solver's stability, speed, and performance across different settings.
  arXiv  Detail & Related papers  (2024-02-12T07:59:25Z)
• Noise in the reverse process improves the approximation capabilities of diffusion models [27.65800389807353]
  In Score based Generative Modeling (SGMs), the state-of-the-art in generative modeling, reverse processes are known to perform better than their deterministic counterparts. This paper delves into the heart of this phenomenon, comparing neural ordinary differential equations (ODEs) and neural dimension equations (SDEs) as reverse processes. We analyze the ability of neural SDEs to approximate trajectories of the Fokker-Planck equation, revealing the advantages of neurality.
  arXiv  Detail & Related papers  (2023-12-13T02:39:10Z)
• 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)
• Gaussian Mixture Solvers for Diffusion Models [84.83349474361204]
  We introduce a novel class of SDE-based solvers called GMS for diffusion models. Our solver outperforms numerous SDE-based solvers in terms of sample quality in image generation and stroke-based synthesis.
  arXiv  Detail & Related papers  (2023-11-02T02:05:38Z)
• A graph convolutional autoencoder approach to model order reduction for parametrized PDEs [0.8192907805418583]
  The present work proposes a framework for nonlinear model order reduction based on a Graph Convolutional Autoencoder (GCA-ROM) We develop a non-intrusive and data-driven nonlinear reduction approach, exploiting GNNs to encode the reduced manifold and enable fast evaluations of parametrized PDEs.
  arXiv  Detail & Related papers  (2023-05-15T12:01:22Z)
• Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions [68.8204255655161]
  We develop a physics-informed neural network (PINN) model for parabolic problems with a sharply perturbed initial condition. Localized large gradients in the ADE solution make the (common in PINN) Latin hypercube sampling of the equation's residual highly inefficient. We propose criteria for weights in the loss function that produce a more accurate PINN solution than those obtained with the weights selected via other methods.
  arXiv  Detail & Related papers  (2022-08-18T05:00:24Z)
• 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)
• Numerical Solution of Stiff Ordinary Differential Equations with Random Projection Neural Networks [0.0]
  We propose a numerical scheme based on Random Projection Neural Networks (RPNN) for the solution of Ordinary Differential Equations (ODEs) We show that our proposed scheme yields good numerical approximation accuracy without being affected by the stiffness, thus outperforming in same cases the textttode45 and textttode15s functions.
  arXiv  Detail & Related papers  (2021-08-03T15:49:17Z)
• 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.