Online learning to accelerate nonlinear PDE solvers: applied to multiphase porous media flow

• URL: http://arxiv.org/abs/2504.18414v1
• Date: Fri, 25 Apr 2025 15:15:45 GMT
• Title: Online learning to accelerate nonlinear PDE solvers: applied to multiphase porous media flow
• Authors: Vinicius L S Silva, Pablo Salinas, Claire E Heaney, Matthew Jackson, Christopher C Pain,
• Abstract summary: We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning.<n>The proposed method rely on four pillars: (i) dimensionless numbers as input parameters for the machine learning model, (ii) simplified numerical model (two-dimensional) for the offline training, (iii) dynamic control of a nonlinear solver tuning parameter (numerical relaxation), and (iv) and online learning for real-time improvement of the machine learning model.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning. It is applied in the context of multiphase flow in porous media. The proposed method rely on four pillars: (i) dimensionless numbers as input parameters for the machine learning model, (ii) simplified numerical model (two-dimensional) for the offline training, (iii) dynamic control of a nonlinear solver tuning parameter (numerical relaxation), (iv) and online learning for real-time improvement of the machine learning model. This strategy decreases the number of nonlinear iterations by dynamically modifying a single global parameter, the relaxation factor, and by adaptively learning the attributes of each numerical model on-the-run. Furthermore, this work performs a sensitivity study in the dimensionless parameters (machine learning features), assess the efficacy of various machine learning models, demonstrate a decrease in nonlinear iterations using our method in more intricate, realistic three-dimensional models, and fully couple a machine learning model into an open-source multiphase flow simulator achieving up to 85\% reduction in computational time.

Related papers

• Random Matrix Theory for Deep Learning: Beyond Eigenvalues of Linear Models [51.85815025140659]
  Modern Machine Learning (ML) and Deep Neural Networks (DNNs) often operate on high-dimensional data.<n>In particular, the proportional regime where the data dimension, sample size, and number of model parameters are all large gives rise to novel and sometimes counterintuitive behaviors.<n>This paper extends traditional Random Matrix Theory (RMT) beyond eigenvalue-based analysis of linear models to address the challenges posed by nonlinear ML models.
  arXiv  Detail & Related papers  (2025-06-16T06:54:08Z)
• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• A parametric framework for kernel-based dynamic mode decomposition using deep learning [0.0]
  The proposed framework consists of two stages, offline and online. The online stage leverages those LANDO models to generate new data at a desired time instant. dimensionality reduction technique is applied to high-dimensional dynamical systems to reduce the computational cost of training.
  arXiv  Detail & Related papers  (2024-09-25T11:13:50Z)
• Efficient Interpretable Nonlinear Modeling for Multiple Time Series [5.448070998907116]
  This paper proposes an efficient nonlinear modeling approach for multiple time series. It incorporates nonlinear interactions among different time-series variables. Experimental results show that the proposed algorithm improves the identification of the support of the VAR coefficients in a parsimonious manner.
  arXiv  Detail & Related papers  (2023-09-29T11:42:59Z)
• Data-driven Nonlinear Parametric Model Order Reduction Framework using Deep Hierarchical Variational Autoencoder [5.521324490427243]
  Data-driven parametric model order reduction (MOR) method using a deep artificial neural network is proposed. LSH-VAE is capable of performing nonlinear MOR for the parametric of a nonlinear dynamic system with a significant number of degrees of freedom.
  arXiv  Detail & Related papers  (2023-07-10T02:44:53Z)
• Training Deep Surrogate Models with Large Scale Online Learning [48.7576911714538]
  Deep learning algorithms have emerged as a viable alternative for obtaining fast solutions for PDEs. Models are usually trained on synthetic data generated by solvers, stored on disk and read back for training. It proposes an open source online training framework for deep surrogate models.
  arXiv  Detail & Related papers  (2023-06-28T12:02:27Z)
• Automatic Parameterization for Aerodynamic Shape Optimization via Deep Geometric Learning [60.69217130006758]
  We propose two deep learning models that fully automate shape parameterization for aerodynamic shape optimization. Both models are optimized to parameterize via deep geometric learning to embed human prior knowledge into learned geometric patterns. We perform shape optimization experiments on 2D airfoils and discuss the applicable scenarios for the two models.
  arXiv  Detail & Related papers  (2023-05-03T13:45:40Z)
• Learning Controllable Adaptive Simulation for Multi-resolution Physics [86.8993558124143]
  We introduce Learning controllable Adaptive simulation for Multi-resolution Physics (LAMP) as the first full deep learning-based surrogate model. LAMP consists of a Graph Neural Network (GNN) for learning the forward evolution, and a GNN-based actor-critic for learning the policy of spatial refinement and coarsening. We demonstrate that our LAMP outperforms state-of-the-art deep learning surrogate models, and can adaptively trade-off computation to improve long-term prediction error.
  arXiv  Detail & Related papers  (2023-05-01T23:20:27Z)
• Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization [3.7565501074323224]
  We propose an exact formulation of the SINDyDy problem using mixed-integer optimization (MIO) to solve the sparsity constrained regression problem to provable optimality in seconds. We illustrate the dramatic improvement of our approach in accurate model discovery while being more sample efficient, robust to noise, and flexible in accommodating physical constraints.
  arXiv  Detail & Related papers  (2022-06-01T01:43:45Z)
• Gradient-Based Trajectory Optimization With Learned Dynamics [80.41791191022139]
  We use machine learning techniques to learn a differentiable dynamics model of the system from data. We show that a neural network can model highly nonlinear behaviors accurately for large time horizons. In our hardware experiments, we demonstrate that our learned model can represent complex dynamics for both the Spot and Radio-controlled (RC) car.
  arXiv  Detail & Related papers  (2022-04-09T22:07:34Z)
• Non-linear manifold ROM with Convolutional Autoencoders and Reduced Over-Collocation method [0.0]
  Non-affine parametric dependencies, nonlinearities and advection-dominated regimes of the model of interest can result in a slow Kolmogorov n-width decay. We implement the non-linear manifold method introduced by Carlberg et al [37] with hyper-reduction achieved through reduced over-collocation and teacher-student training of a reduced decoder. We test the methodology on a 2d non-linear conservation law and a 2d shallow water models, and compare the results obtained with a purely data-driven method for which the dynamics is evolved in time with a long-short term memory network
  arXiv  Detail & Related papers  (2022-03-01T11:16:50Z)
• 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)
• Optimization-driven Machine Learning for Intelligent Reflecting Surfaces Assisted Wireless Networks [82.33619654835348]
  Intelligent surface (IRS) has been employed to reshape the wireless channels by controlling individual scattering elements' phase shifts. Due to the large size of scattering elements, the passive beamforming is typically challenged by the high computational complexity. In this article, we focus on machine learning (ML) approaches for performance in IRS-assisted wireless networks.
  arXiv  Detail & Related papers  (2020-08-29T08:39:43Z)

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.