Recurrent Localization Networks applied to the Lippmann-Schwinger Equation

• URL: http://arxiv.org/abs/2102.00063v1
• Date: Fri, 29 Jan 2021 20:54:17 GMT
• Title: Recurrent Localization Networks applied to the Lippmann-Schwinger Equation
• Authors: Conlain Kelly, Surya R. Kalidindi
• Abstract summary: We present a novel machine learning approach for solving equations of the generalized Lippmann-Schwinger (L-S) type. As part of a learning-based loop unrolling, we use a recurrent convolutional neural network to iteratively solve the governing equations for a field of interest. We demonstrate our learning approach on the two-phase elastic localization problem, where it achieves excellent accuracy on the predictions of the local (i.e., voxel-level) elastic strains.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: The bulk of computational approaches for modeling physical systems in materials science derive from either analytical (i.e. physics based) or data-driven (i.e. machine-learning based) origins. In order to combine the strengths of these two approaches, we advance a novel machine learning approach for solving equations of the generalized Lippmann-Schwinger (L-S) type. In this paradigm, a given problem is converted into an equivalent L-S equation and solved as an optimization problem, where the optimization procedure is calibrated to the problem at hand. As part of a learning-based loop unrolling, we use a recurrent convolutional neural network to iteratively solve the governing equations for a field of interest. This architecture leverages the generalizability and computational efficiency of machine learning approaches, but also permits a physics-based interpretation. We demonstrate our learning approach on the two-phase elastic localization problem, where it achieves excellent accuracy on the predictions of the local (i.e., voxel-level) elastic strains. Since numerous governing equations can be converted into an equivalent L-S form, the proposed architecture has potential applications across a range of multiscale materials phenomena.

Related papers

• LLM4ED: Large Language Models for Automatic Equation Discovery [0.8644909837301149]
  We introduce a new framework that utilizes natural language-based prompts to guide large language models in automatically mining governing equations from data. Specifically, we first utilize the generation capability of LLMs to generate diverse equations in string form, and then evaluate the generated equations based on observations. Experiments are extensively conducted on both partial differential equations and ordinary differential equations.
  arXiv  Detail & Related papers  (2024-05-13T14:03:49Z)
• Self-Adaptive Physics-Informed Quantum Machine Learning for Solving Differential Equations [0.0]
  Chebyshevs have shown significant promise as an efficient tool for both classical and quantum neural networks to solve differential equations. We adapt and generalize this framework in a quantum machine learning setting for a variety of problems. Results indicate a promising approach to the near-term evaluation of differential equations on quantum devices.
  arXiv  Detail & Related papers  (2023-12-14T18:46:35Z)
• 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)
• Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains [0.0]
  Physics-informed neural networks (PINNs) are a new tool for solving boundary value problems. Recent investigations have shown that when designing loss functions for many engineering problems, using first-order derivatives and combining equations from both strong and weak forms can lead to much better accuracy. In this work, we propose applying the mixed formulation to solve multi-physical problems, specifically a stationary thermo-mechanically coupled system of equations.
  arXiv  Detail & Related papers  (2023-02-09T21:56:59Z)
• Deep learning applied to computational mechanics: A comprehensive review, state of the art, and the classics [77.34726150561087]
  Recent developments in artificial neural networks, particularly deep learning (DL), are reviewed in detail. Both hybrid and pure machine learning (ML) methods are discussed. History and limitations of AI are recounted and discussed, with particular attention at pointing out misstatements or misconceptions of the classics.
  arXiv  Detail & Related papers  (2022-12-18T02:03:00Z)
• A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms [64.3064050603721]
  We generalize Runge-Kutta neural network to a recurrent neural network (R2N2) superstructure for the design of customized iterative algorithms. We demonstrate that regular training of the weight parameters inside the proposed superstructure on input/output data of various computational problem classes yields similar iterations to Krylov solvers for linear equation systems, Newton-Krylov solvers for nonlinear equation systems, and Runge-Kutta solvers for ordinary differential equations.
  arXiv  Detail & Related papers  (2022-11-22T16:30:33Z)
• 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)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems [40.20472268839781]
  We generalize the reaction-diffusion equation in statistical physics, Schr"odinger equation in quantum mechanics, Helmholtz equation in paraxial optics. We take finite difference method to discretize NPDE for finding numerical solution. Basic building blocks of deep neural network architecture, including multi-layer perceptron, convolutional neural network and recurrent neural networks, are generated.
  arXiv  Detail & Related papers  (2021-03-10T00:05:46Z)
• 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)
• Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problem [0.0]
  We present a new hybrid physics-based machine-learning approach to reservoir modeling. The methodology relies on a series of deep adversarial neural network architecture with physics-based regularization. The proposed methodology is a simple and elegant way to instill physical knowledge to machine-learning algorithms.
  arXiv  Detail & Related papers  (2020-01-15T08:20:11Z)

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.